MOSEK Optimizer API for Python

420 Pages·2016·1.63 MB·English

MOSEK Optimizer API for Python
Release 8.0.0.94
MOSEK ApS
2017

CONTENTS
1 Introduction 1
1.1 Why the Optimizer API for Python? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 License agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Installation 3
2.1 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Instructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Testing the Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Basic Tutorials 5
3.1 The Basics Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Linear Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Conic Quadratic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Semidefinite Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Quadratic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Integer Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 Optimizer Termination Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8 Problem Modification and Reoptimization . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.9 Solution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.10 Solver Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Nonlinear Tutorials 49
4.1 Separable Convex (SCopt) Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Advanced Tutorials 53
5.1 The Progress Call-back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Solving Linear Systems Involving the Basis Matrix . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Calling BLAS/LAPACK Routines from MOSEK . . . . . . . . . . . . . . . . . . . . . . . 62
5.4 Computing a Sparse Cholesky Factorization . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5 Converting a quadratically constrained problem to conic form . . . . . . . . . . . . . . . 69
5.6 MOSEK OptServer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Guidelines 77
6.1 Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Efficiency Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 The license system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7 Case Studies 81
7.1 Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8 Errors and Warnings 101
8.1 Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9 Managing I/O 103
9.1 Stream I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
i
9.2 File I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.3 Verbosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
10 Problem Formulation and Solutions 107
10.1 Linear Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.2 Conic Quadratic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
10.3 Semidefinite Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.4 Quadratic and Quadratically Constrained Optimization . . . . . . . . . . . . . . . . . . . 114
10.5 General Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
11 The Optimizers for Continuous Problems 117
11.1 Presolve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.2 Linear Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
11.3 Conic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
11.4 Nonlinear Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
11.5 Using Multiple Threads in an Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
12 The Optimizer for Mixed-integer Problems 129
12.1 Some Concepts and Facts Related to Mixed-integer Optimization . . . . . . . . . . . . . 129
12.2 The Mixed-integer Optimizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.3 Termination Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
12.4 Parameters Affecting the Termination of the Integer Optimizer. . . . . . . . . . . . . . . 131
12.5 How to Speed Up the Solution Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
12.6 Understanding Solution Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
13 Problem Analyzer 133
13.1 General Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
13.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13.3 Linear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13.4 Constraint and Variable Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
13.5 Quadratic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
13.6 Conic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
14 Analyzing Infeasible Problems 137
14.1 Example: Primal Infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
14.2 Locating the cause of Primal Infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
14.3 Locating the Cause of Dual Infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
14.4 The Infeasibility Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
14.5 Theory Concerning Infeasible Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
14.6 The Certificate of Primal Infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
14.7 The certificate of dual infeasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
15 Sensitivity Analysis 143
15.1 Sensitivity Analysis for Linear Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
15.2 Sensitivity Analysis with MOSEK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
16 API Reference 153
16.1 API Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
16.2 Functions grouped by topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
16.3 The Interface for Separable Convex Optimization . . . . . . . . . . . . . . . . . . . . . . 165
16.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
16.5 Response codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
16.6 Enumerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.7 Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
16.8 Class Env . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
16.9 Class Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
16.10 Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
17 Supported File Formats 329
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17.1 The LP File Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
17.2 The MPS File Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
17.3 The OPF Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
17.4 The CBF Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
17.5 The XML (OSiL) Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
17.6 The Task Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
17.7 The JSON Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
17.8 The Solution File Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
18 Interface changes 381
18.1 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
18.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
18.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
18.4 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
18.5 Response Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Bibliography 391
API Index 393
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CHAPTER
ONE
INTRODUCTION
The MOSEK Optimization Suite 8.0.0.94 is a powerfull software package capable of solving large-scale
optimization problems of the following kind:
• linear,
• convex quadratic,
• conic quadratic (also known as second-order cone),
• semidefinite,
• and general convex.
Integer constrained variables are supported for all problem classes except for semidefinite and general
convexproblems. InordertoobtainanoverviewoffeaturesintheMOSEKOptimizationSuiteconsult
the product introduction guide.
1.1 Why the Optimizer API for Python?
The Optimizer API for Python provides an object-oriented interface to the MOSEK optimizers. This
objectorienteddesigniscommontoJava, Pythonand.NETandisbasedonathinclass-basedinterface
to the native C optimizer API. The overhead introduced by this mapping is minimal.
The Optimizer API for Python can be used with any application running on recent Python 2 and 3
interpreters. It consists of a single mosek package which can be used in Python scripts and interactive
shells making it suited for fast prototyping and inspection of models.
1.2 License agreement
Before using the MOSEK software, please read the license agreement available in the distribu-
tion at <MSKHOME>/mosek/8/mosek-eula.pdf or on the MOSEK website https://mosek.com/sales/
license-agreement.
1
MOSEK Optimizer API for Python, Release 8.0.0.94
MOSEK uses some third-party open-source libraries. Their license details follows.
zlib
MOSEKincludesthezlib libraryobtainedfromthezlibwebsite. Thelicenseagreementforzlib isshown
in Listing 1.1.
Listing 1.1: zlib license.
zlib.h -- interface of the 'zlib' general purpose compression library
version 1.2.7, May 2nd, 2012
Copyright (C) 1995-2012 Jean-loup Gailly and Mark Adler
This software is provided 'as-is', without any express or implied
warranty. In no event will the authors be held liable for any damages
arising from the use of this software.
Permission is granted to anyone to use this software for any purpose,
including commercial applications, and to alter it and redistribute it
freely, subject to the following restrictions:
1. The origin of this software must not be misrepresented; you must not
claim that you wrote the original software. If you use this software
in a product, an acknowledgment in the product documentation would be
appreciated but is not required.
2. Altered source versions must be plainly marked as such, and must not be
misrepresented as being the original software.
3. This notice may not be removed or altered from any source distribution.
Jean-loup Gailly Mark Adler
[email protected] [email protected]
fplib
MOSEK includes the floating point formatting library developed by David M. Gay obtained from the
netlib website. The license agreement for fplib is shown in Listing 1.2.
Listing 1.2: fplib license.
/****************************************************************
*
* The author of this software is David M. Gay.
*
* Copyright (c) 1991, 2000, 2001 by Lucent Technologies.
*
* Permission to use, copy, modify, and distribute this software for any
* purpose without fee is hereby granted, provided that this entire notice
* is included in all copies of any software which is or includes a copy
* or modification of this software and in all copies of the supporting
* documentation for such software.
*
* THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED
* WARRANTY. IN PARTICULAR, NEITHER THE AUTHOR NOR LUCENT MAKES ANY
* REPRESENTATION OR WARRANTY OF ANY KIND CONCERNING THE MERCHANTABILITY
* OF THIS SOFTWARE OR ITS FITNESS FOR ANY PARTICULAR PURPOSE.
*
***************************************************************/
2 Chapter 1. Introduction
CHAPTER
TWO
INSTALLATION
In this section we discuss how to install and setup the MOSEK Optimizer API for Python.
2.1 Compatibility
The MOSEK Optimizer API for Python requires Python with numpy. Below the supported Python
versions are shown
Platform Python PyPy2.7
Linux 64 bit 2.7, 3.4 and newer Yes
Mac OS 64 bit 2.7, 3.4 and newer Yes
Windows 32 and 64 bit 2.7, 3.4 and newer Yes
2.2 Instructions
2.2.1 Conda Install
Please follow the instructions in the Installation Guide.
2.2.2 Manual Install
Instructions for installing MOSEK Optimization Suite can in located in the Installation Guide.
Subsequently, let <MSKHOME> denote where MOSEK is installed
The relevant files of the MOSEK Optimizer API for Python are organized as reported in Table 2.1.
Table 2.1: Relevant files for the MOSEK Optimizer API for Python.
Relative Path Description Label
<MSKHOME>/mosek/8/tools/platform/<PLATFORM>/python/2 Python 2 install PYTHON2DIR
<MSKHOME>/mosek/8/tools/platform/<PLATFORM>/python/3 Python 3 install PYTHON3DIR
<MSKHOME>/mosek/8/tools/examples/python Examples EXDIR
<MSKHOME>/mosek/8/tools/examples/data Additional data MISCDIR
Install the Package
To install MOSEK run the setup.py script located in <PYTHON2DIR> or <PYTHON3DIR> depending on
the Python version you want to use. For instance, to install MOSEK for Python 3 on a user level, i.e.,
with no special system privilegdes needed, type
3
MOSEK Optimizer API for Python, Release 8.0.0.94
$ python3 <PYTHON3DIR>/setup.py install --user
on Linux and Mac OS or
C:\> python3 <PYTHON3DIR>\setup.py install --user
on Windows.
For a system wide installation drop the --user flag.
2.2.3 PIP Install
A simple PIP installer is available on the MOSEK Github repository. Atlthough there is a MOSEK
package in the official PIP repository then it is not created by MOSEK. It cannot be recommended to
use that package.
2.3 Testing the Installation
The MOSEK Optimizer API for Python installation can be tested by running some of the enclosed
examples. Indeed to run one of the distributed examples, open a terminal and change folder to <EXDIR>
i.e.
cd <EXDIR>
then use your Python interpreter to run the example. For instance to execute example lo1 type
python lo1.py
4 Chapter 2. Installation