The MOSEK command line tool Version 7.0 (Revision 141)
The MOSEK command line tool. Version 7.0 ... - Documentation The MOSEK command line tool. Version 7.0 ... - Documentation
38 CHAPTER 5. THE OPTIMIZERS FOR CONTINUOUS PROBLEMS A x∗ τ ∗ = b, A T y∗ τ ∗ + s∗ = c, τ∗ − c T x∗ y∗ + bT τ ∗ τ ∗ = 0, x ∗ , s ∗ , τ ∗ , κ ∗ ≥ 0. This shows that x∗ τ is a primal optimal solution and ( y∗ ∗ as the optimal interior-point solution since is a primal-dual optimal solution. On other hand, if κ ∗ > 0 then (x, y, s) = τ , s∗ ∗ τ∗ ( ) x ∗ τ ∗ , y∗ τ ∗ , s∗ τ∗ ) is a dual optimal solution; this is reported This implies that at least one of Ax ∗ = 0, A T y ∗ + s ∗ = 0, − c T x ∗ + b T y ∗ = κ ∗ , x ∗ , s ∗ , τ ∗ , κ ∗ ≥ 0. or − c T x ∗ > 0 (5.3) b T y ∗ > 0 (5.4) is satisfied. If (5.3) is satisfied then x ∗ is a certificate of dual infeasibility, whereas if (5.4) is satisfied then y ∗ is a certificate of dual infeasibility. In summary, by computing an appropriate solution to the homogeneous model, all information required for a solution to the original problem is obtained. A solution to the homogeneous model can be computed using a primal-dual interior-point algorithm [5]. 5.2.2.1 Interior-point termination criterion For efficiency reasons it is not practical to solve the homogeneous model exactly. Hence, an exact optimal solution or an exact infeasibility certificate cannot be computed and a reasonable termination criterion has to be employed. In every iteration, k, of the interior-point algorithm a trial solution to homogeneous model is generated where (x k , y k , s k , τ k , κ k )
5.2. LINEAR OPTIMIZATION 39 Whenever the trial solution satisfies the criterion yk AT ∥ ( ∣ ∣ (x k ) T s k ∣∣∣ min (τ k ) 2 , c T x k τ k − bT y k ) ∣∣∣ τ k the interior-point optimizer is terminated and x k , s k , τ k , κ k > 0. ∥ ∥∥∥ ∥ Axk τ k − b ∞ ≤ ɛ p (1 + ‖b‖ ∞ ), ∥ τ k + sk ∥∥∥ τ k − c ∞ ≤ ɛ d (1 + ‖c‖ ∞ ), and ( ≤ ɛ g max 1, min(∣ ∣c T x k∣ ∣ , ∣ ∣b T y k∣ ) ∣ ) τ k , (5.5) (x k , y k , s k ) τ k is reported as the primal-dual optimal solution. The interpretation of (5.5) is that the optimizer is terminated if • xk τ k • is approximately primal feasible, ( ) y k , sk τ k τ k is approximately dual feasible, and • the duality gap is almost zero. On the other hand, if the trial solution satisfies −ɛ i c T x k > ‖c‖ ∞ ∥ ∥ Ax k ∞ max(1, ‖b‖ ∞ ) then the problem is declared dual infeasible and x k is reported as a certificate of dual infeasibility. The motivation for this stopping criterion is as follows: First assume that ∥ ∥ Ax k ∥ ∥ ∞ = 0 ; then x k is an exact certificate of dual infeasibility. Next assume that this is not the case, i.e. and define ∥ Ax k ∥ ∥ ∞ > 0, It is easy to verify that ¯x := ɛ i max(1, ‖b‖ ∞ ) ‖Ax k ‖ ∞ ‖c‖ ∞ x k . ‖A¯x‖ ∞ = ɛ i max(1, ‖b‖ ∞ ) ‖c‖ ∞ and − c T ¯x > 1, which shows ¯x is an approximate certificate of dual infeasibility where ɛ i controls the quality of the approximation. A smaller value means a better approximation.
- Page 3 and 4: Contents 1 Changes and new features
- Page 5 and 6: CONTENTS v 6.5 Termination criterio
- Page 7 and 8: CONTENTS vii 9.1.56 MSK DPAR NONCON
- Page 9 and 10: CONTENTS ix 9.2.79 MSK IPAR MIO FEA
- Page 11 and 12: CONTENTS xi 9.2.171 MSK IPAR SOL RE
- Page 13 and 14: CONTENTS xiii 11.29 Ordering strate
- Page 15 and 16: CONTENTS xv 18.2 arki001 . . . . .
- Page 17 and 18: Contact information Phone +45 3917
- Page 19 and 20: License agreement Before using the
- Page 21 and 22: Chapter 1 Changes and new features
- Page 23 and 24: 1.4. OPTIMIZATION TOOLBOX FOR MATLA
- Page 25 and 26: Chapter 2 What is MOSEK MOSEK is a
- Page 27 and 28: Chapter 3 MOSEK and AMPL AMPL is a
- Page 29 and 30: 3.6. CONSTRAINT AND VARIABLE NAMES
- Page 31 and 32: 3.8. HOT-START 15 Linear dependency
- Page 33 and 34: 3.10. SENSITIVITY ANALYSIS 17 • .
- Page 35 and 36: Chapter 4 Problem formulation and s
- Page 37 and 38: 4.1. LINEAR OPTIMIZATION 21 be a pr
- Page 39 and 40: 4.2. CONIC QUADRATIC OPTIMIZATION 2
- Page 41 and 42: 4.2. CONIC QUADRATIC OPTIMIZATION 2
- Page 43 and 44: 4.3. SEMIDEFINITE OPTIMIZATION 27 4
- Page 45 and 46: 4.5. GENERAL CONVEX OPTIMIZATION 29
- Page 47 and 48: 4.5. GENERAL CONVEX OPTIMIZATION 31
- Page 49 and 50: Chapter 5 The optimizers for contin
- Page 51 and 52: 5.1. HOW AN OPTIMIZER WORKS 35 5.1.
- Page 53: 5.2. LINEAR OPTIMIZATION 37 5.2.2 T
- Page 57 and 58: 5.2. LINEAR OPTIMIZATION 41 5.2.2.3
- Page 59 and 60: 5.2. LINEAR OPTIMIZATION 43 • Rai
- Page 61 and 62: 5.5. NONLINEAR CONVEX OPTIMIZATION
- Page 63 and 64: 5.6. SOLVING PROBLEMS IN PARALLEL 4
- Page 65 and 66: Chapter 6 The optimizers for mixed-
- Page 67 and 68: 6.3. THE MIXED-INTEGER CONIC OPTIMI
- Page 69 and 70: 6.5. TERMINATION CRITERION 53 The f
- Page 71 and 72: 6.7. UNDERSTANDING SOLUTION QUALITY
- Page 73 and 74: Chapter 7 The analyzers 7.1 The pro
- Page 75 and 76: 7.1. THE PROBLEM ANALYZER 59 Constr
- Page 77 and 78: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 79 and 80: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 81 and 82: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 83 and 84: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 85 and 86: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 87 and 88: Chapter 8 Sensitivity analysis 8.1
- Page 89 and 90: 8.4. SENSITIVITY ANALYSIS FOR LINEA
- Page 91 and 92: 8.5. SENSITIVITY ANALYSIS WITH THE
- Page 93 and 94: 8.5. SENSITIVITY ANALYSIS WITH THE
- Page 95 and 96: Chapter 9 Parameters Parameters gro
- Page 97 and 98: 81 • MSK SPAR ITR SOL FILE NAME.
- Page 99 and 100: 83 • MSK DPAR INTPNT NL TOL REL G
- Page 101 and 102: 85 • MSK IPAR LOG MIO. Controls t
- Page 103 and 104: 87 • MSK DPAR INTPNT NL MERIT BAL
5.2. LINEAR OPTIMIZATION 39<br />
Whenever the trial solution satisfies the criterion<br />
yk<br />
AT<br />
∥<br />
( ∣ ∣ (x k ) T s k ∣∣∣<br />
min<br />
(τ k ) 2 , c T x k<br />
τ k − bT y k ) ∣∣∣<br />
τ k<br />
the interior-point optimizer is terminated and<br />
x k , s k , τ k , κ k > 0.<br />
∥ ∥∥∥ ∥ Axk τ k − b ∞ ≤ ɛ p (1 + ‖b‖ ∞ ),<br />
∥<br />
τ k + sk ∥∥∥<br />
τ k − c ∞ ≤ ɛ d (1 + ‖c‖ ∞ ), and<br />
(<br />
≤<br />
ɛ g max<br />
1, min(∣ ∣c T x k∣ ∣ , ∣ ∣b T y k∣ )<br />
∣ )<br />
τ k ,<br />
(5.5)<br />
(x k , y k , s k )<br />
τ k<br />
is reported as the primal-dual optimal solution. <strong>The</strong> interpretation of (5.5) is that the optimizer is<br />
terminated if<br />
• xk<br />
τ k<br />
•<br />
is approximately primal feasible,<br />
( )<br />
y<br />
k<br />
, sk<br />
τ k τ k<br />
is approximately dual feasible, and<br />
• the duality gap is almost zero.<br />
On the other hand, if the trial solution satisfies<br />
−ɛ i c T x k ><br />
‖c‖ ∞<br />
∥ ∥ Ax<br />
k ∞<br />
max(1, ‖b‖ ∞ )<br />
then the problem is declared dual infeasible and x k is reported as a certificate of dual infeasibility.<br />
<strong>The</strong> motivation for this stopping criterion is as follows: First assume that ∥ ∥ Ax<br />
k ∥ ∥ ∞ = 0 ; then x k is<br />
an exact certificate of dual infeasibility. Next assume that this is not the case, i.e.<br />
and define<br />
∥ Ax<br />
k ∥ ∥ ∞ > 0,<br />
It is easy to verify that<br />
¯x := ɛ i<br />
max(1, ‖b‖ ∞ )<br />
‖Ax k ‖ ∞ ‖c‖ ∞<br />
x k .<br />
‖A¯x‖ ∞ = ɛ i<br />
max(1, ‖b‖ ∞ )<br />
‖c‖ ∞<br />
and − c T ¯x > 1,<br />
which shows ¯x is an approximate certificate of dual infeasibility where ɛ i controls the quality of the<br />
approximation. A smaller value means a better approximation.