The MOSEK command line tool Version 7.0 (Revision 141)

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iv CONTENTS 4.2.1 Duality for conic quadratic optimization . . . . . . . . . . . . . . . . . . . . . . . 24 4.2.2 Infeasibility for conic quadratic optimization . . . . . . . . . . . . . . . . . . . . 25 4.3 Semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3.1 Duality for semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3.2 Infeasibility for semidefinite optimization . . . . . . . . . . . . . . . . . . . . . . 27 4.4 Quadratic and quadratically constrained optimization . . . . . . . . . . . . . . . . . . 28 4.4.1 Duality for quadratic and quadratically constrained optimization . . . . . . . . . 28 4.4.2 Infeasibility for quadratic and quadratically constrained optimization . . . . . . . 29 4.5 General convex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5.1 Duality for general convex optimization . . . . . . . . . . . . . . . . . . . . . . . 30 5 The optimizers for continuous problems 33 5.1 How an optimizer works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1.1 Presolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.1.2 Dualizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.1.4 Using multiple threads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2.1 Optimizer selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2.2 The interior-point optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2.3 The simplex based optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2.4 The interior-point or the simplex optimizer? . . . . . . . . . . . . . . . . . . . . . 43 5.2.5 The primal or the dual simplex variant? . . . . . . . . . . . . . . . . . . . . . . . 43 5.3 Linear network optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.3.1 Network flow problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.4 Conic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.4.1 The interior-point optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.5 Nonlinear convex optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.5.1 The interior-point optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.6 Solving problems in parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.6.1 Thread safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.6.2 The parallelized interior-point optimizer . . . . . . . . . . . . . . . . . . . . . . . 47 5.6.3 The concurrent optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6 The optimizers for mixed-integer problems 49 6.1 Some concepts and facts related to mixed-integer optimization . . . . . . . . . . . . . . 49 6.2 The mixed-integer optimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.3 The mixed-integer conic optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.3.1 Presolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.3.2 Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.3.3 The optimization phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.3.4 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.4 The mixed-integer optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.4.1 Presolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.4.2 Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.4.3 The optimization phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
CONTENTS v 6.5 Termination criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.5.1 Relaxed termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.5.2 Important parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.6 How to speed up the solution process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.7 Understanding solution quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7 The analyzers 57 7.1 The problem analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.1.1 General characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.1.3 Linear constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.1.4 Constraint and variable bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.1.5 Quadratic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.1.6 Conic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.2 Analyzing infeasible problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.2.1 Example: Primal infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.2.2 Locating the cause of primal infeasibility . . . . . . . . . . . . . . . . . . . . . . 63 7.2.3 Locating the cause of dual infeasibility . . . . . . . . . . . . . . . . . . . . . . . . 64 7.2.4 The infeasibility report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.2.5 Theory concerning infeasible problems . . . . . . . . . . . . . . . . . . . . . . . . 68 7.2.6 The certificate of primal infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.2.7 The certificate of dual infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . 69 8 Sensitivity analysis 71 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.2 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.4 Sensitivity analysis for linear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.4.1 The optimal objective value function . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.4.2 The basis type sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.4.3 The optimal partition type sensitivity analysis . . . . . . . . . . . . . . . . . . . 74 8.5 Sensitivity analysis with the command line tool . . . . . . . . . . . . . . . . . . . . . . 75 8.5.1 Sensitivity analysis specification file . . . . . . . . . . . . . . . . . . . . . . . . . 76 8.5.2 Example: Sensitivity analysis from command line . . . . . . . . . . . . . . . . . . 77 8.5.3 Controlling log output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 9 Parameters 79 9.1 MSKdparame: Double parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.1.1 MSK DPAR ANA SOL INFEAS TOL . . . . . . . . . . . . . . . . . . . . . . . . 93 9.1.2 MSK DPAR BASIS REL TOL S . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.1.3 MSK DPAR BASIS TOL S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.1.4 MSK DPAR BASIS TOL X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.1.5 MSK DPAR CHECK CONVEXITY REL TOL . . . . . . . . . . . . . . . . . . . 94 9.1.6 MSK DPAR DATA TOL AIJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.1.7 MSK DPAR DATA TOL AIJ HUGE . . . . . . . . . . . . . . . . . . . . . . . . 95 9.1.8 MSK DPAR DATA TOL AIJ LARGE . . . . . . . . . . . . . . . . . . . . . . . . 95 9.1.9 MSK DPAR DATA TOL BOUND INF . . . . . . . . . . . . . . . . . . . . . . . 96
Page 1 and 2: The MOSEK command line tool. Versio
Page 3: Contents 1 Changes and new features
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.
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5.2. LINEAR OPTIMIZATION 39 Wheneve
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5.2. LINEAR OPTIMIZATION 41 5.2.2.3
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5.2. LINEAR OPTIMIZATION 43 • Rai
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5.5. NONLINEAR CONVEX OPTIMIZATION
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5.6. SOLVING PROBLEMS IN PARALLEL 4
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Chapter 6 The optimizers for mixed-
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6.3. THE MIXED-INTEGER CONIC OPTIMI
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6.5. TERMINATION CRITERION 53 The f
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6.7. UNDERSTANDING SOLUTION QUALITY
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Chapter 7 The analyzers 7.1 The pro
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7.1. THE PROBLEM ANALYZER 59 Constr
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7.2. ANALYZING INFEASIBLE PROBLEMS
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Chapter 8 Sensitivity analysis 8.1
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8.4. SENSITIVITY ANALYSIS FOR LINEA
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8.5. SENSITIVITY ANALYSIS WITH THE
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8.5. SENSITIVITY ANALYSIS WITH THE
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Chapter 9 Parameters Parameters gro
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81 • MSK SPAR ITR SOL FILE NAME.
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83 • MSK DPAR INTPNT NL TOL REL G
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85 • MSK IPAR LOG MIO. Controls t
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87 • MSK DPAR INTPNT NL MERIT BAL
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89 • MSK IPAR INFEAS PREFER PRIMA
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91 • MSK IPAR SIM SAVE LU. Contro
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9.1. MSKDPARAME: DOUBLE PARAMETERS
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9.2. MSKIPARAME: INTEGER PARAMETERS
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9.3. MSKSPARAME: STRING PARAMETER T
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Chapter 10 Response codes Response
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205 MSK RES ERR CON Q NOT PSD The q
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207 MSK RES ERR GLOBAL INV CONIC PR
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209 MSK RES ERR INV MARKJ Invalid v
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211 MSK RES ERR INVALID FORMAT TYPE
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213 MSK RES ERR LICENSE NO SERVER S
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215 MSK RES ERR MIO NO OPTIMIZER No
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217 MSK RES ERR NAME MAX LEN A name
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219 MSK RES ERR NUMCONLIM Maximum n
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221 MSK RES ERR QCON UPPER TRIANGLE
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223 MSK RES ERR SYM MAT INVALID COL
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225 MSK RES ERR USER NLO FUNC The u
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227 MSK RES WRN ANA ALMOST INT BOUN
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229 MSK RES WRN MIO INFEASIBLE FINA
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231 MSK RES WRN WRITE DISCARDED CFI
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Chapter 11 API constants 11.1 Const
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11.5. PROGRESS CALL-BACK CODES 235
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11.6. TYPES OF CONVEXITY CHECKS. 24
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11.10. DOUBLE INFORMATION ITEMS 245
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11.10. DOUBLE INFORMATION ITEMS 247
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11.11. FEASIBILITY REPAIR TYPES 249
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11.13. INTEGER INFORMATION ITEMS. 2
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11.17. LANGUAGE SELECTION CONSTANTS
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11.22. MIXED-INTEGER NODE SELECTION
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11.29. ORDERING STRATEGIES 261 MSK
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11.34. PROBLEM STATUS KEYS 263 MSK
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11.38. SENSITIVITY TYPES 265 MSK SC
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11.44. SOLUTION ITEMS 267 MSK SIM S
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11.46. SOLUTION TYPES 269 11.46 Sol
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11.52. INTEGER VALUES 271 11.52 Int
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Chapter 12 MOSEK Command line tool
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12.3. THE PARAMETER FILE 275 -min F
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Chapter 13 The MPS file format MOSE
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13.1. MPS FILE STRUCTURE 279 Extens
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13.1. MPS FILE STRUCTURE 281 [vname
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13.1. MPS FILE STRUCTURE 283 Field
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13.1. MPS FILE STRUCTURE 285 Next d
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13.2. INTEGER VARIABLES 287 13.2 In
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Chapter 14 The LP file format MOSEK
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14.1. THE SECTIONS 291 x1 * x2 Ther
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14.2. LP FORMAT PECULIARITIES 293 1
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14.3. THE STRICT LP FORMAT 295 MSK
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Chapter 15 The OPF format The Optim
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15.2. THE FILE FORMAT 299 [con ’c
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15.2. THE FILE FORMAT 301 - ‘NEAR
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15.4. WRITING OPF FILES FROM MOSEK
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15.5. EXAMPLES 305 [/hints] [variab
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15.5. EXAMPLES 307 x1 x2 [/integer]
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Chapter 16 The XML (OSiL) format MO
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Chapter 17 The solution file format
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17.2. THE INTEGER SOLUTION FILE 313
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Chapter 18 Problem analyzer example
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18.2. ARKI001 317 2 476 45.42 48.19
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18.4. PROBLEM WITH BOTH LINEAR AND
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Bibliography [1] R. Fourer and D. M
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Index AMPL outlev, 13 wantsol, 13 a
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INDEX 325 MSK RES ERR FEASREPAIR IN
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INDEX 327 MSK RES ERR MPS NULL VAR
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INDEX 329 MSK RES TRM MAX TIME, 226
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