The MOSEK optimization toolbox for MATLAB manual Version 7.0 (Revision 141)

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viii CONTENTS 16.4.2 The basis type sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 181 16.4.3 The optimal partition type sensitivity analysis . . . . . . . . . . . . . . . . . . . 182 16.4.4 Example: Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 16.5 Sensitivity analysis in the MATLAB toolbox . . . . . . . . . . . . . . . . . . . . . . . . 186 16.5.1 On bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 16.5.2 Selecting analysis type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 16.5.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 17 Problem formulation and solutions 193 17.1 Linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 17.1.1 Duality for linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 17.1.2 Infeasibility for linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 196 17.2 Conic quadratic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 17.2.1 Duality for conic quadratic optimization . . . . . . . . . . . . . . . . . . . . . . . 198 17.2.2 Infeasibility for conic quadratic optimization . . . . . . . . . . . . . . . . . . . . 199 17.3 Semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 17.3.1 Duality for semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . . . . 200 17.3.2 Infeasibility for semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . 201 17.4 Quadratic and quadratically constrained optimization . . . . . . . . . . . . . . . . . . 202 17.4.1 Duality for quadratic and quadratically constrained optimization . . . . . . . . . 202 17.4.2 Infeasibility for quadratic and quadratically constrained optimization . . . . . . . 203 17.5 General convex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 17.5.1 Duality for general convex optimization . . . . . . . . . . . . . . . . . . . . . . . 204 18 The MPS file format 207 18.1 MPS file structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 18.1.1 Linear example lo1.mps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 18.1.2 NAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 18.1.3 OBJSENSE (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 18.1.4 OBJNAME (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 18.1.5 ROWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 18.1.6 COLUMNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 18.1.7 RHS (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 18.1.8 RANGES (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 18.1.9 QSECTION (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 18.1.10 BOUNDS (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 18.1.11 CSECTION (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 18.1.12 ENDATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 18.2 Integer variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 18.3 General limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 18.4 Interpretation of the MPS format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 18.5 The free MPS format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 19 The LP file format 219 19.1 The sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 19.1.1 The objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 19.1.2 The constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
CONTENTS ix 19.1.3 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 19.1.4 Variable types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 19.1.5 Terminating section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 19.1.6 Linear example lo1.lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 19.1.7 Mixed integer example milo1.lp . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 19.2 LP format peculiarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 19.2.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 19.2.2 Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 19.2.3 Variable bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 19.2.4 MOSEK specific extensions to the LP format . . . . . . . . . . . . . . . . . . . . 224 19.3 The strict LP format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 19.4 Formatting of an LP file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 19.4.1 Speeding up file reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 19.4.2 Unnamed constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 20 The OPF format 227 20.1 Intended use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 20.2 The file format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 20.2.1 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 20.2.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 20.2.3 Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 20.3 Parameters section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 20.4 Writing OPF files from MOSEK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 20.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 20.5.1 Linear example lo1.opf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 20.5.2 Quadratic example qo1.opf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 20.5.3 Conic quadratic example cqo1.opf . . . . . . . . . . . . . . . . . . . . . . . . . . 235 20.5.4 Mixed integer example milo1.opf . . . . . . . . . . . . . . . . . . . . . . . . . . 236 21 The Task format 239 22 The XML (OSiL) format 241 23 Parameters 243 23.1 MSKdparame: Double parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 23.1.1 MSK DPAR ANA SOL INFEAS TOL . . . . . . . . . . . . . . . . . . . . . . . . 257 23.1.2 MSK DPAR BASIS REL TOL S . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 23.1.3 MSK DPAR BASIS TOL S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 23.1.4 MSK DPAR BASIS TOL X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 23.1.5 MSK DPAR CHECK CONVEXITY REL TOL . . . . . . . . . . . . . . . . . . . 258 23.1.6 MSK DPAR DATA TOL AIJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 23.1.7 MSK DPAR DATA TOL AIJ HUGE . . . . . . . . . . . . . . . . . . . . . . . . 259 23.1.8 MSK DPAR DATA TOL AIJ LARGE . . . . . . . . . . . . . . . . . . . . . . . . 259 23.1.9 MSK DPAR DATA TOL BOUND INF . . . . . . . . . . . . . . . . . . . . . . . 260 23.1.10 MSK DPAR DATA TOL BOUND WRN . . . . . . . . . . . . . . . . . . . . . . 260 23.1.11 MSK DPAR DATA TOL C HUGE . . . . . . . . . . . . . . . . . . . . . . . . . . 260 23.1.12 MSK DPAR DATA TOL CJ LARGE . . . . . . . . . . . . . . . . . . . . . . . . 261
Page 1 and 2: The MOSEK optimization toolbox for
Page 3 and 4: Contents 1 Changes and new features
Page 5 and 6: CONTENTS v 7.15.3 Speeding up the s
Page 7: CONTENTS vii 12.3 The mixed-integer
Page 11 and 12: CONTENTS xi 23.1.59 MSK DPAR PRESOL
Page 13 and 14: CONTENTS xiii 23.2.82 MSK IPAR MIO
Page 15 and 16: CONTENTS xv 23.2.174MSK IPAR WARNIN
Page 17 and 18: CONTENTS xvii 25.32 Problem data it
Page 19 and 20: Contact information Phone +45 3917
Page 21 and 22: License agreement Before using the
Page 23 and 24: Chapter 1 Changes and new features
Page 25 and 26: 1.4. API CHANGES 7 1.3.3 Mixed-inte
Page 27 and 28: Chapter 2 Introduction This manual
Page 29 and 30: Chapter 3 Supported MATLAB versions
Page 31 and 32: Chapter 4 Installation In order to
Page 33 and 34: 4.3. TROUBLESHOOTING 15 Search path
Page 35 and 36: Chapter 5 Getting support and help
Page 37 and 38: Chapter 6 MOSEK / MATLAB integratio
Page 39 and 40: Chapter 7 A guided tour 7.1 Introdu
Page 41 and 42: 7.4. LINEAR OPTIMIZATION 23 7.4.1 U
Page 43 and 44: 7.5. CONVEX QUADRATIC OPTIMIZATION
Page 45 and 46: 7.5. CONVEX QUADRATIC OPTIMIZATION
Page 47 and 48: 7.6. CONIC OPTIMIZATION 29 be vecto
Page 49 and 50: 7.6. CONIC OPTIMIZATION 31 minimize
Page 51 and 52: 7.6. CONIC OPTIMIZATION 33 • Quad
Page 53 and 54: 7.7. SEMIDEFINITE OPTIMIZATION 35 m
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7.9. LINEAR LEAST SQUARES AND RELAT
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7.11. MORE ABOUT SOLVING LINEAR LEA
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7.13. GEOMETRIC OPTIMIZATION 49 and
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7.13. GEOMETRIC OPTIMIZATION 51 27
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7.14. SEPARABLE CONVEX OPTIMIZATION
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7.15. MIXED-INTEGER OPTIMIZATION 55
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7.15. MIXED-INTEGER OPTIMIZATION 57
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7.16. SENSITIVITY ANALYSIS 59 24 25
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7.17. INSPECTING A PROBLEM 61 Const
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7.19. VIEWING THE TASK INFORMATION
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7.21. ADVANCED START (HOT-START) 65
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7.22. USING NAMES 71 7.22 Using nam
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7.24. USER CALL-BACK FUNCTIONS 73 U
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7.25. THE LICENSE SYSTEM 75 ture, t
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Chapter 8 Command reference After s
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8.1. DATA STRUCTURES 79 .blx Lower
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8.1. DATA STRUCTURES 81 must be sym
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8.1. DATA STRUCTURES 83 Field name
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8.1. DATA STRUCTURES 85 .type .sub
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8.1. DATA STRUCTURES 87 Comments: .
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8.1. DATA STRUCTURES 89 .rs bl Righ
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8.3. FUNCTIONS PROVIDED BY THE MOSE
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8.3. FUNCTIONS PROVIDED BY THE MOSE
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8.4. MATLAB OPTIMIZATION TOOLBOX CO
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Chapter 9 Case studies 9.1 Robust l
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9.1. ROBUST LINEAR OPTIMIZATION 109
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9.2. GEOMETRIC (POSYNOMIAL) OPTIMIZ
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Chapter 10 The optimizers for conti
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10.1. HOW AN OPTIMIZER WORKS 131 10
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10.2. LINEAR OPTIMIZATION 133 10.2.
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10.2. LINEAR OPTIMIZATION 135 Whene
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10.2. LINEAR OPTIMIZATION 137 10.2.
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10.2. LINEAR OPTIMIZATION 139 • R
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10.5. NONLINEAR CONVEX OPTIMIZATION
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10.6. SOLVING PROBLEMS IN PARALLEL
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Chapter 11 The solution summary All
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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 151 The
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12.7. UNDERSTANDING SOLUTION QUALIT
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Chapter 13 The solution summary for
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Chapter 14 The analyzers 14.1 The p
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14.1. THE PROBLEM ANALYZER 159 Cons
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14.2. ANALYZING INFEASIBLE PROBLEMS
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Chapter 15 Primal feasibility repai
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15.2. AUTOMATIC REPAIR 173 One way
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15.3. FEASIBILITY REPAIR IN MOSEK 1
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15.3. FEASIBILITY REPAIR IN MOSEK 1
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Chapter 16 Sensitivity analysis 16.
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16.4. SENSITIVITY ANALYSIS FOR LINE
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16.5. SENSITIVITY ANALYSIS IN THE M
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Chapter 17 Problem formulation and
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17.1. LINEAR OPTIMIZATION 195 be a
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17.2. CONIC QUADRATIC OPTIMIZATION
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17.2. CONIC QUADRATIC OPTIMIZATION
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17.3. SEMIDEFINITE OPTIMIZATION 201
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17.5. GENERAL CONVEX OPTIMIZATION 2
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17.5. GENERAL CONVEX OPTIMIZATION 2
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Chapter 18 The MPS file format MOSE
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18.1. MPS FILE STRUCTURE 209 Extens
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18.1. MPS FILE STRUCTURE 211 [vname
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18.1. MPS FILE STRUCTURE 213 Field
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18.1. MPS FILE STRUCTURE 215 Next d
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18.2. INTEGER VARIABLES 217 18.2 In
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Chapter 19 The LP file format MOSEK
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19.1. THE SECTIONS 221 x1 * x2 Ther
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19.2. LP FORMAT PECULIARITIES 223 1
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19.3. THE STRICT LP FORMAT 225 MSK
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Chapter 20 The OPF format The Optim
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20.2. THE FILE FORMAT 229 [con ’c
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20.2. THE FILE FORMAT 231 - ‘NEAR
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20.4. WRITING OPF FILES FROM MOSEK
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20.5. EXAMPLES 235 [/hints] [variab
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20.5. EXAMPLES 237 x1 x2 [/integer]
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Chapter 21 The Task format The Task
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Chapter 22 The XML (OSiL) format MO
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Chapter 23 Parameters Parameters gr
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245 • MSK SPAR ITR SOL FILE NAME.
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247 • MSK DPAR INTPNT NL TOL REL
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249 • MSK IPAR LOG MIO. Controls
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251 • MSK DPAR INTPNT NL MERIT BA
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253 • MSK IPAR INFEAS PREFER PRIM
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255 • MSK IPAR SIM SAVE LU. Contr
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23.1. MSKDPARAME: DOUBLE PARAMETERS
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23.2. MSKIPARAME: INTEGER PARAMETER
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23.3. MSKSPARAME: STRING PARAMETER
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Chapter 24 Response codes Response
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369 MSK RES ERR CON Q NOT PSD The q
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371 MSK RES ERR GLOBAL INV CONIC PR
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373 MSK RES ERR INV MARKJ Invalid v
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375 MSK RES ERR INVALID FORMAT TYPE
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377 MSK RES ERR LICENSE NO SERVER S
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379 MSK RES ERR MIO NO OPTIMIZER No
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381 MSK RES ERR NAME MAX LEN A name
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383 MSK RES ERR NUMCONLIM Maximum n
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385 MSK RES ERR QCON UPPER TRIANGLE
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387 MSK RES ERR SYM MAT INVALID COL
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389 MSK RES ERR USER NLO FUNC The u
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391 MSK RES WRN ANA ALMOST INT BOUN
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393 MSK RES WRN MIO INFEASIBLE FINA
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395 MSK RES WRN WRITE DISCARDED CFI
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Chapter 25 API constants 25.1 Const
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25.5. PROGRESS CALL-BACK CODES 399
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25.6. TYPES OF CONVEXITY CHECKS. 40
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25.10. DOUBLE INFORMATION ITEMS 409
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25.10. DOUBLE INFORMATION ITEMS 411
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25.11. FEASIBILITY REPAIR TYPES 413
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25.13. INTEGER INFORMATION ITEMS. 4
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25.17. LANGUAGE SELECTION CONSTANTS
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25.22. MIXED-INTEGER NODE SELECTION
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25.29. ORDERING STRATEGIES 425 MSK
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25.34. PROBLEM STATUS KEYS 427 MSK
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25.38. SENSITIVITY TYPES 429 MSK SC
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25.44. SOLUTION ITEMS 431 MSK SIM S
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25.46. SOLUTION TYPES 433 25.46 Sol
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25.52. INTEGER VALUES 435 25.52 Int
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Chapter 26 Problem analyzer example
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26.2. ARKI001 439 2 476 45.42 48.19
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26.4. PROBLEM WITH BOTH LINEAR AND
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Bibliography [1] Chvátal, V.. Line
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Index basis identification, 136 bin
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INDEX 447 MSK RES ERR DUPLICATE CON
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INDEX 449 MSK RES ERR MPS INVALID O
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INDEX 451 MSK RES ERR XML INVALID P
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