MOSEK - GAMS

MOSEK 

MOSEK ApS, C/O Symbion Science Park, Fruebjergvej 3, Box 16, 2100 Copenhagen Ø, Denmark 

Contents 

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 

1.1 Licensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 

1.2 Reporting of Infeasible/Unbounded Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 

1.3 Solving Problems in Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 

1.3.1 Parallelized Optimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 

1.3.2 Concurrent Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 

1.4 The Infeasibility Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 

1.5 Nonlinear Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 

1.6 Modeling Issues Involving Convex Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 

2 Conic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 

2.2 Implemention of Conic Constraints in GAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 

2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 

3 The MOSEK Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 

4 Summary of MOSEK Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 

4.1 General and Preprocessing Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 

4.2 Problem Data Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 

4.3 Output Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 

4.4 Interior Point Optimizer Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 

4.5 Simplex Optimizer and Basis Identification Options . . . . . . . . . . . . . . . . . . . . . . . . . . 330 

4.6 Mixed Integer Optimizer Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 

4.7 Other Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 

5 Detailed Descriptions of MOSEK Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 

6 The MOSEK Log File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 

6.1 Log Using the Interior Point Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 

6.2 Log Using the Simplex Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 

6.3 Log Using the Mixed Integer Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 

6.4 Log Using the Conic Mixed Integer Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 

1 Introduction 

MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically 

constraint, and convex nonlinear mathematical optimization problems. MOSEK is particularly well suited for solving 

large-scale linear, convex quadratically constraint, and conic quadratic programs using an extremely efficient interior point 

algorithm. 

These problem classes can be solved using an appropriate optimizer built into MOSEK. All the optimizers available in 

MOSEK are built for the solution of large-scale sparse problems. Current optimizers include:

320 MOSEK 

• Interior-point optimizer for all continuous problems 

• Conic interior-point optimizer for conic quadratic problems 

• Simplex optimizer for linear problems 

• Mixed-integer optimizer based on a branch and cut technology 

• Conic Mixed-integer optimizer based on a branch and cut technology 

1.1 Licensing 

Licensing of GAMS/MOSEK is similar to other GAMS solvers. MOSEK is licensed in three different ways: 

• GAMS/MOSEK Base: 

All model types. Using MOSEK’s conic mixed-integer optimizer for the solution of models involving discrete variables. 

• GAMS/MOSEK Extended: 

Same as GAMS/MOSEK Base, but using the mixed-integer optimizer for the solution of models involving discrete 

variables by default 

• GAMS/MOSEK Solver Link: 

Users must have a seperate, licensed MOSEK system. For users who wish to use MOSEK within GAMS and also in 

other environments. 

For more information contact sales@gams.com. For information regarding MOSEK standalone or interfacing MOSEK 

with other applications contact sales@mosek.com. 

1.2 Reporting of Infeasible/Unbounded Models 

MOSEK determines if either the primal or the dual problem is infeasible by means of a Farkas certificate. In such a case 

MOSEK returns a certificate indicating primal or dual infeasibility. A primal infeasibility certificate indicates a primal 

infeasible model and the certificate is reported in the marginals of the equations in the listing file. The primal infeasibility 

certificate for a minimization problem 

is the solution y satisfying: 

minimize c T x 

subject to Ax = b, 

x ≥ 0 

A T y ≤ 0, b T y > 0 

A dual infeasibility certificate is reported in the levels of the variables in the listing file. The dual infeasibility certificate x 

for the same minimization problem is 

Ax = 0, c T x < 0 

Since GAMS reports all model statuses in the primal space, the notion of dual infeasibility does not exist and GAMS reports 

a status of unbounded, which assumes the primal problem is feasible. Although GAMS reports the primal as unbounded, 

there is the possibility that both the primal and dual problem are infeasible. To check if this is the case, the user can set 

appropriate upper and lower bounds on the objective variable, using the (varible).LO and (variable).UP suffixes and 

resolve. 

For more details on primal and dual infeasibility certificates see the MOSEK User’s manual at www.mosek.com. 

1.3 Solving Problems in Parallel 

If a computer has multiple CPUs (or a CPU with multiple cores), then it might be advantageous to use the multiple CPUs 

to solve the optimization problem. For instance if you have two CPUs you may want to exploit the two CPUs to solve the 

problem in the half time. MOSEK can exploit multiple CPUs.

MOSEK 321 

1.3.1 Parallelized Optimizers 

MOSEK’s interior-point and the conic mixed-integer optimizer have been parallelized. 

This implies that whenever the MOSEK interior-point optimizer should solve an optimization problem, then it will try 

to divide the work so each CPU gets a share of the work. The user decides how many CPUs MOSEK should exploit. 

Unfortunately, it is not always easy to divide the work. Also some of the coordination work must occur in sequential. 

Therefore, the speed-up obtained when using multiple CPUs is highly problem dependent. However, as a rule of thumb if 

the problem solves very quickly i.e. in less than 60 seconds, then it is not advantageous of using the parallel option. 

The parameter MSK IPAR NUM THREADS sets the number of threads (and therefore the number of CPU’s) that the interior 

point optimizer will use. 

1.3.2 Concurrent Optimizer 

An alternative to use a parallelized optimizer is the concurrent optimizer. The idea of the concurrent optimizer is to run 

multiple optimizers on the same problem concurrently. For instance the interior-point and the dual simplex optimizers may 

be applied to an linear optimization problem concurrently. The concurrent optimizer terminates when the first optimizer has 

completed and reports the solution of the fastest optimizer. That way a new optimizer has been created which essentially 

has the best performance of the interior-point and the dual simplex optimizer. 

Hence, the concurrent optimizer is the best one to use if there multiple optimizers available in MOSEK for the problem and 

you cannot say beforehand which one is the best one. For more details inspect the MSK IPAR CONCURRENT * options. 

1.4 The Infeasibility Report 

MOSEK has some facilities for diagnosing the cause of a primal or dual infeasibility. They can be turned on using the 

parameter setting MSK IPAR INFEAS REPORT AUTO. This causes MOSEK to print a report about an infeasible subset 

of the constraints, when an infeasibility is encountered. Moreover, the parameter MSK IPAR INFEAS REPORT LEVEL 

controls the amount info presented in the infeasibility report. We will use the trnsport.gms example from the GAMS Model 

Library with increased demand (b(j)=1.6*b(j)) to make the model infeasible. MOSEK produces the following infeasibility 

report 

MOSEK PRIMAL INFEASIBILITY REPORT. 

Problem status: The problem is primal infeasible 

The following constraints are involved in the primal infeasibility. 

Index Name Lower bound Upper bound Dual lower Dual upper 

1 supply(seattle) none 3.500000e+002 0.000000e+000 1.000000e+000 

2 supply(san-diego) none 6.000000e+002 0.000000e+000 1.000000e+000 

3 demand(new-york) 5.200000e+002 none 1.000000e+000 0.000000e+000 

4 demand(chicago) 4.800000e+002 none 1.000000e+000 0.000000e+000 

5 demand(topeka) 4.400000e+002 none 1.000000e+000 0.000000e+000 

The following bound constraints are involved in the infeasibility. 

Index Name Lower bound Upper bound Dual lower Dual upper 

which indicates which constraints and bounds that are important for the infeasibility i.e. causing the infeasibility. The 

infeasibility report is divided into two sections where the first section shows which constraints that are important for the 

infeasibility. In this case the important constraints are supply and demand. The values in the columns Dual lower and 

Dual upper are also useful, because if the dual lower value is different from zero for a constraint, then it implies that the 

lower bound on the constraint is important for the infeasibility. Similarly, if the dual upper value is different from zero on a 

constraint, then this implies the upper bound on the constraint is important for infeasibility.


1.5 Nonlinear Programs 

MOSEK can efficiently solve convex programs, but is not intended for nonconvex optimization. For nonconvex programs, 

MOSEK can detect some nonconvexities and will print out a warning message and terminate. If MOSEK does not detect 

nonconvexities for a nonconvex model, the optimizer may continue but stagnate. Hence care must be taken when solving 

nonlinear programs if convexity is not immediately known. 

1.6 Modeling Issues Involving Convex Programs 

It is often preferable to model convex programs in seperable form, if it is possible. Consider the following example of 

minizing an objective function f (x): 

f (x) = log(a ′ ∗ x) 

where a ∈ ℜ n is a parameter and x ∈ ℜ n the decision variable. The equation implies an implicit constraint of a ′ ∗ x > 0. 

Unfortunately, domain violations can still occur because no restrictions are set on a ′ ∗ x. A better approach is to introduce 

an intermediate variable y: 

f (x) = log(y) 

y = a ′ ∗ x 

y ≥ 0 

This accomplishes two things. It implies an explicit bound on a ′ ∗ x, thereby reducing the risk of domain violations. Secondly, 

it speeds up computation since computations of gradients and Hessians in the first (non-seperable) form are more 

expensive. Finally, it reduces the amount of memory needed (see the section on “Memory Options”) 

2 Conic Programming 

MOSEK is well suited for solving generalized linear programs involving nonlinear conic constraints. Conic programming 

is useful in a wide variety of application areas 1 including engineering and financial management . Conic programming 

has been used, for example, in antenna array weight design, grasping force optimization, finite impulse response (FIR) 

filter design, and portfolio optimization. The Mosek web site (http://mosek.com/resources/doc) has an excellent modeling 

manual for conic problems. 

This section gives an overview of conic programming and how conic constraints are implemented in GAMS. 

2.1 Introduction 

Conic programs can be thought of as generalized linear programs with the additional nonlinear constraint x ∈ C, where C is 

required to be a convex cone. The resulting class of problems is known as conic optimization and has the following form: 

minimize c T x 

subject to Ax ≤ r c , 

x ∈ [l x ,u x ] 

x ∈ C 

where A ∈ ℜ m×n is the constraint matrix, x ∈ ℜ n the decision variable, and c ∈ ℜ n the objective function cost coefficients. 

The vector r c ∈ ℜ m represents the right hand side and the vectors l x ,u x ∈ ℜ n are lower and upper bounds on the decision 

variable x. 

1 See M. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, Applications of second-order cone programming Linear Algebra and its Applications, 

284:193-228, Special Issue on Linear Algebra in Control, Signals and Image Processing. November, 1998.


Now partition the set of decision variables x into sets St ,t = 1,...,k, such that each decision variables x is a member of at 

most one set St . For example, we could have 

⎡ ⎤ 

⎡ ⎤ 

S 1 = 

⎣ x1 

x4 

Let x S t denote the variables x belonging to set S t . Then define 

where Ct must have one of the following forms: 

x7 

⎦and S 2 ⎢ 

= ⎢ 

⎣ 

• Quadratic cone: (also referred to as Lorentz or ice cream cone) 

⎧ 

 

⎨ 

 

 

Ct = x ∈ ℜnt : x1 ≥ 

⎩ 

• Rotated quadratic cone: (also referred to as hyberbolic constraints) 

 

x6 

x5 

x3 

x2 

⎥ 

⎦ . (2.1) 

C := {x ∈ ℜ n : x S t ∈ Ct,t = 1,...,k} (2.2) 

Ct = 

x ∈ ℜ nt 

: 2x1x2 ≥ 

nt ∑ 

j=3 

nt 

∑ x 

j=2 

2 j 

x 2 j, x1,x2 ≥ 0 

⎫ 

⎬ 

. (2.3) 

⎭ 

 

. (2.4) 

These two types of cones allow the formulation of quadratic, quadratically constrained, and many other classes of nonlinear 

convex optimization problems. 

2.2 Implemention of Conic Constraints in GAMS 

GAMS handles conic equations using the =C= equation type. The conic cases are written as: 

• Quadratic cone: 

• Rotated quadratic cone: 

x(‘1‘) = C = sum(i$[not sameas(i,‘1‘)],x(i)); (2.5) 

x(‘1‘) + x(‘2‘) = C = sum(i$[not sameas(i,‘1‘) and not sameas(i,‘2‘)],x(i)); (2.6) 

Note that the resulting nonlinear conic constraints result in “linear” constraints in GAMS. Thus the original nonlinear 

formulation is in fact a linear model in GAMS. We remark that we could formulate conic problems as regular NLP using 

constraints: 

• Quadratic cone: 

x(‘1‘) = G = sqrt[sum(i$[not sameas(i,‘1‘)],sqr[x(i)])]; (2.7) 

• Rotated quadratic cone: x( ′ 1 ′ ) and x( ′ 2 ′ ) are positive variables 

2 ∗ x(‘1‘) ∗ x(‘2‘) = G = sum(i$[not sameas(i,‘1‘) and not sameas(i,‘2‘)],sqr[x(i)]); (2.8) 

The example below illustrates the different formulations for conic programming problems. Note that the conic optimizer in 

MOSEK usually outperforms a general NLP method for the reformulated (NLP) cone problems. 

2.3 Example 

Consider the following example (cone2.gms) which illustrates the use of rotated conic constraints. We will give reformulations 

of the original problem in regular NLP form using conic constraints and in conic form.


The original problem is: 

minimize ∑i di 

xi 

subject to a t x ≤ b 

xi ∈ [li,ui], li > 0, di ≥ 0, i = 1,2,...,n 

where x ∈ ℜ n is the decision variable, d,a,l,u ∈ ℜ n parameters, and b ∈ ℜ a scalar parameter. The original model (2.9) can 

be written in GAMS using the equations: 

defobj.. sum(n, d(n)/x(n)) =E= obj; 

e1.. sum(n, a(n)*x(n)) =L= b; 

Model orig /defobj, e1/; 

x.lo(n) = l(n); 

x.up(n) = u(n); 

We can write an equivalent NLP formulation, replacing the objective function and adding another constraint: 

minimize ∑i diti 

subject to a t x ≤ b 

2tixi ≥ 2, i = 1,...,n 

x ∈ [l,u], l > 0, di ≥ 0 

where t ∈ ℜ n is a new decision variable. The GAMS formulation of this NLP (model cnlp) is: 

defobjc.. sum(n, d(n)*t(n)) =E= obj; 

e1.. sum(n, a(n)*x(n)) =L= b; 

conenlp(n).. 2*t(n)*x(n) =G= 2; 

Model cnlp /defobjc, e1, conenlp/; 

x.lo(n) = l(n); 

x.up(n) = u(n); 

We can change the equality to an inequality since the parameter di ≥ 0 and we are dealing with a minimization problem. 

Also, note that the constraint conenlp(n) is almost in rotated conic form. If we introduce a variable z ∈ ℜ n ,zi = √ 2, then 

we can reformulate the problem using conic constraints as: 

The GAMS formulation using conic equations =C= is: 


e1.. sum(n, a(n)*x(n)) =L= b; 

e2(n).. z(n) =E= sqrt(2); 

cone(n).. x(n) + t(n) =C= z(n); 

Model clp /defobjc, e1, e2, cone/; 

x.lo(n) = l(n); 

x.up(n) = u(n); 

minimize ∑i diti 

subject to atx ≤ b 

zi = √ 2 

2tixi ≥ z2 i , i = 1,...,n 

x ∈ [l,u], l > 0, di ≥ 0 

Note that this formulation is a linear program in GAMS, although the constraints cone(n)... represent the nonlinear 

rotated quadratic cone constraint. 

The complete model is listed below: 

(2.9) 

(2.10) 

(2.11)


Set n / n1*n10 /; 

Parameter d(n), a(n), l(n), u(n); 

Scalar b; 

d(n) = uniform(1,2); 

a(n) = uniform (10,50); 

l(n) = uniform(0.1,10); 

u(n) = l(n) + uniform(0,12-l(n)); 

Variables x(n); 

x.l(n) = uniform(l(n), u(n)); 

b = sum(n, x.l(n)*a(n)); 

Variables t(n), z(n), obj; 

Equations defobjc, defobj, e1, e2(n), cone(n), conenlp(n); 


defobj.. sum(n, d(n)/x(n)) =E= obj; 

e1.. sum(n, a(n)*x(n)) =L= b; 

e2(n).. z(n) =E= sqrt(2); 

cone(n).. x(n) + t(n) =C= z(n); 

conenlp(n).. 2*t(n)*x(n) =G= 2; 

Model clp /defobjc, e1, e2, cone/; 

Model cnlp /defobjc, e1, conenlp/; 

Model orig /defobj, e1/; 

x.lo(n) = l(n); 

x.up(n) = u(n); 

Solve clp min obj using lp; 

Solve cnlp min obj using nlp; 

Solve orig min obj using nlp; 

3 The MOSEK Options 

MOSEK works like other GAMS solvers, and many options can be set in the GAMS model. The most relevant GAMS 

options are reslim, nodlim, optca, optcr, and optfile. The option iterlim works only for the simplex optimizer. 

A description of all available GAMS options can be found in Chapter ”Using Solver Specific Options”. 

We remark that MOSEK contains many complex solver options, many of which require a deep understanding of the algorithms 

used. For a complete description of the many MOSEK options, consult the MOSEK User’s Guide, available online 

at www.mosek.com. 

If you specify ”.optfile = 1;” before the SOLVE statement in your GAMS model, MOSEK will then 

look for and read an option file with the name mosek.opt (see ”Using Solver Specific Options” for general use of solver 

option files). The syntax for the MOSEK option file is 

optname value 

with one option on each line. 

For example, 

MSK_IPAR_INTPNT_MAX_ITERATIONS 20 

MSK_IPAR_INTPNT_SCALING MSK_SCALING_NONE 

The first option specifies the maximum number of interior-point iterations, in this case 20. The second option indicates that


no scaling should be used. 

4 Summary of MOSEK Options 

4.1 General and Preprocessing Options 

MSK DPAR ANA SOL INFEAS TOL 

Report constraint bound violation if it exceeds this tolerance. 

MSK DPAR BASIS REL TOL S 

Maximum relative dual bound violation allowed in an optimal basic solution. 

MSK DPAR BASIS TOL S 

Maximum absolute dual bound violation in an optimal basic solution. 

MSK DPAR BASIS TOL X 

Maximum absolute primal bound violation allowed in an optimal basic solution. 

MSK DPAR CHECK CONVEXITY REL TOL 

This parameter controls when the full convexity check declares a problem to be non-convex. 

MSK DPAR OPTIMIZER MAX TIME 

Maximum amount of time the optimizer is allowed to spent on the optimization. 

MSK DPAR PRESOLVE TOL ABS LINDEP 

Absolute tolerance employed by the linear dependency checker. 

MSK DPAR PRESOLVE TOL AIJ 

Absolute zero tolerance employed for a i j in the presolve. 

MSK DPAR PRESOLVE TOL REL LINDEP 

Relative tolerance employed by the linear dependency checker. 

MSK DPAR PRESOLVE TOL S 

Absolute zero tolerance employed for s i in the presolve. 

MSK DPAR PRESOLVE TOL X 

Absolute zero tolerance employed for x j in the presolve. 

MSK IPAR AUTO SORT A BEFORE OPT 

Controls whether the elements in each column of A are sorted before an optimization is performed. 

MSK IPAR CONCURRENT NUM OPTIMIZERS 

The maximum number of simultaneous optimizations that will be started by the concurrent optimizer. 

MSK IPAR CONCURRENT PRIORITY DUAL SIMPLEX 

Priority of the dual simplex algorithm when selecting solvers for concurrent optimization. 

MSK IPAR CONCURRENT PRIORITY FREE SIMPLEX 

Priority of the free simplex optimizer when selecting solvers for concurrent optimization. 

MSK IPAR CONCURRENT PRIORITY INTPNT 

Priority of the interior-point algorithm when selecting solvers for concurrent optimization. 

MSK IPAR CONCURRENT PRIORITY PRIMAL SIMPLEX 

Priority of the primal simplex algorithm when selecting solvers for concurrent optimization. 

MSK IPAR INFEAS REPORT AUTO 

Controls infeasibility report production. 

MSK IPAR INFEAS REPORT LEVEL 

Controls the amount of information presented in an infeasibility report. 

MSK IPAR NUM THREADS 

Controls the number of threads employed by the optimizer. 

MSK IPAR OPTIMIZER 

The paramter controls which optimizer is used to optimize the task. 

MSK IPAR PRESOLVE ELIMINATOR MAX NUM TRIES 

Control the maximum number of times the eliminator is tried. 

MSK IPAR PRESOLVE ELIMINATOR USE 

Controls whether free or implied free variables are eliminated from the problem. 

MSK IPAR PRESOLVE ELIM FILL


Controls the maximum amount of fill-in in the elimination phase of the presolve. 

MSK IPAR PRESOLVE LEVEL 

Currently not used. 

MSK IPAR PRESOLVE LINDEP ABS WORK TRH 

Controls the linear dependency check which is potentially computationally expensive. 

MSK IPAR PRESOLVE LINDEP REL WORK TRH 


MSK IPAR PRESOLVE LINDEP USE 

Controls whether the linear constraints are checked for linear dependencies. 

MSK IPAR PRESOLVE MAX NUM REDUCTIONS 

Controls the maximum number reductions performed by the presolve. 

MSK IPAR PRESOLVE USE 

Controls whether the presolve is applied to a problem before it is optimized. 

MSK IPAR PRIMAL REPAIR OPTIMIZER 

Controls which optimizer that is used to find the optimal repair. 

MSK SPAR PARAM READ FILE NAME 

Modifications to the parameter database is read from this file. 

MSK SPAR PARAM WRITE FILE NAME 

The parameter database is written to this file. 

MSK SPAR DATA FILE NAME 

Problem data is written to this file. File extension determines format. 

4.2 Problem Data Options 

MSK DPAR DATA TOL AIJ 

Absolute zero tolerance for elements in A. 

MSK DPAR DATA TOL AIJ HUGE 

An element in A which is larger than this value in absolute size causes an error. 

MSK DPAR DATA TOL AIJ LARGE 

An element in A which is larger than this value in absolute size causes a warning message. 

MSK DPAR DATA TOL BOUND INF 

Any bound which in absolute value is greater than this parameter is considered infinite. 

MSK DPAR DATA TOL BOUND WRN 

If a bound value is larger than this value in absolute size, then a warning message is issued. 

MSK DPAR DATA TOL CJ LARGE 

An element in c which is larger than this value in absolute terms causes a warning message. 

MSK DPAR DATA TOL C HUGE 

An element in c larger than the value of this parameter in absolute terms generates an error. 

MSK DPAR DATA TOL QIJ 

Absolute zero tolerance for elements in Q matrices. 

MSK DPAR DATA TOL X 

Zero tolerance for constraints and variables. 

MSK DPAR LOWER OBJ CUT 

Lower objective limit. 

MSK DPAR LOWER OBJ CUT FINITE TRH 

Lower objective limit threshold. 

MSK DPAR UPPER OBJ CUT 

Upper objective limit. 

MSK DPAR UPPER OBJ CUT FINITE TRH 

Upper objective limit threshold. 

MSK IPAR CHECK CONVEXITY 

Specify the level of convexity check on quadratic problems


4.3 Output Options 

MSK IPAR LOG 

Controls the amount of log information. 

MSK IPAR LOG BI 

Controls the amount of output printed by the basis identification procedure. 

MSK IPAR LOG BI FREQ 

Controls logging frequency. 

MSK IPAR LOG CHECK CONVEXITY 

Controls logging in convexity check on quadratic problems. 

MSK IPAR LOG CONCURRENT 

Controls amount of output printed by the concurrent optimizer. 

MSK IPAR LOG FACTOR 

If turned on, then the factor log lines are added to the log. 

MSK IPAR LOG FEAS REPAIR 

Controls the amount of output printed when performing feasibility repair. 

MSK IPAR LOG HEAD 

If turned on, then a header line is added to the log. 

MSK IPAR LOG INFEAS ANA 

Controls amount of output printed by the infeasibility analyzer procedures. 

MSK IPAR LOG INTPNT 

Controls amount of output printed printed by the interior-point optimizer. 

MSK IPAR LOG MIO 

Controls the log level for the mixed-integer optimizer. 

MSK IPAR LOG MIO FREQ 

Controls how frequent the mixed-integer optimizer prints the log line. 

MSK IPAR LOG NONCONVEX 

Controls amount of output printed by the nonconvex optimizer. 

MSK IPAR LOG OPTIMIZER 

Controls the amount of general optimizer information that is logged. 

MSK IPAR LOG ORDER 

If turned on, then factor lines are added to the log. 

MSK IPAR LOG PARAM 

Controls the amount of information printed out about parameter changes. 

MSK IPAR LOG PRESOLVE 

Controls amount of output printed by the presolve procedure. 

MSK IPAR LOG RESPONSE 

Controls amount of output printed when response codes are reported. 

MSK IPAR LOG SENSITIVITY 

Controls the amount of logging during the sensitivity analysis. 

MSK IPAR LOG SENSITIVITY OPT 

Controls the amount of logging from the optimizers employed during the sensitivity analysis. 

MSK IPAR LOG SIM 

Controls amount of output printed by the simplex optimizer. 

MSK IPAR LOG SIM FREQ 

Controls simplex logging frequency. 

MSK IPAR LOG STORAGE 

When turned on, MOSEK prints messages regarding the storage usage and allocation. 

MSK IPAR MAX NUM WARNINGS 

Waning level. A higher value results in more warnings. 

MSK IPAR WARNING LEVEL 

Warning level. A higher value implies more warnings. 

MSK IPAR WRITE IGNORE INCOMPATIBLE CONIC ITEMS 

Controls ignoration of incompatible conic items. 

MSK IPAR WRITE IGNORE INCOMPATIBLE ITEMS


Controls if the writer ignores incompatible problem items when writing files. 

MSK IPAR WRITE IGNORE INCOMPATIBLE NL ITEMS 

Controls if the writer ignores general non-linear terms or produces an error. 

MSK IPAR WRITE IGNORE INCOMPATIBLE PSD ITEMS 

Controls ignoration of incompatible psd items. 

MSK IPAR WRITE LP LINE WIDTH 

Maximum width of line in an LP file written by MOSEK. 

MSK IPAR WRITE LP QUOTED NAMES 

If this option is turned on, then MOSEK will quote invalid LP names when writing an LP file. 

MSK IPAR WRITE LP STRICT FORMAT 

Controls whether LP output files satisfy the LP format strictly. 

MSK IPAR WRITE LP TERMS PER LINE 

Maximum number of terms on a single line in an LP file written by MOSEK. 0 means unlimited. 

MSK IPAR WRITE MPS INT 

Controls writing of marker records to the MPS file to indicate if variables are integer restricted. 

MSK IPAR WRITE PRECISION 

Controls the precision with which double numbers are printed in the MPS data file. 

MSK IPAR WRITE XML MODE 

Controls output mode for XML file. 

4.4 Interior Point Optimizer Options 

MSK DPAR FEASREPAIR TOL 

Tolerance for constraint enforcing upper bound on sum of weighted violations in feasibility repair. 

MSK DPAR INTPNT CO TOL DFEAS 

Dual feasibility tolerance used by the conic interior-point optimizer. 

MSK DPAR INTPNT CO TOL INFEAS 

Controls when the conic interior-point optimizer declares the model primal or dual infeasible. 

MSK DPAR INTPNT CO TOL MU RED 

Optimality tolerance for the conic solver. 

MSK DPAR INTPNT CO TOL NEAR REL 

Termination tolerance multiplier that is used if no accurate solution can be found. 

MSK DPAR INTPNT CO TOL PFEAS 

Primal feasibility tolerance used by the conic interior-point optimizer. 

MSK DPAR INTPNT CO TOL REL GAP 

Relative gap termination tolerance used by the conic interior-point optimizer. 

MSK DPAR INTPNT NL MERIT BAL 

Controls if the complementarity and infeasibility is converging to zero at about equal rates. 

MSK DPAR INTPNT NL TOL DFEAS 

Dual feasibility tolerance used when a nonlinear model is solved. 

MSK DPAR INTPNT NL TOL MU RED 

Relative complementarity gap tolerance. 

MSK DPAR INTPNT NL TOL NEAR REL 


MSK DPAR INTPNT NL TOL PFEAS 

Primal feasibility tolerance used when a nonlinear model is solved. 

MSK DPAR INTPNT NL TOL REL GAP 

Relative gap termination tolerance for nonlinear problems. 

MSK DPAR INTPNT NL TOL REL STEP 

Relative step size to the boundary for general nonlinear optimization problems. 

MSK DPAR INTPNT TOL DFEAS 

Dual feasibility tolerance used for linear and quadratic optimization problems. 

MSK DPAR INTPNT TOL DSAFE


Controls the initial dual starting point used by the interior-point optimizer. 

MSK DPAR INTPNT TOL INFEAS 

Controls when the optimizer declares the model primal or dual infeasible. 

MSK DPAR INTPNT TOL MU RED 


MSK DPAR INTPNT TOL PATH 

Controls how close the interior-point optimizer follows the central path. 

MSK DPAR INTPNT TOL PFEAS 

Primal feasibility tolerance used for linear and quadratic optimization problems. 

MSK DPAR INTPNT TOL PSAFE 

Controls the initial primal starting point used by the interior-point optimizer. 

MSK DPAR INTPNT TOL REL GAP 

Relative gap termination tolerance. 

MSK DPAR INTPNT TOL REL STEP 

Relative step size to the boundary for linear and quadratic optimization problems. 

MSK DPAR INTPNT TOL STEP SIZE 

Step size tolerance value. 

MSK IPAR INTPNT BASIS 

Controls whether the interior-point optimizer also computes an optimal basis. 

MSK IPAR INTPNT DIFF STEP 

Controls whether different step sizes are allowed in the primal and dual space. 

MSK IPAR INTPNT FACTOR DEBUG LVL 

Controls factorization debug level. 

MSK IPAR INTPNT FACTOR METHOD 

Controls the method used to factor the Newton equation system. 

MSK IPAR INTPNT MAX ITERATIONS 

Controls the maximum number of iterations allowed in the interior-point optimizer. 

MSK IPAR INTPNT MAX NUM COR 

Controls the maximum number of correctors allowed by the multiple corrector procedure. 

MSK IPAR INTPNT MAX NUM REFINEMENT STEPS 

Maximum number of steps to be used by the iterative refinement of the search direction. 

MSK IPAR INTPNT OFF COL TRH 

Controls how many offending columns are detected in the Jacobian of the constraint matrix. 

MSK IPAR INTPNT ORDER METHOD 

Controls the ordering strategy. 

MSK IPAR INTPNT REGULARIZATION USE 

Controls whether regularization is allowed. 

MSK IPAR INTPNT SCALING 

Controls how the problem is scaled before the interior-point optimizer is used. 

MSK IPAR INTPNT SOLVE FORM 

Controls whether the primal or the dual problem is solved. 

MSK IPAR INTPNT STARTING POINT 

Starting point used by the interior-point optimizer. 

4.5 Simplex Optimizer and Basis Identification Options 

MSK DPAR SIMPLEX ABS TOL PIV 

Absolute pivot tolerance employed by the simplex optimizers. 

MSK DPAR SIM LU TOL REL PIV 

Relative pivot tolerance for LU factorization in simplex and basis identification. 

MSK IPAR BI CLEAN OPTIMIZER 

Controls which optimizer is used in the clean-up phase. 

MSK IPAR BI IGNORE MAX ITER


Controls if basis identification is performed under certain conditions. 

MSK IPAR BI IGNORE NUM ERROR 

Turns on basis identification if interior-point optimizer is terminated due to a numerical problem. 

MSK IPAR BI MAX ITERATIONS 

Maximum number of iterations after basis identification. 

MSK IPAR SIM BASIS FACTOR USE 

Controls whether a (LU) factorization of the basis is used in a hot-start. 

MSK IPAR SIM DEGEN 

Controls how aggressively degeneration is handled. 

MSK IPAR SIM DUAL CRASH 

Controls whether crashing is performed in the dual simplex optimizer. 

MSK IPAR SIM DUAL RESTRICT SELECTION 

Controls how aggressively restricted selection is used. 

MSK IPAR SIM DUAL SELECTION 

Controls the dual simplex strategy. 

MSK IPAR SIM EXPLOIT DUPVEC 

Controls if the simplex optimizers are allowed to exploit duplicated columns. 

MSK IPAR SIM HOTSTART 

Controls the type of hot-start that the simplex optimizer perform. 

MSK IPAR SIM HOTSTART LU 

Determines if the simplex optimizer should exploit the initial factorization. 

MSK IPAR SIM MAX ITERATIONS 

Maximum number of iterations that can be used by a simplex optimizer. 

MSK IPAR SIM MAX NUM SETBACKS 

Controls how many set-backs are allowed within a simplex optimizer. 

MSK IPAR SIM NON SINGULAR 

Controls if the simplex optimizer ensures a non-singular basis, if possible. 

MSK IPAR SIM PRIMAL CRASH 

Controls whether crashing is performed in the primal simplex optimizer. 

MSK IPAR SIM PRIMAL RESTRICT SELECTION 


MSK IPAR SIM PRIMAL SELECTION 

Controls the primal simplex strategy. 

MSK IPAR SIM REFACTOR FREQ 

Controls how frequent the basis is refactorized. 

MSK IPAR SIM REFORMULATION 

Controls if the simplex optimizers are allowed to reformulate the problem. 

MSK IPAR SIM SAVE LU 

Controls storage of LU factorization. 

MSK IPAR SIM SCALING 

Controls how much effort is used in scaling the problem before a simplex optimizer is used. 

MSK IPAR SIM SCALING METHOD 

Controls how the problem is scaled before a simplex optimizer is used. 

MSK IPAR SIM SOLVE FORM 

Controls whether the primal or the dual problem is solved by the primal-/dual- simplex optimizer. 

MSK IPAR SIM STABILITY PRIORITY 

Controls how high priority the numerical stability should be given. 

MSK IPAR SIM SWITCH OPTIMIZER 

Controls the simplex behavior. 

USE BASIS EST 

Use MOSEK basis estimation in case of an interior solution.


4.6 Mixed Integer Optimizer Options 

MSK DPAR MIO DISABLE TERM TIME 

Disables termination criteria. 

MSK DPAR MIO HEURISTIC TIME 

Maximum amount of time to be used in the heuristic search for a good feasible integer solution. 

MSK DPAR MIO MAX TIME 

This parameter limits the maximum time spent by the mixed-integer optimizer. 

MSK DPAR MIO MAX TIME APRX OPT 

Number of seconds spent by the MIO before the MIO TOL REL RELAX INT is applied. 

MSK DPAR MIO NEAR TOL ABS GAP 

Relaxed absolute optimality tolerance employed by the mixed-integer optimizer. 

MSK DPAR MIO NEAR TOL REL GAP 

Relaxed relative optimality tolerance employed by the mixed integer optimizer. 

MSK DPAR MIO REL ADD CUT LIMITED 

Controls how many cuts the mixed-integer optimizer is allowed to add to the problem. 

MSK DPAR MIO REL GAP CONST 

This value is used to compute the relative gap for the solution to an integer optimization problem. 

MSK DPAR MIO TOL ABS GAP 

Absolute optimality tolerance employed by the mixed-integer optimizer. 

MSK DPAR MIO TOL ABS RELAX INT 

Absolute relaxation tolerance of the integer constraints. 

MSK DPAR MIO TOL FEAS 

Feasibility tolerance for mixed integer solver. 

MSK DPAR MIO TOL REL GAP 

Relative optimality tolerance employed by the mixed-integer optimizer. 

MSK DPAR MIO TOL REL RELAX INT 

Relative relaxation tolerance of the integer constraints. 

MSK DPAR MIO TOL X 

Absolute solution tolerance used in mixed-integer optimizer. 

MSK IPAR MIO BRANCH DIR 

Controls whether the mixed-integer optimizer is branching up or down by default. 

MSK IPAR MIO CONSTRUCT SOL 

Controls construction of an initial mixed integer solution from the values of the integer variables 

MSK IPAR MIO CONT SOL 

Controls the meaning of the interior-point and basic solutions in mixed integer problems. 

MSK IPAR MIO CUT LEVEL ROOT 

Controls the cut level employed by the mixed-integer optimizer at the root node. 

MSK IPAR MIO CUT LEVEL TREE 

Controls the cut level employed by the mixed-integer optimizer at the tree. 

MSK IPAR MIO FEASPUMP LEVEL 

Controls the feasibility pump heuristic used to construct a good initial feasible solution. 

MSK IPAR MIO HEURISTIC LEVEL 

Controls the heuristic employed by the MIO to locate an initial good integer feasible solution. 

MSK IPAR MIO HOTSTART 

Controls whether the integer optimizer is hot-started. 

MSK IPAR MIO KEEP BASIS 

Controls whether the integer presolve keeps bases in memory. 

MSK IPAR MIO LOCAL BRANCH NUMBER 

Controls the size of the local search space when doing local branching. 

MSK IPAR MIO MAX NUM BRANCHES 

Maximum number of branches allowed during the branch and bound search. 

MSK IPAR MIO MAX NUM RELAXS 

Maximum number of relaxations allowed during the branch and bound search. 

MSK IPAR MIO MAX NUM SOLUTIONS


Controls how many feasible solutions the mixed-integer optimizer investigates. 

MSK IPAR MIO NODE OPTIMIZER 

Controls which optimizer is employed at the non-root nodes in the mixed-integer optimizer. 

MSK IPAR MIO NODE SELECTION 

Controls the node selection strategy employed by the mixed-integer optimizer. 

MSK IPAR MIO PRESOLVE AGGREGATE 

Controls whether the presolve used by the MIO tries to aggregate the constraints. 

MSK IPAR MIO PRESOLVE PROBING 

Controls whether the mixed-integer presolve performs probing which can be very time consuming. 

MSK IPAR MIO PRESOLVE USE 

Controls whether presolve is performed by the mixed-integer optimizer. 

MSK IPAR MIO ROOT OPTIMIZER 

Controls which optimizer is employed at the root node in the mixed-integer optimizer. 

MSK IPAR MIO STRONG BRANCH 

The value specifies the depth from the root in which strong branching is used. 

MSK IPAR MIO USE MULTITHREADED OPTIMIZER 

Controls wheter the new multithreaded optimizer should be used for Mixed integer problems. 

SOLVEFINAL 

Switch to solve the problem with fixed discrete variables. Overwrites MSK IPAR MIO CONT SOL. 

FIXOPTFILE 

Name of option file which is read just before solving the fixed problem. 

4.7 Other Options 

MSK DPAR NONCONVEX TOL FEAS 

Feasibility tolerance used by the nonconvex optimizer. 

MSK DPAR NONCONVEX TOL OPT 

Optimality tolerance used by the nonconvex optimizer. 

MSK DPAR QCQO REFORMULATE REL DROP TOL 

Determines when columns are dropped in incomplete cholesky factorization. 

MSK IPAR ANA SOL BASIS 

Controls whether the basis matrix is analyzed in solution analyzer. 

MSK IPAR ANA SOL PRINT VIOLATED 

Controls whether a list of violated constraints is printed when calling task ANALYZESOLUTION. 

MSK IPAR FEASREPAIR OPTIMIZE 

Controls which type of feasibility analysis is to be performed. 

MSK IPAR INFEAS GENERIC NAMES 

Controls whether generic names are used when an infeasible subproblem is created. 

MSK IPAR INFEAS PREFER PRIMAL 

Controls which certificate is used if primal- and dual- certificate of infeasibility is available. 

MSK IPAR MT SPINCOUNT 

Set the number of iterations to spin before sleeping. 

MSK IPAR NONCONVEX MAX ITERATIONS 

Maximum number of iterations that can be used by the nonconvex optimizer. 

MSK IPAR OPF MAX TERMS PER LINE 

The maximum number of terms (linear and quadratic) per line when an OPF file is written. 

MSK IPAR OPF WRITE PARAMETERS 

Write a parameter section in an OPF file. 

MSK IPAR QO SEPARABLE REFORMULATION 

Determine if Quadratic programing problems should be reformulated to separable form. 

MSK IPAR SENSITIVITY OPTIMIZER 

Controls which optimizer is used for optimal partition sensitivity analysis. 

MSK IPAR SENSITIVITY TYPE


Controls which type of sensitivity analysis is to be performed. 

MSK IPAR TIMING LEVEL 

Controls the a amount of timing performed inside MOSEK. 

MSK IPAR WRITE DATA FORMAT 

Controls the file format when writing task data to a file. 

MSK IPAR WRITE GENERIC NAMES 

Controls whether the generic names or user-defined names are used in the data file. 

MSK IPAR WRITE GENERIC NAMES IO 

Index origin used in generic names. 

READFILE 

Read secondary option file. 

5 Detailed Descriptions of MOSEK Options 

MSK DPAR ANA SOL INFEAS TOL (real) 

Report constraint bound violation if it exceeds this tolerance. 

If a constraint violates its bound with an amount larger than this value, the constraint name, index and violation will 

be printed by the solution analyzer. 

(default = 1e-6) 

MSK DPAR BASIS REL TOL S (real) 

Maximum relative dual bound violation allowed in an optimal basic solution. 

(default = 1.0e-12) 

MSK DPAR BASIS TOL S (real) 

Maximum absolute dual bound violation in an optimal basic solution. 


MSK DPAR BASIS TOL X (real) 

Maximum absolute primal bound violation allowed in an optimal basic solution. 


MSK DPAR CHECK CONVEXITY REL TOL (real) 

This parameter controls when the full convexity check declares a problem to be non-convex. 

Increasing this tolerance relaxes the criteria for declaring the problem non-convex. A problem is declared non-convex 

if negative (positive) pivot elements are detected in the cholesky factor of a matrix which is required to be PSD (NSD). 

This parameter controles how much this non-negativity requirement may be violated. If d i is the pivot element for 

column i, then the matrix Q is considered to not be PSD if: 

d i


MSK DPAR PRESOLVE TOL AIJ (real) 

Absolute zero tolerance employed for a i j in the presolve. 


MSK DPAR PRESOLVE TOL REL LINDEP (real) 

Relative tolerance employed by the linear dependency checker. 


MSK DPAR PRESOLVE TOL S (real) 

Absolute zero tolerance employed for s i in the presolve. 


MSK DPAR PRESOLVE TOL X (real) 

Absolute zero tolerance employed for x j in the presolve. 


MSK IPAR AUTO SORT A BEFORE OPT (string) 

Controls whether the elements in each column of A are sorted before an optimization is performed. 

This is not required but makes the optimization more deterministic. 

(default = MSK OFF) 

MSK OFF Switch the option off. 

MSK ON Switch the option on. 

MSK IPAR CONCURRENT NUM OPTIMIZERS (integer) 

The maximum number of simultaneous optimizations that will be started by the concurrent optimizer. 

(default = 2) 

MSK IPAR CONCURRENT PRIORITY DUAL SIMPLEX (integer) 

Priority of the dual simplex algorithm when selecting solvers for concurrent optimization. 

(default = 2) 

MSK IPAR CONCURRENT PRIORITY FREE SIMPLEX (integer) 

Priority of the free simplex optimizer when selecting solvers for concurrent optimization. 

(default = 3) 

MSK IPAR CONCURRENT PRIORITY INTPNT (integer) 

Priority of the interior-point algorithm when selecting solvers for concurrent optimization. 

(default = 4) 

MSK IPAR CONCURRENT PRIORITY PRIMAL SIMPLEX (integer) 

Priority of the primal simplex algorithm when selecting solvers for concurrent optimization. 

(default = 1) 

MSK IPAR INFEAS REPORT AUTO (string) 

Controls infeasibility report production. 

Controls whether an infeasibility report is automatically produced after the optimization if the problem is primal or 

dual infeasible. 



MSK ON Switch the option on.


MSK IPAR INFEAS REPORT LEVEL (integer) 

Controls the amount of information presented in an infeasibility report. 

Higher values imply more information. 

(default = 1) 

MSK IPAR NUM THREADS (integer) 

Controls the number of threads employed by the optimizer. 

If set to 0 the number of threads used will be equal to the number of cores detected on the machine. 

(default = 0) 

MSK IPAR OPTIMIZER (string) 

The paramter controls which optimizer is used to optimize the task. 

(default = MSK OPTIMIZER FREE) 

MSK OPTIMIZER CONCURRENT 

The optimizer for nonconvex nonlinear problems. 

MSK OPTIMIZER CONIC 

The optimizer for problems having conic constraints. 

MSK OPTIMIZER DUAL SIMPLEX 

The dual simplex optimizer is used. 

MSK OPTIMIZER FREE 

The optimizer is chosen automatically. 

MSK OPTIMIZER FREE SIMPLEX 

One of the simplex optimizers is used. 

MSK OPTIMIZER INTPNT 

The interior-point optimizer is used. 

MSK OPTIMIZER MIXED INT 

The mixed-integer optimizer. 

MSK OPTIMIZER MIXED INT CONIC 

The mixed-integer optimizer for conic and linear problems. 

MSK OPTIMIZER NETWORK PRIMAL SIMPLEX 

The network primal simplex optimizer is used. It is only applicable to pute network problems. 

MSK OPTIMIZER NONCONVEX 


MSK OPTIMIZER PRIMAL DUAL SIMPLEX 

The primal dual simplex optimizer is used. 

MSK OPTIMIZER PRIMAL SIMPLEX 

The primal simplex optimizer is used. 

MSK IPAR PRESOLVE ELIMINATOR MAX NUM TRIES (integer) 

Control the maximum number of times the eliminator is tried. 

(default = -1) 

MSK IPAR PRESOLVE ELIMINATOR USE (string) 

Controls whether free or implied free variables are eliminated from the problem. 

(default = MSK ON) 



MSK IPAR PRESOLVE ELIM FILL (integer) 

Controls the maximum amount of fill-in in the elimination phase of the presolve. 

This parameter times the number of variables plus the number of constraints denotes the amount of fill in. 

(default = 1)


MSK IPAR PRESOLVE LEVEL (integer) 

Currently not used. 


MSK IPAR PRESOLVE LINDEP ABS WORK TRH (integer) 


(default = 100) 

MSK IPAR PRESOLVE LINDEP REL WORK TRH (integer) 



MSK IPAR PRESOLVE LINDEP USE (string) 

Controls whether the linear constraints are checked for linear dependencies. 




MSK IPAR PRESOLVE MAX NUM REDUCTIONS (integer) 

Controls the maximum number reductions performed by the presolve. 

The value of the parameter is normally only changed in connection with debugging. A negative value implies that an 

infinite number of reductions are allowed. 


MSK IPAR PRESOLVE USE (string) 

Controls whether the presolve is applied to a problem before it is optimized. 

(default = MSK PRESOLVE MODE FREE) 

MSK PRESOLVE MODE FREE 

It is decided automatically whether to presolve before the problem is optimized. 

MSK PRESOLVE MODE OFF 

The problem is not presolved before it is optimized. 

MSK PRESOLVE MODE ON 

The problem is presolved before it is optimized. 

MSK IPAR PRIMAL REPAIR OPTIMIZER (string) 

Controls which optimizer that is used to find the optimal repair. 















The mixed-integer optimizer.












MSK SPAR PARAM READ FILE NAME (string) 

Modifications to the parameter database is read from this file. 

MSK SPAR PARAM WRITE FILE NAME (string) 

The parameter database is written to this file. 

MSK SPAR DATA FILE NAME (string) 

Problem data is written to this file. File extension determines format. 

MSK DPAR DATA TOL AIJ (real) 

Absolute zero tolerance for elements in A. 

If any value in the constraint matrix is smaller than this parameter in absolute terms MOSEK will treat the values as 

zero and generate a warning. 

Range: [1.0e-16,1.0e-6] 


MSK DPAR DATA TOL AIJ HUGE (real) 

An element in A which is larger than this value in absolute size causes an error. 

(default = 1.0e20) 

MSK DPAR DATA TOL AIJ LARGE (real) 

An element in A which is larger than this value in absolute size causes a warning message. 


MSK DPAR DATA TOL BOUND INF (real) 

Any bound which in absolute value is greater than this parameter is considered infinite. 


MSK DPAR DATA TOL BOUND WRN (real) 

If a bound value is larger than this value in absolute size, then a warning message is issued. 


MSK DPAR DATA TOL CJ LARGE (real) 

An element in c which is larger than this value in absolute terms causes a warning message. 


MSK DPAR DATA TOL C HUGE (real) 

An element in c larger than the value of this parameter in absolute terms generates an error. 

c is the objective and coefficients larger than this value are considered to be huge. 

(default = 1.0e16)


MSK DPAR DATA TOL QIJ (real) 

Absolute zero tolerance for elements in Q matrices. 


MSK DPAR DATA TOL X (real) 

Zero tolerance for constraints and variables. 

That means if the distance between the lower and upper bound is less than this value, then the lower and lower bound 

is considered identical. 


MSK DPAR LOWER OBJ CUT (real) 

Lower objective limit. 

If either a primal or dual feasible solution is found proving that the optimal objective value is outside, the interval 

(MSK DPAR LOWER OBJ CUT, MSK DPAR UPPER OBJ CUT), then MOSEK is terminated. 

(default = -1.0e30) 

See also: MSK DPAR LOWER OBJ CUT FINITE TRH. 

MSK DPAR LOWER OBJ CUT FINITE TRH (real) 

Lower objective limit threshold. 

If the lower objective cut (MSK DPAR LOWER OBJ CUT) is less than MSK DPAR LOWER OBJ CUT FINITE TRH, then the 

lower objective cut MSK DPAR LOWER OBJ CUT is treated as infinity. 

(default = -0.5e30) 

MSK DPAR UPPER OBJ CUT (real) 

Upper objective limit. 

If either a primal or dual feasible solution is found proving that the optimal objective value is outside, the interval 

(MSK DPAR LOWER OBJ CUT, MSK DPAR UPPER OBJ CUT), then MOSEK is terminated. 


See also: MSK DPAR UPPER OBJ CUT FINITE TRH. 

MSK DPAR UPPER OBJ CUT FINITE TRH (real) 

Upper objective limit threshold. 

If the upper objective cut (MSK DPAR UPPER OBJ CUT) is greater than MSK DPAR UPPER OBJ CUT FINITE TRH, then 

the upper objective cut MSK DPAR UPPER OBJ CUT is treated as infinity. 


MSK IPAR CHECK CONVEXITY (string) 

Specify the level of convexity check on quadratic problems 

(default = MSK CHECK CONVEXITY FULL) 

MSK CHECK CONVEXITY FULL 

Perform a full convexity check. 

MSK CHECK CONVEXITY NONE 

No convexity check. 

MSK CHECK CONVEXITY SIMPLE 

Perform simple and fast convexity check. 

MSK IPAR LOG (integer) 

Controls the amount of log information. 

The value 0 implies that all log information is suppressed. A higher level implies that more information is logged. 

(default = 10)


MSK IPAR LOG BI (integer) 

Controls the amount of output printed by the basis identification procedure. 

A higher level implies that more information is logged. 

(default = 4) 

MSK IPAR LOG BI FREQ (integer) 

Controls logging frequency. 

In detail thios option controls how frequent the optimizer outputs information about the basis identification and how 

frequent the user-defined call-back function is called. 


MSK IPAR LOG CHECK CONVEXITY (integer) 

Controls logging in convexity check on quadratic problems. 

Set to a positive value to turn logging on. If a quadratic coefficient matrix is found to violate the requirement of PSD 

(NSD) then a list of negative (positive) pivot elements is printed. The absolute value of the pivot elements is also 

shown. 

(default = 0) 

MSK IPAR LOG CONCURRENT (integer) 

Controls amount of output printed by the concurrent optimizer. 

(default = 1) 

MSK IPAR LOG FACTOR (integer) 

If turned on, then the factor log lines are added to the log. 

(default = 1) 

MSK IPAR LOG FEAS REPAIR (integer) 

Controls the amount of output printed when performing feasibility repair. 

(default = 1) 

MSK IPAR LOG HEAD (integer) 

If turned on, then a header line is added to the log. 

(default = 1) 

MSK IPAR LOG INFEAS ANA (integer) 

Controls amount of output printed by the infeasibility analyzer procedures. 


(default = 1) 

MSK IPAR LOG INTPNT (integer) 

Controls amount of output printed printed by the interior-point optimizer. 


(default = 4) 

MSK IPAR LOG MIO (integer) 

Controls the log level for the mixed-integer optimizer. 


(default = 4)


MSK IPAR LOG MIO FREQ (integer) 

Controls how frequent the mixed-integer optimizer prints the log line. 

It will print a line every time MSK INTPAR LOG MIO FREQ relaxations have been solved. 


MSK IPAR LOG NONCONVEX (integer) 

Controls amount of output printed by the nonconvex optimizer. 

(default = 1) 

MSK IPAR LOG OPTIMIZER (integer) 

Controls the amount of general optimizer information that is logged. 

(default = 1) 

MSK IPAR LOG ORDER (integer) 

If turned on, then factor lines are added to the log. 

(default = 1) 

MSK IPAR LOG PARAM (integer) 

Controls the amount of information printed out about parameter changes. 

(default = 0) 

MSK IPAR LOG PRESOLVE (integer) 

Controls amount of output printed by the presolve procedure. 


(default = 1) 

MSK IPAR LOG RESPONSE (integer) 

Controls amount of output printed when response codes are reported. 


(default = 0) 

MSK IPAR LOG SENSITIVITY (integer) 

Controls the amount of logging during the sensitivity analysis. 

• 0: No logging information is produced. 

• 1: Timing information is printed. 

• 2: Sensitivity results are printed. 

(default = 1) 

MSK IPAR LOG SENSITIVITY OPT (integer) 

Controls the amount of logging from the optimizers employed during the sensitivity analysis. 

0 means no logging information is produced. 

(default = 0) 

MSK IPAR LOG SIM (integer) 

Controls amount of output printed by the simplex optimizer. 


(default = 4)


MSK IPAR LOG SIM FREQ (integer) 

Controls simplex logging frequency. 

In detail this option controls how frequent the simplex optimizer outputs information about the optimization and how 

frequent the user-defined call-back function is called. 


MSK IPAR LOG STORAGE (integer) 

When turned on, MOSEK prints messages regarding the storage usage and allocation. 

(default = 0) 

MSK IPAR MAX NUM WARNINGS (integer) 

Waning level. A higher value results in more warnings. 


MSK IPAR WARNING LEVEL (integer) 

Warning level. A higher value implies more warnings. 

(default = 1) 

MSK IPAR WRITE IGNORE INCOMPATIBLE CONIC ITEMS (string) 

Controls ignoration of incompatible conic items. 

If the output format is not compatible with conic quadratic problems this parameter controls if the writer ignores the 

conic parts or produces an error. 




MSK IPAR WRITE IGNORE INCOMPATIBLE ITEMS (string) 

Controls if the writer ignores incompatible problem items when writing files. 




MSK IPAR WRITE IGNORE INCOMPATIBLE NL ITEMS (string) 

Controls if the writer ignores general non-linear terms or produces an error. 




MSK IPAR WRITE IGNORE INCOMPATIBLE PSD ITEMS (string) 

Controls ignoration of incompatible psd items. 

If the output format is not compatible with semidefinite problems this parameter controls if the writer ignores the psd 

parts or produces an error. 




MSK IPAR WRITE LP LINE WIDTH (integer) 

Maximum width of line in an LP file written by MOSEK. 

(default = 80)


MSK IPAR WRITE LP QUOTED NAMES (string) 

If this option is turned on, then MOSEK will quote invalid LP names when writing an LP file. 




MSK IPAR WRITE LP STRICT FORMAT (string) 

Controls whether LP output files satisfy the LP format strictly. 




MSK IPAR WRITE LP TERMS PER LINE (integer) 

Maximum number of terms on a single line in an LP file written by MOSEK. 0 means unlimited. 


MSK IPAR WRITE MPS INT (string) 

Controls writing of marker records to the MPS file to indicate if variables are integer restricted. 




MSK IPAR WRITE PRECISION (integer) 

Controls the precision with which double numbers are printed in the MPS data file. 

In general it is not worthwhile to use a value higher than 15. 

(default = 8) 

MSK IPAR WRITE XML MODE (string) 

Controls output mode for XML file. 

In detail this option controls if linear coefficients should be written by row or column when writing in the XML file 

format. 

(default = MSK WRITE XML MODE ROW) 

MSK DPAR FEASREPAIR TOL (real) 

Tolerance for constraint enforcing upper bound on sum of weighted violations in feasibility repair. 

Range: [1.0e-16,1.0e+16] 


MSK DPAR INTPNT CO TOL DFEAS (real) 

Dual feasibility tolerance used by the conic interior-point optimizer. 

Range: [0.0,1.0] 


See also: MSK DPAR INTPNT CO TOL NEAR REL. 

MSK DPAR INTPNT CO TOL INFEAS (real) 

Controls when the conic interior-point optimizer declares the model primal or dual infeasible. 

A small number means the optimizer gets more conservative about declaring the model infeasible. 

Range: [0.0,1.0] 

(default = 1.0e-10)


MSK DPAR INTPNT CO TOL MU RED (real) 

Optimality tolerance for the conic solver. 

Relative complementarity gap tolerance feasibility tolerance used by the conic interior-point optimizer. 

Range: [0.0,1.0] 


MSK DPAR INTPNT CO TOL NEAR REL (real) 


If MOSEK cannot compute a solution that has the prescribed accuracy, then it will multiply the termination tolerances 

with value of this parameter. If the solution then satisfies the termination criteria, then the solution is denoted near 

optimal, near feasible and so forth. 


MSK DPAR INTPNT CO TOL PFEAS (real) 

Primal feasibility tolerance used by the conic interior-point optimizer. 

Range: [0.0,1.0] 



MSK DPAR INTPNT CO TOL REL GAP (real) 

Relative gap termination tolerance used by the conic interior-point optimizer. 

Range: [0.0,1.0] 



MSK DPAR INTPNT NL MERIT BAL (real) 

Controls if the complementarity and infeasibility is converging to zero at about equal rates. 

Range: [0.0,0.99] 


MSK DPAR INTPNT NL TOL DFEAS (real) 

Dual feasibility tolerance used when a nonlinear model is solved. 

Range: [0.0,1.0] 


MSK DPAR INTPNT NL TOL MU RED (real) 


Range: [0.0,1.0] 


MSK DPAR INTPNT NL TOL NEAR REL (real) 


If MOSEK nonlinear interior-point optimizer cannot compute a solution that has the prescribed accuracy, then it will 

multiply the termination tolerances with value of this parameter. If the solution then satisfies the termination criteria, 

then the solution is denoted near optimal, near feasible and so forth. 

(default = 1000.0) 

MSK DPAR INTPNT NL TOL PFEAS (real) 

Primal feasibility tolerance used when a nonlinear model is solved. 

Range: [0.0,1.0] 

(default = 1.0e-8)


MSK DPAR INTPNT NL TOL REL GAP (real) 

Relative gap termination tolerance for nonlinear problems. 


MSK DPAR INTPNT NL TOL REL STEP (real) 

Relative step size to the boundary for general nonlinear optimization problems. 

Range: [1.0e-4,0.9999999] 

(default = 0.995) 

MSK DPAR INTPNT TOL DFEAS (real) 

Dual feasibility tolerance used for linear and quadratic optimization problems. 

Range: [0.0,1.0] 


MSK DPAR INTPNT TOL DSAFE (real) 

Controls the initial dual starting point used by the interior-point optimizer. 

If the interior-point optimizer converges slowly and/or the constraint or variable bounds are very large, then it might 

be worthwhile to increase this value. 

(default = 1.0) 

MSK DPAR INTPNT TOL INFEAS (real) 

Controls when the optimizer declares the model primal or dual infeasible. 

A small number means the optimizer gets more conservative about declaring the model infeasible. 

Range: [0.0,1.0] 


MSK DPAR INTPNT TOL MU RED (real) 


Range: [0.0,1.0] 


MSK DPAR INTPNT TOL PATH (real) 

Controls how close the interior-point optimizer follows the central path. 

A large value of this parameter means the central is followed very closely. On numerical unstable problems it might 

worthwhile to increase this parameter. 

Range: [0.0,0.9999] 


MSK DPAR INTPNT TOL PFEAS (real) 

Primal feasibility tolerance used for linear and quadratic optimization problems. 

Range: [0.0,1.0] 


MSK DPAR INTPNT TOL PSAFE (real) 

Controls the initial primal starting point used by the interior-point optimizer. 

If the interior-point optimizer converges slowly and/or the constraint or variable bounds are very large, then it might 

be worthwhile to increase this value. 

(default = 1.0)


MSK DPAR INTPNT TOL REL GAP (real) 

Relative gap termination tolerance. 


MSK DPAR INTPNT TOL REL STEP (real) 

Relative step size to the boundary for linear and quadratic optimization problems. 

Range: [1.0e-4,0.999999] 

(default = 0.9999) 

MSK DPAR INTPNT TOL STEP SIZE (real) 

Step size tolerance value. 

If the step size falls below the value of this parameter, then the interior-point optimizer assumes it is stalled. It it does 

not not make any progress. 

Range: [0.0,1.0] 


MSK IPAR INTPNT BASIS (string) 

Controls whether the interior-point optimizer also computes an optimal basis. 

(default = MSK BI ALWAYS) 

MSK BI ALWAYS 

Always perform basis identification.. 

Basis identification is always performed even if the interior-point optimizer terminates abnormally. 

MSK BI IF FEASIBLE 

Only perform Basis identification if problem status is feasible.. 

Basis identification is not performed if the interior-point optimizer terminates with a problem 

status saying that the problem is primal or dual infeasible. 

MSK BI NEVER 

Never do basis identification. 

MSK BI NO ERROR 

Basis identification is performed if the interior-point optimizer terminates without an error. 

See also: MSK IPAR BI IGNORE MAX ITER, MSK IPAR BI IGNORE NUM ERROR. 

MSK IPAR INTPNT DIFF STEP (string) 

Controls whether different step sizes are allowed in the primal and dual space. 




MSK IPAR INTPNT FACTOR DEBUG LVL (integer) 

Controls factorization debug level. 

(default = 0) 

MSK IPAR INTPNT FACTOR METHOD (integer) 

Controls the method used to factor the Newton equation system. 

(default = 0) 

MSK IPAR INTPNT MAX ITERATIONS (integer) 

Controls the maximum number of iterations allowed in the interior-point optimizer. 

(default = 400)


MSK IPAR INTPNT MAX NUM COR (integer) 

Controls the maximum number of correctors allowed by the multiple corrector procedure. 

A negative value means that MOSEK is making the choice. 


MSK IPAR INTPNT MAX NUM REFINEMENT STEPS (integer) 

Maximum number of steps to be used by the iterative refinement of the search direction. 

A negative value implies that the optimizer chooses the maximum number of iterative refinement steps. 


MSK IPAR INTPNT OFF COL TRH (integer) 

Controls how many offending columns are detected in the Jacobian of the constraint matrix. 

1 means aggressive detection, higher values mean less aggressive detection. 0 means no detection. 


MSK IPAR INTPNT ORDER METHOD (string) 

Controls the ordering strategy. 

This refers to the ordering strategy used by the interior-point optimizer when factorizing the Newton equation system. 

(default = MSK ORDER METHOD FREE) 

MSK ORDER METHOD APPMINLOC 

Approximate minimum local fill-in ordering is employed. 

MSK ORDER METHOD EXPERIMENTAL 

This option should not be used. 

MSK ORDER METHOD FORCE GRAPHPAR 

Always use the graph partitioning based ordering.. 

Use the graph partitioning based ordering even if it is worse than the approximate minimum 

local fill ordering. 

MSK ORDER METHOD FREE 

The ordering method is chosen automatically. 

MSK ORDER METHOD NONE 

No ordering is used. 

MSK ORDER METHOD TRY GRAPHPAR 

Always try the the graph partitioning based ordering. 

MSK IPAR INTPNT REGULARIZATION USE (string) 

Controls whether regularization is allowed. 




MSK IPAR INTPNT SCALING (string) 

Controls how the problem is scaled before the interior-point optimizer is used. 

(default = MSK SCALING FREE) 

MSK SCALING AGGRESSIVE 

A very aggressive scaling is performed. 

MSK SCALING FREE 

The optimizer chooses the scaling heuristic. 

MSK SCALING MODERATE 

A conservative scaling is performed. 

MSK SCALING NONE 

No scaling is performed.


MSK IPAR INTPNT SOLVE FORM (string) 

Controls whether the primal or the dual problem is solved. 

(default = MSK SOLVE FREE) 

MSK SOLVE DUAL 

The optimizer should solve the dual problem. 

MSK SOLVE FREE 

The optimizer is free to solve either the primal or the dual problem. 

MSK SOLVE PRIMAL 

The optimizer should solve the primal problem. 

MSK IPAR INTPNT STARTING POINT (string) 

Starting point used by the interior-point optimizer. 

(default = MSK STARTING POINT FREE) 

MSK STARTING POINT CONSTANT 

The optimizer constructs a starting point by assigning a constant value to all variables.. 

The constant value is assigned to all primal and dual variables. This starting point is 

normally robust. 

MSK STARTING POINT FREE 

The starting point is chosen automatically. 

MSK STARTING POINT GUESS 

The optimizer guesses a starting point. 

MSK STARTING POINT SATISFY BOUNDS 

The starting point is choosen to satisfy all the simple bounds on nonlinear variables.. 

If this starting point is employed, then more care than usual should employed when choosing the 

bounds on the nonlinear variables. In particular very tight bounds should be avoided. 

MSK DPAR SIMPLEX ABS TOL PIV (real) 

Absolute pivot tolerance employed by the simplex optimizers. 


MSK DPAR SIM LU TOL REL PIV (real) 

Relative pivot tolerance for LU factorization in simplex and basis identification. 

This tolerance is employed when computing the LU factorization of the basis in the simplex optimizers and in the 

basis identification procedure. A value closer to 1.0 generally improves numerical stability but typically also implies 

an increase in the computational work. 

Range: [1.0e-6,0.999999] 


MSK IPAR BI CLEAN OPTIMIZER (string) 

Controls which optimizer is used in the clean-up phase. 











One of the simplex optimizers is used.
















MSK IPAR BI IGNORE MAX ITER (string) 

Controls if basis identification is performed under certain conditions. 

If the parameter MSK IPAR INTPNT BASIS has the value MSK BI NO ERROR and the interior-point optimizer has terminated 

due to maximum number of iterations, then basis identification is performed if this parameter has the value 

MSK ON. 




MSK IPAR BI IGNORE NUM ERROR (string) 

Turns on basis identification if interior-point optimizer is terminated due to a numerical problem. 

If the parameter MSK IPAR INTPNT BASIS has the value MSK BI NO ERROR and the interior-point optimizer has terminated 

due to a numerical problem, then basis identification is performed if this parameter has the value MSK ON. 




MSK IPAR BI MAX ITERATIONS (integer) 

Maximum number of iterations after basis identification. 

Controls the maximum number of simplex iterations allowed to optimize a basis after the basis identification. 

(default = 1000000) 

MSK IPAR SIM BASIS FACTOR USE (string) 

Controls whether a (LU) factorization of the basis is used in a hot-start. 

Forcing a refactorization sometimes improves the stability of the simplex optimizers, but in most cases there is a 

performance penalty. 




MSK IPAR SIM DEGEN (string) 

Controls how aggressively degeneration is handled. 

(default = MSK SIM DEGEN FREE)


MSK SIM DEGEN AGGRESSIVE 

The simplex optimizer should use an aggressive degeneration strategy. 

MSK SIM DEGEN FREE 

The simplex optimizer chooses the degeneration strategy. 

MSK SIM DEGEN MINIMUM 

The simplex optimizer should use a minimum degeneration strategy. 

MSK SIM DEGEN MODERATE 

The simplex optimizer should use a moderate degeneration strategy. 

MSK SIM DEGEN NONE 

The simplex optimizer should use no degeneration strategy. 

MSK IPAR SIM DUAL CRASH (integer) 

Controls whether crashing is performed in the dual simplex optimizer. 

In general if a basis consists of more than (100*MSK IPAR SIM DUAL CRASH) percent fixed variables, then a crash 

will be performed. 

(default = GAMS BRatio) 

MSK IPAR SIM DUAL RESTRICT SELECTION (integer) 


The dual simplex optimizer can use a so-called restricted selection/pricing strategy to chooses the outgoing variable. 

Hence, if restricted selection is applied, then the dual simplex optimizer first choose a subset of all the potential 

outgoing variables. Next, for some time it will choose the outgoing variable only among the subset. From time to 

time the subset is redefined. 

A larger value of this parameter implies that the optimizer will be more aggressive in its restriction strategy, i.e. a 

value of 0 implies that the restriction strategy is not applied at all. 

Range: [0,100] 


MSK IPAR SIM DUAL SELECTION (string) 

Controls the dual simplex strategy. 

In detail this option controls the choice of the incoming variable, known as the selection strategy, in the dual simplex 

optimizer. 

(default = MSK SIM SELECTION FREE) 

MSK SIM SELECTION ASE 

The optimizer uses approximate steepest-edge pricing. 

MSK SIM SELECTION DEVEX 

The optimizer uses devex steepest-edge pricing.. 

If it is not available an approximate steep-edge selection is chosen. 

MSK SIM SELECTION FREE 

The optimizer chooses the pricing strategy. 

MSK SIM SELECTION FULL 

The optimizer uses full pricing. 

MSK SIM SELECTION PARTIAL 

The optimizer uses a partial selection approach.. 

The approach is usually beneficial if the number of variables is much larger than the number 

of constraints. 

MSK SIM SELECTION SE 

The optimizer uses steepest-edge selection.. 

If it is not available an approximate steep-edge selection is chosen 

MSK IPAR SIM EXPLOIT DUPVEC (string) 

Controls if the simplex optimizers are allowed to exploit duplicated columns. 

(default = MSK SIM EXPLOIT DUPVEC OFF)


MSK SIM EXPLOIT DUPVEC FREE 

The simplex optimizer can choose freely. 

MSK SIM EXPLOIT DUPVEC OFF 

Disallow the simplex optimizer to exploit duplicated columns. 

MSK SIM EXPLOIT DUPVEC ON 

Allow the simplex optimizer to exploit duplicated columns. 

MSK IPAR SIM HOTSTART (string) 

Controls the type of hot-start that the simplex optimizer perform. 

(default = MSK SIM HOTSTART FREE) 

MSK SIM HOTSTART FREE 

The simplex optimize chooses the hot-start type. 

MSK SIM HOTSTART NONE 

The simplex optimizer performs a coldstart. 

MSK SIM HOTSTART STATUS KEYS 

Only the status keys of the constraints and variables are used to choose the type of hot-start. 

MSK IPAR SIM HOTSTART LU (string) 

Determines if the simplex optimizer should exploit the initial factorization. 




MSK IPAR SIM MAX ITERATIONS (integer) 

Maximum number of iterations that can be used by a simplex optimizer. 

(default = GAMS IterLim) 

MSK IPAR SIM MAX NUM SETBACKS (integer) 

Controls how many set-backs are allowed within a simplex optimizer. 

A setback is an event where the optimizer moves in the wrong direction. This is impossible in theory but may happen 

due to numerical problems. 


MSK IPAR SIM NON SINGULAR (string) 

Controls if the simplex optimizer ensures a non-singular basis, if possible. 




MSK IPAR SIM PRIMAL CRASH (integer) 

Controls whether crashing is performed in the primal simplex optimizer. 

In general if a basis consists of more than (100*MSK IPAR SIM PRIMAL CRASH) percent fixed variables, then a crash 

will be performed. 

(default = GAMS BRatio) 

MSK IPAR SIM PRIMAL RESTRICT SELECTION (integer) 


The primal simplex optimizer can use a so-called restricted selection/pricing strategy to chooses the outgoing variable. 

Hence, if restricted selection is applied, then the primal simplex optimizer first choose a subset of all the potential


incoming variables. Next, for some time it will choose the incoming variable only among the subset. From time to 

time the subset is redefined. 

A larger value of this parameter implies that the optimizer will be more aggressive in its restriction strategy, i.e. a 

value of 0 implies that the restriction strategy is not applied at all. 

Range: [0,100] 


MSK IPAR SIM PRIMAL SELECTION (string) 

Controls the primal simplex strategy. 

In detail this option controls the choice of the incoming variable, known as the selection strategy, in the primal simplex 

optimizer. 

(default = MSK SIM SELECTION FREE) 

MSK SIM SELECTION ASE 

The optimizer uses approximate steepest-edge pricing. 

MSK SIM SELECTION DEVEX 

The optimizer uses devex steepest-edge pricing.. 

If it is not available an approximate steep-edge selection is chosen. 

MSK SIM SELECTION FREE 

The optimizer chooses the pricing strategy. 

MSK SIM SELECTION FULL 

The optimizer uses full pricing. 

MSK SIM SELECTION PARTIAL 

The optimizer uses a partial selection approach.. 

The approach is usually beneficial if the number of variables is much larger than the number 

of constraints. 

MSK SIM SELECTION SE 

The optimizer uses steepest-edge selection.. 

If it is not available an approximate steep-edge selection is chosen 

MSK IPAR SIM REFACTOR FREQ (integer) 

Controls how frequent the basis is refactorized. 

The value 0 means that the optimizer determines the best point of refactorization. It is strongly recommended NOT 

to change this parameter. 

(default = 0) 

MSK IPAR SIM REFORMULATION (string) 

Controls if the simplex optimizers are allowed to reformulate the problem. 

(default = MSK SIM REFORMULATION OFF) 

MSK SIM REFORMULATION AGGRESSIVE 

The simplex optimizer should use an aggressive reformulation strategy. 

MSK SIM REFORMULATION FREE 

The simplex optimizer can choose freely. 

MSK SIM REFORMULATION OFF 

Disallow the simplex optimizer to reformulate the problem. 

MSK SIM REFORMULATION ON 

Allow the simplex optimizer to reformulate the problem. 

MSK IPAR SIM SAVE LU (string) 

Controls storage of LU factorization. 

In detail this option controls if the LU factorization stored should be replaced with the LU factorization corresponding 

to the initial basis. 

(default = MSK OFF)




MSK IPAR SIM SCALING (string) 

Controls how much effort is used in scaling the problem before a simplex optimizer is used. 

(default = MSK SCALING FREE) 

MSK SCALING AGGRESSIVE 

A very aggressive scaling is performed. 

MSK SCALING FREE 


MSK SCALING MODERATE 

A conservative scaling is performed. 

MSK SCALING NONE 

No scaling is performed. 

MSK IPAR SIM SCALING METHOD (string) 

Controls how the problem is scaled before a simplex optimizer is used. 

(default = MSK SCALING METHOD POW2) 

MSK SCALING METHOD FREE 


MSK SCALING METHOD POW2 

Scales only with power of 2 leaving the mantissa untouched. 

MSK IPAR SIM SOLVE FORM (string) 

Controls whether the primal or the dual problem is solved by the primal-/dual- simplex optimizer. 

(default = MSK SOLVE FREE) 

MSK SOLVE DUAL 

The optimizer should solve the dual problem. 

MSK SOLVE FREE 

The optimizer is free to solve either the primal or the dual problem. 

MSK SOLVE PRIMAL 

The optimizer should solve the primal problem. 

MSK IPAR SIM STABILITY PRIORITY (integer) 

Controls how high priority the numerical stability should be given. 

Range: [0,100] 


MSK IPAR SIM SWITCH OPTIMIZER (string) 

Controls the simplex behavior. 

The simplex optimizer sometimes chooses to solve the dual problem instead of the primal problem. This implies that 

if you have chosen to use the dual simplex optimizer and the problem is dualized, then it actually makes sense to use 

the primal simplex optimizer instead. If this parameter is on and the problem is dualized and furthermore the simplex 

optimizer is chosen to be the primal (dual) one, then it is switched to the dual (primal). 





MSK DPAR MIO DISABLE TERM TIME (real) 

Disables termination criteria. 

The termination criteria governed by 

• MSK IPAR MIO MAX NUM RELAXS 

• MSK IPAR MIO MAX NUM BRANCHES 

• MSK DPAR MIO NEAR TOL ABS GAP 

• MSK DPAR MIO NEAR TOL REL GAP 

are disabled the first n seconds. This parameter specifies the number n. A negative value is identical to infinity i.e. 

the termination criteria are never checked.’ 

(default = -1.0) 

See also: MSK DPAR MIO NEAR TOL ABS GAP, MSK DPAR MIO NEAR TOL REL GAP, 

MSK IPAR MIO MAX NUM BRANCHES, MSK IPAR MIO MAX NUM RELAXS. 

MSK DPAR MIO HEURISTIC TIME (real) 

Maximum amount of time to be used in the heuristic search for a good feasible integer solution. 

A negative values implies that the optimizer decides the amount of time to be spend in the heuristic. 


MSK DPAR MIO MAX TIME (real) 

This parameter limits the maximum time spent by the mixed-integer optimizer. 

A negative number means infinity. 


MSK DPAR MIO MAX TIME APRX OPT (real) 

Number of seconds spent by the MIO before the MIO TOL REL RELAX INT is applied. 


MSK DPAR MIO NEAR TOL ABS GAP (real) 

Relaxed absolute optimality tolerance employed by the mixed-integer optimizer. 

This termination criteria is delayed. 

(default = GAMS OptCa) 

See also: MSK DPAR MIO DISABLE TERM TIME. 

MSK DPAR MIO NEAR TOL REL GAP (real) 

Relaxed relative optimality tolerance employed by the mixed integer optimizer. 

The mixed integer optimizer is terminated when this tolerance is satisfied. This termination criteria is delayed. See 

MSK DPAR MIO DISABLE TERM TIME for details. 

(default = GAMS OptCr) 


MSK DPAR MIO REL ADD CUT LIMITED (real) 

Controls how many cuts the mixed-integer optimizer is allowed to add to the problem. 

Let m be the number of constraints, then the mixed-integer optimizer is allowed to add 

MSK DPAR MIO REL ADD CUT LIMITED*m cuts. 

Range: [0.0,2.0] 

(default = 0.75)


MSK DPAR MIO REL GAP CONST (real) 

This value is used to compute the relative gap for the solution to an integer optimization problem. 


MSK DPAR MIO TOL ABS GAP (real) 

Absolute optimality tolerance employed by the mixed-integer optimizer. 


MSK DPAR MIO TOL ABS RELAX INT (real) 

Absolute relaxation tolerance of the integer constraints. 

That means if the fractional part of a discrete variable is less than the tolerance, the integer restrictions assumed to be 

satisfied. 


MSK DPAR MIO TOL FEAS (real) 

Feasibility tolerance for mixed integer solver. 

Any solution with maximum infeasibility below this value will be considered feasible. 


MSK DPAR MIO TOL REL GAP (real) 

Relative optimality tolerance employed by the mixed-integer optimizer. 


MSK DPAR MIO TOL REL RELAX INT (real) 

Relative relaxation tolerance of the integer constraints. 

That means if the fractional part of a discrete variable is less than the tolerance times the level of that variable, the 

integer restrictions assumed to be satisfied. 


MSK DPAR MIO TOL X (real) 

Absolute solution tolerance used in mixed-integer optimizer. 


MSK IPAR MIO BRANCH DIR (string) 

Controls whether the mixed-integer optimizer is branching up or down by default. 

(default = MSK BRANCH DIR FREE) 

MSK BRANCH DIR DOWN 

The mixed-integer optimizer always chooses the down branch first. 

MSK BRANCH DIR FREE 

The mixed-integer optimizer decides which branch to choose. 

MSK BRANCH DIR UP 

The mixed-integer optimizer always chooses the up branch first. 

MSK IPAR MIO CONSTRUCT SOL (string) 

Controls construction of an initial mixed integer solution from the values of the integer variables 

If set to MSK ON and all integer variables have been given a value for which a feasible mixed integer solution exists, 

then MOSEK generates an initial solution to the mixed integer problem by fixing all integer values and solving the 

remaining problem. 





MSK IPAR MIO CONT SOL (string) 

Controls the meaning of the interior-point and basic solutions in mixed integer problems. 

(default = MSK MIO CONT SOL NONE) 

MSK MIO CONT SOL ITG 

The solution is to the problem with all discrete variables fixed. 

MSK MIO CONT SOL NONE 

The fixed problem is skipped. The primal integer solution is reported. 

MSK MIO CONT SOL ROOT 

The LP solution to the root node problem is reported. 

MSK MIO CONT SOL ITG REL 

Reports the solution to the fixed problem for feasible problems otherwise the root solution. 

MSK IPAR MIO CUT LEVEL ROOT (integer) 

Controls the cut level employed by the mixed-integer optimizer at the root node. 

A negative value means a default value determined by the mixed integer optimizer is used. By adding the appropriate 

values from the following table the employed cut types can be controlled. 

GUB cover +2 

Flow cover +4 

Lifting +8 

Plant location +16 

Disaggregation +32 

Knapsack cover +64 

Lattice +128 

Gomory +256 

Coefficient reduction +512 

GCD +1024 

Obj. integrality +2048 


MSK IPAR MIO CUT LEVEL TREE (integer) 

Controls the cut level employed by the mixed-integer optimizer at the tree. 

See MSK IPAR MIO CUT LEVEL ROOT for an explanation of the parameter values. 


MSK IPAR MIO FEASPUMP LEVEL (integer) 

Controls the feasibility pump heuristic used to construct a good initial feasible solution. 

A value of 0 implies that the feasibility pump heuristic is not used. A value of -1 implies that the mixed integer 

optimizer decides how the feasibility pump heuristic is used. A larger value than 1 implies that the feasibility pump 

is employed more aggressively. Normally a value beyond 3 is not worthwhile. 

Range: [minint,3] 


MSK IPAR MIO HEURISTIC LEVEL (integer) 

Controls the heuristic employed by the MIO to locate an initial good integer feasible solution. 

A value of zero means the heuristic is not used at all. A larger value than 0 means that a gradually more sophisticated 

heuristic is used which is computationally more expensive. A negative value implies that the optimizer chooses the 

heuristic. Normally a value around 3 to 5 should be optimal. 

(default = -1)


MSK IPAR MIO HOTSTART (string) 

Controls whether the integer optimizer is hot-started. 




MSK IPAR MIO KEEP BASIS (string) 

Controls whether the integer presolve keeps bases in memory. 

This speeds on the solution process at cost of bigger memory consumption. 




MSK IPAR MIO LOCAL BRANCH NUMBER (integer) 

Controls the size of the local search space when doing local branching. 


MSK IPAR MIO MAX NUM BRANCHES (integer) 

Maximum number of branches allowed during the branch and bound search. 

A negative value means infinite. 



MSK IPAR MIO MAX NUM RELAXS (integer) 

Maximum number of relaxations allowed during the branch and bound search. 

A negative value means infinite. 



MSK IPAR MIO MAX NUM SOLUTIONS (integer) 

Controls how many feasible solutions the mixed-integer optimizer investigates. 

The mixed integer optimizer can be terminated after a certain number of different feasible solutions have been located. 

If this parameter has the value n and n is strictly positive, then the mixed integer optimizer will be terminated when n 

feasible solutions have been located. 



MSK IPAR MIO NODE OPTIMIZER (string) 

Controls which optimizer is employed at the non-root nodes in the mixed-integer optimizer. 










MSK OPTIMIZER FREE SIMPLEX

















MSK IPAR MIO NODE SELECTION (string) 

Controls the node selection strategy employed by the mixed-integer optimizer. 

(default = MSK MIO NODE SELECTION FREE) 

MSK MIO NODE SELECTION BEST 

The optimizer employs a best bound node selection strategy. 

MSK MIO NODE SELECTION FIRST 

The optimizer employs a depth first node selection strategy. 

MSK MIO NODE SELECTION FREE 

The optimizer decides the node selection strategy. 

MSK MIO NODE SELECTION HYBRID 

The optimizer employs a hybrid strategy. 

MSK MIO NODE SELECTION PSEUDO 

The optimizer employs selects the node based on a pseudo cost estimate. 

MSK MIO NODE SELECTION WORST 

The optimizer employs a worst bound node selection strategy. 

MSK IPAR MIO PRESOLVE AGGREGATE (string) 

Controls whether the presolve used by the MIO tries to aggregate the constraints. 




MSK IPAR MIO PRESOLVE PROBING (string) 

Controls whether the mixed-integer presolve performs probing which can be very time consuming. 




MSK IPAR MIO PRESOLVE USE (string) 

Controls whether presolve is performed by the mixed-integer optimizer. 





MSK IPAR MIO ROOT OPTIMIZER (string) 

Controls which optimizer is employed at the root node in the mixed-integer optimizer. 


























MSK IPAR MIO STRONG BRANCH (integer) 

The value specifies the depth from the root in which strong branching is used. 

A negative value means the optimizer chooses a default value automatically. 


MSK IPAR MIO USE MULTITHREADED OPTIMIZER (string) 

Controls wheter the new multithreaded optimizer should be used for Mixed integer problems. 




MSK DPAR NONCONVEX TOL FEAS (real) 

Feasibility tolerance used by the nonconvex optimizer. 


MSK DPAR NONCONVEX TOL OPT (real) 

Optimality tolerance used by the nonconvex optimizer. 


MSK DPAR QCQO REFORMULATE REL DROP TOL (real) 

Determines when columns are dropped in incomplete cholesky factorization. 

This parameter is applied when doing reformulation of quadratic problems. 

(default = 1e-15)


MSK IPAR ANA SOL BASIS (string) 

Controls whether the basis matrix is analyzed in solution analyzer. 




MSK IPAR ANA SOL PRINT VIOLATED (string) 

Controls whether a list of violated constraints is printed when calling task ANALYZESOLUTION. 

All constraints violated by more than the value set by the parameter MSK DPAR ANA SOL INFEAS TOL will be printed. 




MSK IPAR FEASREPAIR OPTIMIZE (string) 

Controls which type of feasibility analysis is to be performed. 

(default = MSK FEASREPAIR OPTIMIZE NONE) 

MSK FEASREPAIR OPTIMIZE COMBINED 

Minimize with original objective subject to minimal weighted violation of bounds. 

MSK FEASREPAIR OPTIMIZE NONE 

Do not optimize the feasibility repair problem. 

MSK FEASREPAIR OPTIMIZE PENALTY 

Minimize weighted sum of violations. 

MSK IPAR INFEAS GENERIC NAMES (string) 

Controls whether generic names are used when an infeasible subproblem is created. 




MSK IPAR INFEAS PREFER PRIMAL (string) 

Controls which certificate is used if primal- and dual- certificate of infeasibility is available. 

If both certificates of primal and dual infeasibility are supplied then only the primal is used when this option is turned 

on. 




MSK IPAR MT SPINCOUNT (integer) 

Set the number of iterations to spin before sleeping. 

Range: [0,1000000000] 

(default = 0) 

MSK IPAR NONCONVEX MAX ITERATIONS (integer) 

Maximum number of iterations that can be used by the nonconvex optimizer. 

(default = 100000) 

MSK IPAR OPF MAX TERMS PER LINE (integer) 

The maximum number of terms (linear and quadratic) per line when an OPF file is written. 

(default = 5)


MSK IPAR OPF WRITE PARAMETERS (string) 

Write a parameter section in an OPF file. 




MSK IPAR QO SEPARABLE REFORMULATION (string) 

Determine if Quadratic programing problems should be reformulated to separable form. 




MSK IPAR SENSITIVITY OPTIMIZER (string) 

Controls which optimizer is used for optimal partition sensitivity analysis. 

(default = MSK OPTIMIZER FREE SIMPLEX) 

























MSK IPAR SENSITIVITY TYPE (string) 

Controls which type of sensitivity analysis is to be performed. 

(default = MSK SENSITIVITY TYPE BASIS) 

MSK SENSITIVITY TYPE BASIS 

Basis sensitivity analysis is performed. 

MSK SENSITIVITY TYPE OPTIMAL PARTITION 

Optimal partition sensitivity analysis is performed. 

MSK IPAR TIMING LEVEL (integer) 

Controls the a amount of timing performed inside MOSEK. 

(default = 1)


MSK IPAR WRITE DATA FORMAT (string) 

Controls the file format when writing task data to a file. 

(default = MSK DATA FORMAT EXTENSION) 

MSK IPAR WRITE GENERIC NAMES (string) 

Controls whether the generic names or user-defined names are used in the data file. 




MSK IPAR WRITE GENERIC NAMES IO (integer) 

Index origin used in generic names. 

(default = 1) 

READFILE (string) 

Read secondary option file. 

USE BASIS EST (integer) 

Use MOSEK basis estimation in case of an interior solution. 

(default = 0) 

SOLVEFINAL (integer) 

Switch to solve the problem with fixed discrete variables. Overwrites MSK IPAR MIO CONT SOL. 

(default = 1) 

0 Do not solve the fixed problem 

1 Solve the fixed problem and return duals 

FIXOPTFILE (string) 

Name of option file which is read just before solving the fixed problem. 

6 The MOSEK Log File 

The MOSEK log output gives much useful information about the current solver progress and individual phases. 

6.1 Log Using the Interior Point Optimizer 

The following is a MOSEK log output from running the transportation model trnsport.gms from the GAMS Model 

Library: 

Optimizer started. 

Interior-point optimizer started. 

Presolve started. 

Linear dependency checker started. 

Linear dependency checker terminated. 

Eliminator - tries : 0 time : 0.00 

Eliminator - elim’s : 0 

Lin. dep. - tries : 1 time : 0.00 

Lin. dep. - number : 0 

Presolve terminated. Time: 0.00 

Optimizer - threads : 1


Optimizer - solved problem : the primal 

Optimizer - Constraints : 5 

Optimizer - Cones : 0 

Optimizer - Scalar variables : 11 conic : 0 

Optimizer - Semi-definite variables: 0 scalarized : 0 

Factor - setup time : 0.00 dense det. time : 0.00 

Factor - ML order time : 0.00 GP order time : 0.00 

Factor - nonzeros before factor : 11 after factor : 12 

Factor - dense dim. : 0 flops : 1.60e+002 

The first part gives information about the presolve (if used). The main log follows: 

ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME 

0 6.0e+002 1.0e+002 1.0e+002 1.00e+002 1.053000000e+000 0.000000000e+000 1.0e+002 0.00 

1 5.9e+002 1.3e+002 2.6e+002 0.00e+000 3.063797526e+000 2.650041612e+002 2.4e+002 0.01 

2 4.6e+001 1.0e+001 2.0e+001 -9.59e-001 3.650704301e+001 2.594816940e+002 1.9e+001 0.01 

3 3.9e-001 8.7e-002 1.7e-001 -4.39e-001 1.604589379e+002 2.276036163e+002 1.6e-001 0.01 

4 2.7e-002 6.0e-003 1.2e-002 9.62e-001 1.627664502e+002 1.676438787e+002 1.1e-002 0.01 

5 2.2e-003 4.9e-004 9.7e-004 1.04e+000 1.585004810e+002 1.591499235e+002 8.9e-004 0.01 

6 3.1e-004 6.9e-005 1.4e-004 1.01e+000 1.546312243e+002 1.547272945e+002 1.2e-004 0.01 

7 2.9e-005 6.5e-006 1.3e-005 1.01e+000 1.536906429e+002 1.536999628e+002 1.2e-005 0.01 

8 7.6e-008 1.7e-008 3.4e-008 1.00e+000 1.536751995e+002 1.536752387e+002 3.1e-008 0.01 

9 7.6e-012 1.7e-012 3.4e-012 1.00e+000 1.536750000e+002 1.536750000e+002 3.1e-012 0.01 

Basis identification started. 

Primal basis identification phase started. 

ITER TIME 

1 0.00 

Primal basis identification phase terminated. Time: 0.00 

Dual basis identification phase started. 

ITER TIME 

0 0.00 

Dual basis identification phase terminated. Time: 0.00 

Basis identification terminated. Time: 0.00 

Interior-point optimizer terminated. Time: 0.01. 

Optimizer terminated. Time: 0.03 

Interior-point solution summary 

Problem status : PRIMAL_AND_DUAL_FEASIBLE 

Solution status : OPTIMAL 

Primal. obj: 1.5367500002e+002 Viol. con: 3e-010 var: 0e+000 

Dual. obj: 1.5367500002e+002 Viol. con: 0e+000 var: 6e-011 

Basic solution summary 



Primal. obj: 1.5367500000e+002 Viol. con: 0e+000 var: 0e+000 

Dual. obj: 1.5367500002e+002 Viol. con: 0e+000 var: 5e-011 

Return code - 0 [MSK_RES_OK]: No error occurred. 

The last section gives details about the model and solver status, primal and dual feasibilities, as well as solver resource 

times. Furthermore, the log gives information about the basis identification phase. Some of this information is listed in the 

GAMS solve summary in the model listing (.LST) file as well. 

The fields in the main MOSEK log output are:


Field Description 

ITE The number of the current iteration. 

PFEAS Primal feasibility. 

DFEAS Dual feasibility. 

GFEAS The numbers in this column should converge monotonically toward to zero but may stall at low 

level due to rounding errors. 

PRSTATUS This number converge to 1 if the problem has an optimal solution whereas it converge to -1 if that 

is not the case. 

POBJ Current objective function value of primal problem. 

DOBJ Current objective function value of dual problem. 

MU Relative complementary gap. 

TIME Current elapsed resource time in seconds. 

6.2 Log Using the Simplex Optimizer 

Below is a log output running the model trnsport.gms from the GAMS model library using the MOSEK simplex optimizer. 

Reading parameter(s) from "c:\tmp\mosek.opt" 

>> MSK_IPAR_OPTIMIZER MSK_OPTIMIZER_DUAL_SIMPLEX 

Finished reading from "c:\tmp\mosek.opt" 


Simplex optimizer started. 









Dual simplex optimizer started. 

Dual simplex optimizer setup started. 

Dual simplex optimizer setup terminated. 

Optimizer - solved problem : the primal 

Optimizer - Constraints : 5 

Optimizer - Scalar variables : 6 conic : 0 

Optimizer - hotstart : no 

ITER DEGITER(%) PFEAS DFEAS POBJ DOBJ TIME TOTTIME 

0 0.00 NA 0.00e+000 NA 0.0000000000e+000 0.00 0.02 

4 20.00 NA 0.00e+000 NA 1.5367501014e+002 0.00 0.02 

Dual simplex optimizer terminated. 

Simplex optimizer terminated. Time: 0.00. 






Dual. obj: 1.5367500000e+002 Viol. con: 0e+000 var: 0e+000 


The fields in the main MOSEK log output are:



ITER Current number of iterations. 

DEGITER% Current percentage of degenerate iterations. 

DPFEAS Current primal and dual infeasibility. 

D/POBJ Current dual abd primal objective 


TOTTIME Total elapsed resource time in seconds. 

6.3 Log Using the Mixed Integer Optimizer 

Below is a log output running the model cube.gms from the GAMS model library using the MOSEK mixed-integer optimizer. 


Mixed integer optimizer started. 

BRANCHES RELAXS ACT_NDS BEST_INT_OBJ BEST_RELAX_OBJ REL_GAP(%) TIME 

0 0 0 1.8000000000e+001 NA NA 0.0 

0 1 0 1.8000000000e+001 0.0000000000e+000 100.00 0.1 

0 1 0 7.0000000000e+000 0.0000000000e+000 100.00 0.1 

0 10 0 6.0000000000e+000 0.0000000000e+000 100.00 0.1 

0 15 0 4.0000000000e+000 0.0000000000e+000 100.00 0.1 

355 1003 42 4.0000000000e+000 0.0000000000e+000 100.00 2.1 

392 1081 45 4.0000000000e+000 5.0000000000e-001 87.50 2.2 

A near optimal solution satisfying the absolute gap tolerance of 3.90e+000 has been located. 

The fields in the main MOSEK log output are: 


BRANCHES Current number of branches in tree. 

RELAXS Current number of nodes in branch and bound tree. 

ACT NDS Current number of active nodes. 

BEST INT OBJ. Current best integer solution 

BEST RELAX OBJ Current best relaxed solution. 

REL GAP(%) Relative gap between current BEST INT OBJ. and BEST RELAX OBJ. 


The log then gives information about solving the model with discrete variables fixed in order to determine marginals and 

some summary of the branch and cut algorithm. We also get information about crossover to determine a basic solution, and 

finally MOSEK provides information about using the Simplex Method to determine an optimal basic solution. 

Interior-point optimizer started. 









Interior-point optimizer terminated. Time: 0.00. 

Objective of best integer solution : 4.000000000000e+000 

Construct solution objective : Not employed 

User objective cut value : Not employed


Number of cuts generated : 24 

Number of Gomory cuts : 9 

Number of contra cuts : 2 

Number of branches : 392 

Number of relaxations solved : 1081 

Number of interior point iterations: 0 

Number of simplex iterations : 17710 

Time spend presolving the root : 0.02 

Time spend in the heuristic : 0.56 

Time spend in the sub optimizers : 1.59 

Time spend optimizing the root : 0.00 

Mixed integer optimizer terminated. Time: 2.23 


Interior-point solution summary 

Problem status : PRIMAL_FEASIBLE 

Solution status : PRIMAL_FEASIBLE 

Primal. obj: 4.0000000000e+000 Viol. con: 0e+000 var: 0e+000 itg: 0e+000 

Dual. obj: -6.0000000000e+001 Viol. con: 0e+000 var: 0e+000 



Solution status : PRIMAL_FEASIBLE 


Dual. obj: -6.0000000000e+001 Viol. con: 0e+000 var: 0e+000 

Integer solution solution summary 


Solution status : NEAR_INTEGER_OPTIMAL 


Return code - 10004 [MSK_RES_TRM_MIO_NEAR_ABS_GAP]: The mixed-integer optimizer terminated 

because the near optimal absolute gap tolerance was satisfied. 

MIP Solution: 4.000000 (0 iterations, 0 nodes) 

Best possible: 0.500000 

Absolute gap: 3.500000 

Relative gap: 3.500000 

6.4 Log Using the Conic Mixed Integer Optimizer 

Below is a log output running the model cube.gms from the GAMS model library using the MOSEK conic mixed-integer 

optimizer. The columns of the iteration log are identical to the regular mixed-integer optimizer. 

Reading parameter(s) from "c:\tmp\mosek.opt" 

>> MSK_IPAR_OPTIMIZER MSK_OPTIMIZER_MIXED_INT_CONIC 

Finished reading from "c:\tmp\mosek.opt" 


Mixed integer optimizer started. 

Optimizer - threads : 1 

BRANCHES RELAXS ACT_NDS BEST_INT_OBJ BEST_RELAX_OBJ REL_GAP(%) TIME 

0 1 0 NA 0.0000000000e+000 NA 0.0 

0 2 0 NA 0.0000000000e+000 NA 0.1


0 2 2 NA 0.0000000000e+000 NA 0.1 

0 2 2 1.2000000000e+001 0.0000000000e+000 100.00 0.1 

0 2 2 7.0000000000e+000 0.0000000000e+000 100.00 0.1 

0 2 2 7.0000000000e+000 0.0000000000e+000 100.00 0.1 

1 3 2 7.0000000000e+000 0.0000000000e+000 100.00 0.1 

3 5 4 7.0000000000e+000 0.0000000000e+000 100.00 0.1 

7 9 8 7.0000000000e+000 0.0000000000e+000 100.00 0.1 

15 17 16 7.0000000000e+000 0.0000000000e+000 100.00 0.1 

31 33 28 6.0000000000e+000 0.0000000000e+000 100.00 0.1 

59 61 48 6.0000000000e+000 0.0000000000e+000 100.00 0.1 

89 91 74 6.0000000000e+000 0.0000000000e+000 100.00 0.1 

121 123 90 6.0000000000e+000 0.0000000000e+000 100.00 0.1 

151 153 112 5.0000000000e+000 0.0000000000e+000 100.00 0.2 

183 184 134 5.0000000000e+000 3.3333333333e-001 93.33 0.2 

215 216 160 5.0000000000e+000 3.3333333333e-001 93.33 0.2 

247 248 184 5.0000000000e+000 5.0000000000e-001 90.00 0.2 

280 280 209 5.0000000000e+000 5.0000000000e-001 90.00 0.2 

313 312 232 5.0000000000e+000 5.0000000000e-001 90.00 0.2 

346 345 251 5.0000000000e+000 5.8823529412e-001 88.24 0.3 

378 377 279 5.0000000000e+000 7.1428571429e-001 85.71 0.3 

410 409 299 5.0000000000e+000 1.0000000000e+000 80.00 0.3 

443 441 318 5.0000000000e+000 1.0000000000e+000 80.00 0.3 

477 475 340 5.0000000000e+000 1.0000000000e+000 80.00 0.3 

510 506 365 4.0000000000e+000 1.0000000000e+000 75.00 0.3 

A near optimal solution satisfying the absolute gap tolerance of 3.90e+000 has been located. 

Objective of best integer solution : 4.000000000000e+000 

Construct solution objective : Not employed 

User objective cut value : Not employed 

Number of cuts generated : 3 

Number of branches : 510 

Number of relaxations solved : 506 

Number of interior point iterations: 5 

Number of simplex iterations : 5581 

Time spend presolving the root : 0.00 

Time spend in the heuristic : 0.00 

Time spend in the sub optimizers : 0.00 

Time spend optimizing the root : 0.03 

Mixed integer optimizer terminated. Time: 0.34 

The mixed-integer optimizer terminated because the near optimal absolute gap tolerance 

was satisfied. (*10004*) 


Integer solution solution summary 


Solution status : NEAR_INTEGER_OPTIMAL 


Return code - 10004 [MSK_RES_TRM_MIO_NEAR_ABS_GAP]: The mixed-integer optimizer terminated 

because the near optimal absolute gap tolerance was satisfied. 

The log then gives information about solving the model with discrete variables fixed in order to determine marginals. For 

this secltion, the fixed problem is solved as a regular LP with warm start information. So the log looks identical to the 

MOSEK simplex optimizer for linear programs:


Solving fixed problem... 

MOSEK 24.1.0 r39955 ALFA Released 27Apr13 VS8 x86/MS Windows 

M O S E K version 7.0.0.52(BETA) (Build date: 2013-4-15 08:20:40) 

Copyright (C) MOSEK ApS, Fruebjergvej 3, Box 16 

DK-2100 Copenhagen, Denmark 

http://www.mosek.com 


Simplex optimizer started. 









Simplex optimizer terminated. Time: 0.00. 






Dual. obj: 4.0000000000e+000 Viol. con: 0e+000 var: 0e+000 


The log ends with the regular summary for MIP problems printing objective values, bounds and gaps: 

MIP Solution: 4.000000 (0 iterations, 0 nodes) 

Final Solve: 4.000000 (0 iterations) 

Best possible: 4.000000 

Absolute gap: 0.000000 

Relative gap: 0.000000

MOSEK - GAMS

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