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

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vi CONTENTS 11.4.1 The interior-point optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 11.5 Nonlinear convex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.5.1 The interior-point optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.6 Solving problems in parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 11.6.1 Thread safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 11.6.2 The parallelized interior-point optimizer . . . . . . . . . . . . . . . . . . . . . . . 149 11.6.3 The concurrent optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 11.6.4 A more flexible concurrent optimizer . . . . . . . . . . . . . . . . . . . . . . . . . 151 12 The optimizers for mixed-integer problems 155 12.1 Some concepts and facts related to mixed-integer optimization . . . . . . . . . . . . . . 155 12.2 The mixed-integer optimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 12.3 The mixed-integer conic optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.3.1 Presolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.3.2 Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.3.3 The optimization phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 12.3.4 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 12.4 The mixed-integer optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 12.4.1 Presolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 12.4.2 Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 12.4.3 The optimization phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12.5 Termination criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12.5.1 Relaxed termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12.5.2 Important parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 12.6 How to speed up the solution process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 12.7 Understanding solution quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 13 The analyzers 163 13.1 The problem analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 13.1.1 General characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 13.1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 13.1.3 Linear constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 13.1.4 Constraint and variable bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 13.1.5 Quadratic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 13.1.6 Conic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 13.2 Analyzing infeasible problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 13.2.1 Example: Primal infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 13.2.2 Locating the cause of primal infeasibility . . . . . . . . . . . . . . . . . . . . . . 169 13.2.3 Locating the cause of dual infeasibility . . . . . . . . . . . . . . . . . . . . . . . . 170 13.2.4 The infeasibility report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 13.2.5 Theory concerning infeasible problems . . . . . . . . . . . . . . . . . . . . . . . . 174 13.2.6 The certificate of primal infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . 174 13.2.7 The certificate of dual infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . 175 14 Primal feasibility repair 177 14.1 Manual repair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 14.2 Automatic repair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
CONTENTS vii 14.2.1 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 14.3 Feasibility repair in MOSEK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 14.3.1 An example using the command line tool . . . . . . . . . . . . . . . . . . . . . . 180 14.3.2 Feasibility repair using the API . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 15 Sensitivity analysis 185 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 15.2 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 15.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 15.4 Sensitivity analysis for linear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 15.4.1 The optimal objective value function . . . . . . . . . . . . . . . . . . . . . . . . . 186 15.4.2 The basis type sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 187 15.4.3 The optimal partition type sensitivity analysis . . . . . . . . . . . . . . . . . . . 188 15.4.4 Example: Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 15.5 Sensitivity analysis from the MOSEK API . . . . . . . . . . . . . . . . . . . . . . . . . 192 15.6 Sensitivity analysis with the command line tool . . . . . . . . . . . . . . . . . . . . . . 195 15.6.1 Sensitivity analysis specification file . . . . . . . . . . . . . . . . . . . . . . . . . 196 15.6.2 Example: Sensitivity analysis from command line . . . . . . . . . . . . . . . . . . 197 15.6.3 Controlling log output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 A API reference 199 A.1 Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 A.2 Class Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.2.1 Task.analyzenames() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.2.2 Task.analyzeproblem() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.2.3 Task.analyzesolution() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.2.4 Task.appendbarvars() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A.2.5 Task.appendcone() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 A.2.6 Task.appendconeseq() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 A.2.7 Task.appendconesseq() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.2.8 Task.appendcons() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.2.9 Task.appendsparsesymmat() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 A.2.10 Task.appendstat() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 A.2.11 Task.appendvars() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 A.2.12 Task.basiscond() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 A.2.13 Task.checkconvexity() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.2.14 Task.checkmem() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.2.15 Task.chgbound() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.2.16 Task.commitchanges() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 A.2.17 Task.deletesolution() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 A.2.18 Task.dualsensitivity() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 A.2.19 Task.getacol() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 A.2.20 Task.getacolnumnz() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 A.2.21 Task.getacolslicetrip() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 A.2.22 Task.getaij() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 A.2.23 Task.getapiecenumnz() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
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Page 25 and 26: License agreement Before using the
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5.3. CONIC QUADRATIC OPTIMIZATION 3
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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 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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Page 591 and 592:
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
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F.1. THE MPS FILE FORMAT 593 Constr
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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
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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
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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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