Review of Linear Programming Software
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1 to LP/QP/MIP Review of Software Peter He, Yang Li, Zhenghue Nie, Nael El Shawwa Instructor: Prof. Tamas Terlaky School of Computational Engineering and Science McMaster University Jan. 12, 2007
2 Outline to LP/QP/MIP 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
3 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Outline 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
4 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What are LP? Standard LP form: min s.t. c T x Ax = b x 0 Practical form: min s.t. Main Methods: Simplex Methods Interior Point Methods c T x b Ax b l x u
5 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What are LP? Standard LP form: min s.t. c T x Ax = b x 0 Practical form: min s.t. Main Methods: Simplex Methods Interior Point Methods c T x b Ax b l x u
6 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What are LP? Standard LP form: min s.t. c T x Ax = b x 0 Practical form: min s.t. Main Methods: Simplex Methods Interior Point Methods c T x b Ax b l x u
7 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What are LP? Standard LP form: min s.t. c T x Ax = b x 0 Practical form: min s.t. Main Methods: Simplex Methods Interior Point Methods c T x b Ax b l x u
8 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What are LP? Standard LP form: min s.t. c T x Ax = b x 0 Practical form: min s.t. Main Methods: Simplex Methods Interior Point Methods c T x b Ax b l x u
9 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Simplex Method (Pivot Method) Ax = b defines a polytope, namely, Simplex Jump from vertex to vertex Perform well for small and medium scale, but bad for large scale problem, O(2 n ) Efficient for MIP Degeneracy Source: terlaky/htm/talks/pivot-vs-ipm.pdf
10 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Simplex Method (Pivot Method) Ax = b defines a polytope, namely, Simplex Jump from vertex to vertex Perform well for small and medium scale, but bad for large scale problem, O(2 n ) Efficient for MIP Degeneracy Source: terlaky/htm/talks/pivot-vs-ipm.pdf
11 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Simplex Method (Pivot Method) Ax = b defines a polytope, namely, Simplex Jump from vertex to vertex Perform well for small and medium scale, but bad for large scale problem, O(2 n ) Efficient for MIP Degeneracy Source: terlaky/htm/talks/pivot-vs-ipm.pdf
12 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Simplex Method (Pivot Method) Ax = b defines a polytope, namely, Simplex Jump from vertex to vertex Perform well for small and medium scale, but bad for large scale problem, O(2 n ) Efficient for MIP Degeneracy Source: terlaky/htm/talks/pivot-vs-ipm.pdf
13 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Simplex Method (Pivot Method) Ax = b defines a polytope, namely, Simplex Jump from vertex to vertex Perform well for small and medium scale, but bad for large scale problem, O(2 n ) Efficient for MIP Degeneracy Source: terlaky/htm/talks/pivot-vs-ipm.pdf
14 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Interior Point Methods Ax = b A T y + s = c x T s = µe follow the central path in the feasible region Source: terlaky/htm/talks/pivot-vs-ipm.pdf
15 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Interior Point Methods Ax = b A T y + s = c x T s = µe follow the central path in the feasible region Source: terlaky/htm/talks/pivot-vs-ipm.pdf
16 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Interior Point Methods Very efficient for large-scale problem, total computational complexity O(n 3 L) Bad performance in MIP No troubles caused by degeneracy Source: terlaky/htm/talks/pivot-vs-ipm.pdf
17 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Interior Point Methods Very efficient for large-scale problem, total computational complexity O(n 3 L) Bad performance in MIP No troubles caused by degeneracy Source: terlaky/htm/talks/pivot-vs-ipm.pdf
18 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Interior Point Methods Very efficient for large-scale problem, total computational complexity O(n 3 L) Bad performance in MIP No troubles caused by degeneracy Source: terlaky/htm/talks/pivot-vs-ipm.pdf
19 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Outline 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
20 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What is (Mixed) Integer Programming Integer Programming: min c T x s.t. Ax = b x i Z or subset of Z for all i Branch and Bound & Cutting Plane.
21 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What is (Mixed) Integer Programming Mixed Integer Programming: min c T x s.t. Ax = b x i Z or subset of Z for some i Branch and Bound & Cutting Plane.
22 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming Outline 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
23 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What is Quadratic Programming? Standard QP form: min c T x x T Qx s.t. Ax = b x 0 If Q is semi-definite, the problem is convex. Otherwise, local convergence.
24 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming What is Quadratic Programming? Standard QP form: min c T x x T Qx s.t. Ax = b x 0 If Q is semi-definite, the problem is convex. Otherwise, local convergence.
25 Outline to LP/QP/MIP 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
26 Developers View to LP/QP/MIP Programs Algorithms Modeling systems Cost Free codes Commercial products - providing the demo version Others products type interface platforms and/or operating systems development environment modeling system parallel...
27 Outline to LP/QP/MIP 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
28 to LP/QP/MIP Overview of Created by Robert E. Bixby using C language in 1987 Been Acquired by ILOG, inc. in 1997 Widespread 95% of papers that mention a solver mention at any conference where papers are presented Standard solver in supply-chain applications; Industry leaders including SAP, Oracle, i2, JD Edwards, and Manugistics Most of the major airlines including United, American, Delta, Continental, Northwest, and Southwest 2004 INFORMS Impact Award Latest version: 10.1 (Parallel, Indictor Constraints, etc. )
29 Problems Covered to LP/QP/MIP Mixed Integer Quadratic Programming Mixed Integer Quadratic Programming Quadratic Constrained Programming Mixed Integer Quadratic Constrained Programming Millions of constraints and variables
30 Algorithms to LP/QP/MIP LP: Primal simplex Dual simplex Network simplex Primal/dual log barrier QP: Primal simplex Dual simplex Primal dual log barrier QCP: Primal dual log barrier
31 to LP/QP/MIP ILOG Optimization Suite Simplex Optimizers (Primal, Dual, Network) for LP and QP Barrier Optimizer for LP, QP and QCP Mixed Integer Optimizer for MIP, MIQP and MIQCP Interactive Optimizer Component Libraries Callable Library (C and VB6 APIs) ILOG Concert Technology (C++, Java,.NET APIs) Source:
32 Embedding to LP/QP/MIP Callable Library uses matrices to represent a problem ILOG Concert Technology uses objects and methods to represent a problem ILOG OPL Interfaces allow OPL models to be embedded inside an application Source:
33 Embedding to LP/QP/MIP Callable Library uses matrices to represent a problem ILOG Concert Technology uses objects and methods to represent a problem ILOG OPL Interfaces allow OPL models to be embedded inside an application Source:
34 Embedding to LP/QP/MIP Callable Library uses matrices to represent a problem ILOG Concert Technology uses objects and methods to represent a problem ILOG OPL Interfaces allow OPL models to be embedded inside an application Source:
35 to LP/QP/MIP Modeling System, Platforms and Parallel Modeling System: AMPL, MPL, OPL Studio, and GAMS Platforms: PC/Windows, PC/Linux, Sun/Solaris, HP, IBM Power Parallel: Yes
36 Outline to LP/QP/MIP 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
37 to LP/QP/MIP Overview of Xpress-MP Developed by Dash Optimization Used in industry: Finance and investment Italcementi Group - Cement plant supply chain optimisation Honeywell - Flowsheet Decomposition Heuristic for Scheduling Power Optimization Limited - Generator Scheduling in the Electricity Industry
38 Problems covered to LP/QP/MIP LP, QP MIP, MIQP
39 Algorithms to LP/QP/MIP Primal and Dual simplex methods Interior points method for LP and QP Branch and Bound techniques to solve MIP and MIQP Techniques to reduce problem size
40 Platforms and OS to LP/QP/MIP Windows Linux, Unix Mac (PowerPC) HP, AIX
41 Modeling Systems to LP/QP/MIP Mosel AMPL GAMS Frontline MPL
42 Interfaces to LP/QP/MIP command line C/ C++ Java Fortran VB6,.NET ODBC I/O driver for external spreadsheets and databases
43 Other Features to LP/QP/MIP Size: no fixed limit Input form: Industry standard formats - LP,MPS Supports parallel computing Available add-ons to solve other problem types - stochastic problems, NLP total user control of optimization process fast re-solve time Development tools - GUI development, problem formulation, solving and analysis
44 Cost to LP/QP/MIP $ 6,000 per license (basic) $ 22,500 per license (bundle) $ 1,000 per additional licence Academic: 90% off, free with academic partnership program Other Mosel and Optimizer add-ons available separately Prices as of January Dash Optimization Price List
45 Outline to LP/QP/MIP 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
46 to LP/QP/MIP Overviews of Erling D. Andersen and Knud D. Andersen Implement language: C Latest Version: 4.0
47 to LP/QP/MIP Overviews of Erling D. Andersen and Knud D. Andersen Implement language: C Latest Version: 4.0
48 to LP/QP/MIP Overviews of Erling D. Andersen and Knud D. Andersen Implement language: C Latest Version: 4.0
49 Problems covered to LP/QP/MIP Linear optimization Convex quadratic optimization. Quadratically constrained convex optimization. Conic quadratic optimization (also known as second order optimization). Convex optimization. Geometric optimization (posynomial case). Handle integer valued variables in linearly and quadratically constrained optimization problems
50 Algorithms to LP/QP/MIP Interior-point optimizer for all continuous problems. Primal or dual simplex optimizer for linear problems. Conic interior-point optimizer for conic quadratic problems. Mixed-integer optimizer based on a branch and cut technology.
51 Platforms and OS to LP/QP/MIP Linux 32 and 64 bit MAC OSX Solaris 32 and 64 bit Windows 32 and 64 bit
52 Modeling Systems to LP/QP/MIP AMPL GAMS AIMMS
53 Interfaces to LP/QP/MIP C/C++ API Command line interface Java API Microsoft. NET API MATLAB
54 Costs: to LP/QP/MIP A: Linux and Windows 32 bit B: Other Platforms, like Solaris, Mac OS, Linux and Windows 64 bit. Additional maintenance fee charged Discount for academic use: 70% to 90%
55 Other Features to LP/QP/MIP Input format: MPS, LP, MBT( binary task format) Parallel Computing
56 Outline to LP/QP/MIP 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
57 to LP/QP/MIP Overviews of Michel Berkelaar and Jeroen Dirks Implement language: ANSI C Latest Version: Version 5.5
58 to LP/QP/MIP Overviews of Michel Berkelaar and Jeroen Dirks Implement language: ANSI C Latest Version: Version 5.5
59 to LP/QP/MIP Overviews of Michel Berkelaar and Jeroen Dirks Implement language: ANSI C Latest Version: Version 5.5
60 Problems covered to LP/QP/MIP (Large-scale) Mixed Integer Programming
61 Algorithms to LP/QP/MIP Revised Simplex Method for A Branch and Bound Method for Mixed Integer Programming
62 Platforms and OS to LP/QP/MIP Linux/Unix Windows MAC OS
63 Modeling Systems to LP/QP/MIP AMPL GAMS AIMMS
64 Interfaces to LP/QP/MIP C/C++ API Visual Basic Microsoft.NET Delphi Java MATLAB Excel
65 Other Features to LP/QP/MIP Sizes: Basically, no limit on model size Input form: LP format and MPS Parallel Computing: No Prices: Free
66 Outline to LP/QP/MIP 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
67 to to LP/QP/MIP means Coin OR Linear Porgramming.It is a high quality simplex code under the term of the Common Public License Author: John Forrest; Published Time: 2002; The first vesion:version 0.90; Current version:clp ( is written in C++ and it is primarily intended to be used as a callable library.
68 to to LP/QP/MIP means Coin OR Linear Porgramming.It is a high quality simplex code under the term of the Common Public License Author: John Forrest; Published Time: 2002; The first vesion:version 0.90; Current version:clp ( is written in C++ and it is primarily intended to be used as a callable library.
69 to to LP/QP/MIP means Coin OR Linear Porgramming.It is a high quality simplex code under the term of the Common Public License Author: John Forrest; Published Time: 2002; The first vesion:version 0.90; Current version:clp ( is written in C++ and it is primarily intended to be used as a callable library.
70 to LP/QP/MIP What can solve? Linear Optimization Regular Large Scale It has been proved that LP with 1.5 million constraints still keeps reliability. Sparse Coeffient Matrixes Boolean Matrixes Network Matrixes Some Nonlinear Optimization:Such as Quadratic Programming Piecewise Linear Convex function as the objective function
71 to LP/QP/MIP What can solve? Linear Optimization Regular Large Scale It has been proved that LP with 1.5 million constraints still keeps reliability. Sparse Coeffient Matrixes Boolean Matrixes Network Matrixes Some Nonlinear Optimization:Such as Quadratic Programming Piecewise Linear Convex function as the objective function
72 to LP/QP/MIP What Are Algorithms Being Used? Prime simplex method : the column pivot choice Dual simplex method: Dantzig and Steepest edge row pivot choice barrier method: solving convex QPs as well as LPs
73 to LP/QP/MIP What Are Algorithms Being Used? Prime simplex method : the column pivot choice Dual simplex method: Dantzig and Steepest edge row pivot choice barrier method: solving convex QPs as well as LPs
74 to LP/QP/MIP What Are Algorithms Being Used? Prime simplex method : the column pivot choice Dual simplex method: Dantzig and Steepest edge row pivot choice barrier method: solving convex QPs as well as LPs
75 to LP/QP/MIP What Input Form Is Often Used? often reads or loads MPS file.mps (Mathematical Programming System) is a file format for LO and MIP.
76 to LP/QP/MIP What Platforms Are Available? Linux using g++ version (or later) Windows using Microsoft Visual C++ 6 (or later) Windows using cygwin AIX using xic (not supported in the current Makefile)
77 Features of to LP/QP/MIP Apparently as reliable as OSL Slightly slower that OSL on calculation speed Barrier code is not as mature as the simplex code But, CPL is easier than OSL, free or low cost, and the high ratio of the function to the price.
78 to LP/QP/MIP Benchmark Analysis Comparison Outline 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
79 to LP/QP/MIP Benchmark Analysis Comparison Benchmark Run on a Linux-PC(3.2GHz P4, 4GB RDRAM, Linx-2.6) Version: -10.0; ; ; Times are user times in seconds including input and crossover to a feasible basis for all codes. B/D/P:barrier/daul/primal simplex source:
80 to LP/QP/MIP Benchmark Analysis Comparison Parallel on MIP problems was run in default mode on a single and on a dual processor 2.4GHz Opteron (64-bit, Linux), and a 2.2GHz dual-core Opteron (64-bit,Linux) Times given are elapsed CPU times in seconds. "c" problem convex. AMPL or MPS input. source:
81 to LP/QP/MIP Benchmark Analysis Comparison parallel benchmark shows significant improvement with dual processors Some problems may have dense coefficient matrix bad for parallel computing Commercial packages were more robust and efficient than open-source software BUT there are cases were algorithms failed to converge while and did. There are tests where has reported in-feasible solution while converges IPM is more robust than Primal/Dual for large scale problems
82 to LP/QP/MIP Benchmark Analysis Comparison parallel benchmark shows significant improvement with dual processors Some problems may have dense coefficient matrix bad for parallel computing Commercial packages were more robust and efficient than open-source software BUT there are cases were algorithms failed to converge while and did. There are tests where has reported in-feasible solution while converges IPM is more robust than Primal/Dual for large scale problems
83 to LP/QP/MIP Benchmark Analysis Comparison parallel benchmark shows significant improvement with dual processors Some problems may have dense coefficient matrix bad for parallel computing Commercial packages were more robust and efficient than open-source software BUT there are cases were algorithms failed to converge while and did. There are tests where has reported in-feasible solution while converges IPM is more robust than Primal/Dual for large scale problems
84 to LP/QP/MIP Benchmark Analysis Comparison parallel benchmark shows significant improvement with dual processors Some problems may have dense coefficient matrix bad for parallel computing Commercial packages were more robust and efficient than open-source software BUT there are cases were algorithms failed to converge while and did. There are tests where has reported in-feasible solution while converges IPM is more robust than Primal/Dual for large scale problems
85 to LP/QP/MIP Benchmark Analysis Comparison parallel benchmark shows significant improvement with dual processors Some problems may have dense coefficient matrix bad for parallel computing Commercial packages were more robust and efficient than open-source software BUT there are cases were algorithms failed to converge while and did. There are tests where has reported in-feasible solution while converges IPM is more robust than Primal/Dual for large scale problems
86 to LP/QP/MIP Benchmark Analysis Comparison parallel benchmark shows significant improvement with dual processors Some problems may have dense coefficient matrix bad for parallel computing Commercial packages were more robust and efficient than open-source software BUT there are cases were algorithms failed to converge while and did. There are tests where has reported in-feasible solution while converges IPM is more robust than Primal/Dual for large scale problems
87 to LP/QP/MIP Benchmark Analysis Comparison Outline 1 to LP/QP/MIP (Mixed) Integer Programming Quadratic Programming 2 3 Benchmark Analysis Comparison
88 to LP/QP/MIP Benchmark Analysis Comparison Price Name Commercial Academic Evaluation Call $995 Free student $6,000 $600 Free student 30 day demo $1,750 $175 Free student 15 day demo
89 to LP/QP/MIP Benchmark Analysis Comparison Platform Supported Product solver enginee PC/Windows y y y y y PC/Linux y y y y y Sun/Solaris y y y y y Mac/Mac OS - Power PC - y y Other HP,IBM power HP,AIX Shared Memory Parallel y y y - - source:
90 to LP/QP/MIP Benchmark Analysis Comparison Size of Problems solvable by the system - Internal Max No. of Available Available Product Restrictions Constraints Memory Disk Space - - y - XPRESS Solver Engine - - y - Solver Engine - - y y y - source:
91 to LP/QP/MIP Benchmark Analysis Comparison Algorithms - Primal Dual Interior Branch-and- Branch-and- Product Simplex Simplex Point Cut Price y y y y - XPRESS Solver Engine y y y y y Solver Engine y y y y - y y CPL y y y y - source:
92 to LP/QP/MIP Summary All of them can solve the LP programs, but commercial programs provide more flexible interface; support more modeling systems; provide development tools; provide parallel computation; Parallel computation provide a new way to improve the efficiency and solve large-scale problems, but is limited by the algorithm used and matrix sparsity. Which program is the fastest one?
93 Reference Reference T. Illes, and T. Terlaky Pivot versus interior point methods: Pros and cons European Journal of Operational Research, Volume 140, Number 2, 16 July 2002, pp (21) : XPRESS: : : algorith/implement/lpsolve/implement.shtml : OR/MS Today 2005 Software Survey: Hans Mittelmann s Benchmarks for Optimization Software: Software for Optimization: A Buyer s Guide Optimization Software Guide:
94 Thanks &Questions Please... Thanks! Questions?
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