The MOSEK Python optimizer API manual Version 7.0 (Revision 141)

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iv CONTENTS 5.3.1 Example: Conic quadratic optimization . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 Semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.4.1 Example: Semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.5 Quadratic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.5.1 Example: Quadratic objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.5.2 Example: Quadratic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.6 The solution summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.7 Integer optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.7.1 Example: Mixed integer linear optimization . . . . . . . . . . . . . . . . . . . . . 54 5.7.2 Specifying an initial solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.7.3 Example: Specifying an integer solution . . . . . . . . . . . . . . . . . . . . . . . 57 5.8 The solution summary for mixed integer problems . . . . . . . . . . . . . . . . . . . . 59 5.9 Response handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.10 Problem modification and reoptimization . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.10.1 Example: Production planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.10.2 Changing the A matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.10.3 Appending variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.10.4 Reoptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.10.5 Appending constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.11 Solution analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.11.1 Retrieving solution quality information with the API . . . . . . . . . . . . . . . . 67 5.12 Efficiency considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.12.1 API overhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.13 Conventions employed in the API . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.13.1 Naming conventions for arguments . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.13.2 Vector formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.13.3 Matrix formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.13.4 Array objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.13.5 Typical problems using the Python API . . . . . . . . . . . . . . . . . . . . . . . 75 5.14 The license system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.14.1 Waiting for a free license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6 Nonlinear API tutorial 77 6.1 Separable convex (SCopt) interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.1.1 Adding separable terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.1.2 Example: Simple separable problem . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.1.3 Ensuring convexity and differentiability . . . . . . . . . . . . . . . . . . . . . . . 82 6.1.4 SCopt Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7 Advanced API tutorial 85 7.1 The progress call-back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.1.1 Source code example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 Solving linear systems involving the basis matrix . . . . . . . . . . . . . . . . . . . . . 88 7.2.1 Identifying the basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.2.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.2.3 Solving arbitrary linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
CONTENTS v 8 A case study 97 8.1 Portfolio optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.1.2 A basic portfolio optimization model . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.1.3 The efficient frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8.1.4 Improving the computational efficiency . . . . . . . . . . . . . . . . . . . . . . . 109 8.1.5 Slippage cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 9 Usage guidelines 119 9.1 Verifying the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.1.1 Verifying primal feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.1.2 Verifying optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.2 Turn on logging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.3 Writing task data to a file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9.4 Important API limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9.4.1 Thread safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 9.5 Bug reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10 Problem formulation and solutions 123 10.1 Linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.1.1 Duality for linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.1.2 Infeasibility for linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.2 Conic quadratic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 10.2.1 Duality for conic quadratic optimization . . . . . . . . . . . . . . . . . . . . . . . 128 10.2.2 Infeasibility for conic quadratic optimization . . . . . . . . . . . . . . . . . . . . 129 10.3 Semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 10.3.1 Duality for semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . . . . 130 10.3.2 Infeasibility for semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . 131 10.4 Quadratic and quadratically constrained optimization . . . . . . . . . . . . . . . . . . 132 10.4.1 Duality for quadratic and quadratically constrained optimization . . . . . . . . . 132 10.4.2 Infeasibility for quadratic and quadratically constrained optimization . . . . . . . 133 11 The optimizers for continuous problems 135 11.1 How an optimizer works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 11.1.1 Presolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.1.2 Dualizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11.1.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 11.1.4 Using multiple threads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 11.2 Linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 11.2.1 Optimizer selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 11.2.2 The interior-point optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 11.2.3 The simplex based optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 11.2.4 The interior-point or the simplex optimizer? . . . . . . . . . . . . . . . . . . . . . 145 11.2.5 The primal or the dual simplex variant? . . . . . . . . . . . . . . . . . . . . . . . 145 11.3 Linear network optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 11.3.1 Network flow problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 11.4 Conic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
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5.2. LINEAR OPTIMIZATION 33 51 mose
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5.3. CONIC QUADRATIC OPTIMIZATION 3
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5.4. SEMIDEFINITE OPTIMIZATION 39 m
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5.4. SEMIDEFINITE OPTIMIZATION 41 4
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5.4. SEMIDEFINITE OPTIMIZATION 43 1
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5.5. QUADRATIC OPTIMIZATION 45 with
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5.5. QUADRATIC OPTIMIZATION 47 68 #
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5.5. QUADRATIC OPTIMIZATION 49 5.5.
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5.5. QUADRATIC OPTIMIZATION 51 81 a
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5.6. THE SOLUTION SUMMARY 53 • Th
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5.7. INTEGER OPTIMIZATION 55 21 # f
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5.7. INTEGER OPTIMIZATION 57 137 sy
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5.8. THE SOLUTION SUMMARY FOR MIXED
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5.9. RESPONSE HANDLING 61 Note that
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5.10. PROBLEM MODIFICATION AND REOP
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5.10. PROBLEM MODIFICATION AND REOP
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5.11. SOLUTION ANALYSIS 67 151 # Pu
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5.13. CONVENTIONS EMPLOYED IN THE A
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Chapter 6 Nonlinear API tutorial Th
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6.1. SEPARABLE CONVEX (SCOPT) INTER
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Chapter 7 Advanced API tutorial Thi
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7.1. THE PROGRESS CALL-BACK 87 71 p
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7.2. SOLVING LINEAR SYSTEMS INVOLVI
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Chapter 8 A case study 8.1 Portfoli
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8.1. PORTFOLIO OPTIMIZATION 99 e T
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8.1. PORTFOLIO OPTIMIZATION 101 is
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8.1. PORTFOLIO OPTIMIZATION 103 15
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8.1. PORTFOLIO OPTIMIZATION 107 8.1
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8.1. PORTFOLIO OPTIMIZATION 111 z j
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8.1. PORTFOLIO OPTIMIZATION 113 Var
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Chapter 9 Usage guidelines The purp
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9.3. WRITING TASK DATA TO A FILE 12
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Chapter 10 Problem formulation and
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10.1. LINEAR OPTIMIZATION 125 be a
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10.2. CONIC QUADRATIC OPTIMIZATION
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10.2. CONIC QUADRATIC OPTIMIZATION
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10.3. SEMIDEFINITE OPTIMIZATION 131
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10.4. QUADRATIC AND QUADRATICALLY C
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Chapter 11 The optimizers for conti
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11.1. HOW AN OPTIMIZER WORKS 137 11
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11.2. LINEAR OPTIMIZATION 139 11.2.
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11.2. LINEAR OPTIMIZATION 141 Whene
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11.2. LINEAR OPTIMIZATION 143 11.2.
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11.2. LINEAR OPTIMIZATION 145 • R
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11.5. NONLINEAR CONVEX OPTIMIZATION
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11.6. SOLVING PROBLEMS IN PARALLEL
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Chapter 12 The optimizers for mixed
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12.3. THE MIXED-INTEGER CONIC OPTIM
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12.5. TERMINATION CRITERION 159 •
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12.7. UNDERSTANDING SOLUTION QUALIT
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Chapter 13 The analyzers 13.1 The p
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13.1. THE PROBLEM ANALYZER 165 Cons
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13.2. ANALYZING INFEASIBLE PROBLEMS
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Chapter 14 Primal feasibility repai
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14.2. AUTOMATIC REPAIR 179 One way
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14.3. FEASIBILITY REPAIR IN MOSEK 1
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14.3. FEASIBILITY REPAIR IN MOSEK 1
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Chapter 15 Sensitivity analysis 15.
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15.4. SENSITIVITY ANALYSIS FOR LINE
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15.5. SENSITIVITY ANALYSIS FROM THE
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15.6. SENSITIVITY ANALYSIS WITH THE
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15.6. SENSITIVITY ANALYSIS WITH THE
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Appendix A API reference This chapt
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201 • Task.relaxprimal Obtain inf
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A.1. EXCEPTIONS 203 • Task.putvar
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A.2. CLASS TASK 205 Arguments which
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A.2. CLASS TASK 207 See also • Ro
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A.2. CLASS TASK 209 Description: Ap
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A.2. CLASS TASK 211 Description: If
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A.2. CLASS TASK 213 A.2.16 Task.com
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A.2. CLASS TASK 215 subj : int[] In
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A.2. CLASS TASK 217 firsti : int In
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A.2. CLASS TASK 219 A.2.27 Task.get
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A.2. CLASS TASK 221 valijkl : Descr
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A.2. CLASS TASK 225 idx : long Inde
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A.2. CLASS TASK 229 i : int Index o
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A.2. CLASS TASK 233 conetype : cone
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A.2. CLASS TASK 235 Description: Ob
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A.2. CLASS TASK 237 sub : int[] Ind
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A.2. CLASS TASK 241 Computes the vi
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A.2. CLASS TASK 253 k : int Index o
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A.2. CLASS TASK 255 A.2.103 Task.ge
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A.2. CLASS TASK 257 normalize : int
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A.2. CLASS TASK 259 Description: Le
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A.2. CLASS TASK 261 max(l x j τ
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A.2. CLASS TASK 265 last : int Last
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A.2. CLASS TASK 269 Arguments snx :
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A.2. CLASS TASK 271 slc : double[]
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A.2. CLASS TASK 273 Arguments accmo
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A.2. CLASS TASK 277 last : int Valu
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A.2. CLASS TASK 279 subi : int[] Ro
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A.2. CLASS TASK 283 Arguments taskn
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A.2. CLASS TASK 287 vartype : varia
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A.2. CLASS TASK 289 whichsol : solt
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A.2. CLASS TASK 293 A.2.162 Task.is
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A.2. CLASS TASK 295 whichstream : s
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A.2. CLASS TASK 297 A.2.170 Task.pr
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A.2. CLASS TASK 299 markj : mark Th
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A.2. CLASS TASK 301 Prints a part o
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A.2. CLASS TASK 303 A.2.175 Task.pu
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A.2. CLASS TASK 305 j : int Index o
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A.2. CLASS TASK 309 Description: Th
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A.2. CLASS TASK 313 Changes the bou
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A.2. CLASS TASK 315 Modifies one li
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A.2. CLASS TASK 317 bkc : boundkey
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A.2. CLASS TASK 319 Arguments j : i
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A.2. CLASS TASK 321 parvalue : int
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A.2. CLASS TASK 325 Sets a string p
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A.2. CLASS TASK 327 Description: Re
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A.2. CLASS TASK 329 qosubj : int[]
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A.2. CLASS TASK 331 Sets the status
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A.2. CLASS TASK 335 Arguments sux :
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A.2. CLASS TASK 337 y : double[] Ve
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A.2. CLASS TASK 341 last : int Last
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A.2. CLASS TASK 343 See also • Ta
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A.2. CLASS TASK 347 Description: Se
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A.2. CLASS TASK 349 Sets a slice of
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A.2. CLASS TASK 351 A.2.254 Task.re
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A.2. CLASS TASK 353 wux : double[]
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A.2. CLASS TASK 355 See also • Ta
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A.2. CLASS TASK 357 isdef : int Is
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A.2. CLASS TASK 359 A.2.269 Task.st
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A.2. CLASS TASK 361 Description: Wr
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A.3. CLASS ENV 363 A.2.278 Task.wri
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A.3. CLASS ENV 365 Arguments code :
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A.3. CLASS ENV 367 Arguments keepdl
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A.4. CALLBACK FUNCTIONS AND RELATED
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A.6. ALL FUNCTIONS BY NAME 371 Task
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Appendix B Parameters Parameters gr
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391 • iparam.log file. If turned
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393 • dparam.intpnt nl tol rel st
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395 • iparam.mio construct sol. C
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397 • dparam.intpnt nl tol mu red
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399 • iparam.mio mode. Turns on/o
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401 • iparam.sim solve form. Cont
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B.1. DPARAM: DOUBLE PARAMETERS 403
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B.2. IPARAM: INTEGER PARAMETERS 427
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B.3. SPARAM: STRING PARAMETER TYPES
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Appendix C Response codes Response
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517 rescode.err con q not psd The q
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519 rescode.err global inv conic pr
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521 rescode.err inv markj Invalid v
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523 rescode.err invalid format type
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525 rescode.err license no server s
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527 rescode.err mio no optimizer No
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529 rescode.err name max len A name
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531 rescode.err numconlim Maximum n
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533 rescode.err qcon upper triangle
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535 rescode.err sym mat invalid col
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537 rescode.err user nlo func The u
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539 rescode.wrn ana almost int boun
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541 rescode.wrn mio infeasible fina
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543 rescode.wrn write discarded cfi
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Appendix D API constants D.1 Constr
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D.5. PROGRESS CALL-BACK CODES 547 c
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D.6. TYPES OF CONVEXITY CHECKS. 555
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D.10. DOUBLE INFORMATION ITEMS 557
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D.10. DOUBLE INFORMATION ITEMS 559
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D.11. FEASIBILITY REPAIR TYPES 561
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D.13. INTEGER INFORMATION ITEMS. 56
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D.16. INPUT/OUTPUT MODES 569 intpnt
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D.20. CONTINUOUS MIXED-INTEGER SOLU
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D.26. OBJECTIVE SENSE TYPES 573 D.2
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D.31. PRESOLVE METHOD. 575 paramete
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D.35. RESPONSE CODE TYPE 577 D.35 R
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D.42. PROBLEM REFORMULATION. 579 si
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D.46. SOLUTION TYPES 581 solsta.dua
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D.50. STREAM TYPES 583 startpointty
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Appendix E Troubleshooting When cre
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Appendix F Mosek file formats MOSEK
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F.1. THE MPS FILE FORMAT 589 Fields
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F.1. THE MPS FILE FORMAT 591 must b
Page 615 and 616:
F.1. THE MPS FILE FORMAT 593 Constr
Page 617 and 618:
F.1. THE MPS FILE FORMAT 595 v 1 is
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F.1. THE MPS FILE FORMAT 597 Please
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F.2. THE LP FILE FORMAT 599 minimiz
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F.2. THE LP FILE FORMAT 601 st defi
Page 625 and 626:
F.2. THE LP FILE FORMAT 603 bounds
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F.3. THE OPF FORMAT 605 iparam.writ
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F.3. THE OPF FORMAT 607 [tag "value
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F.3. THE OPF FORMAT 609 Note that a
Page 633 and 634:
F.3. THE OPF FORMAT 611 F.3.2.3 Nam
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F.3. THE OPF FORMAT 613 [bounds] [b
Page 637 and 638:
F.4. THE TASK FORMAT 615 This can b
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F.7. THE SOLUTION FILE FORMAT 617 c
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Appendix G Problem analyzer example
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G.2. ARKI001 621 2 476 45.42 48.19
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G.4. PROBLEM WITH BOTH LINEAR AND C
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Bibliography [1] Chvátal, V.. Line
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Index analyzenames (Task method), 2
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INDEX 629 getpcni (Task method), 25
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INDEX 631 putbarablocktriplet (Task
Page 655 and 656:
INDEX 633 rescode.err inv numj, 521
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INDEX 635 rescode.err sen bound inv
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