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SparsePOP (A Sparse SemiDefinite Programming Relaxation of Polynomial Optimization Problems) Hayato Waki, Sunyoung Kim, Masakazu Kojima, Masakazu Muramatsu, Hiroshi Sugimoto and Makoto Yamashita ---------------------------------------- Index 1. Overview 2. System Requirements 3. Installation Guide 4. Usage of SparsePOP 5. Acknowledgements 6. License 7. E-mail address 8. History ---------------------------------------- 1. Overview SparesPOP is a MATLAB package for a sparse semidefinite programming (SDP) relaxation method for approximating a global optimal value and solution of a polynomial optimization problem (POP) proposed by Waki et al. The sparse SDP relaxation exploits a sparse structure of polynomials in POPs when applying "a hierarchy of LMI relaxations of increasing dimensions" by Lasserre. The efficiency of SparsePOP to approximate optimal solutions of POPs is thus increased, and larger scale POPs can be handled. Two versions of SparsePOP are available in this package: (A) MATLAB only version (B) MATLAB with C++ version (A) will always be installed, while (B) is optional. The functions and capabilities of the two versions are identical. Some of time-consuming parts of (B) is coded in C++, thus, it is faster in generating SDP problems than (A). We recommend to use (B) whenever possible, although in many cases (A) is sufficient. 2. System Requirements The following software packages are required for SparsePOP. i. MATLAB R2009b or later for Mac OSX MATLAB R2008a or later for Linux ii. SeDuMi, SDPT3, SDPA or CDP to call SeDuMi from SparsePOP for solving an SDP relaxation problem If you want to use (B), then you need appropriate C++ compilers, compatible with MATLAB, to compile the programs. NOTE: If Symbolic Math Toolbox is available on your computer, SparsePOP can handle a gms file which contains parentheses, e.g. Bex2_1_2.gms. NOTE2: If Optimization Toolbox 4.0 or later is available on your computer, you can refine approximated solutions of SparsePOP by using some functions in Optimization Toolbox. See Section 3.4 in User Manual for more detail. NOTE3: We recommend to install Xcode to compile mex files if you use SparsePOP on Mac OSX. In particular, llvm-gcc and llvm-g++ in Xcode are available for compiling mex files in SparsePOP. See the following webpage for the detail: http://www.mathworks.com/matlabcentral/answers/103258-mex-on-mavericks-with-r2012b NOTE4: SeDuMi in Github (2014-08-14) has two bugs to use it. We list two modifications of SeDuMi: (1) Remove % in the head of the 792nd line in sedumi.m (2) Replace the 77th line in eigK.m by lab(li+1:li+nl) = x(xi+1:xi+nl); 3. Installation Guide See install.txt in the top directory of SparsePOP. After installing SparsePOP, add the path of SparsePOP in your MATLAB search path. 4. Usage of SparsePOP We assume that SDPA and/or SeDuMi have been already installed on your computer and that the paths of their folders and subfolders have been already added in your Matlab search path. The usage of SparsePOP is explained in the user manual in the top directory of SparsePOP. 5. Acknowledgments The authors are grateful to Dr. Mevissen Martine, Mr. Dan Gugenheim, Mr. Victor Magron, Dr. Hanne Ackermann and Mr. Ye Wang for bug reports. 6. License This software is distributed under GNU General Public License 2.0 7. E-mail address kojima-spop@is.titech.ac.jp Please send a message if you have any questions, ideas, or bug reports. 8. History Version 3.00 (September 30, 2014) (1) SparsePOP returns a lucid message when it cannot allocate an array due to the integer overflow. (2) Modified simplifyPolynomial.m. Since output of built-in function `unique' in MATLAB changed from R2013a or later, this function returned a wrong result. (3) Implemented a reduction method described in the following paper: ``AN EXTENSION OF THE ELIMINATION METHOD FOR A SPARSE SOS POLYNOMIALby H. Waki and M. Muramatsu'' This method works in only MATLAB by setting param.reduceMomentMatSW = 2. ** This method works in only MATLAB when param.reduceMomentMatSW = 2 ** ** is set. We recommend to use this parameter for small POPs with ** ** the number of variables involved in POP up to 70. ** (4) Add a parameter for reducing the moment matrices more aggressively. Any candidate for solution of POP may not be retrieved, while a smaller SDP relaxation problems may be obtained. In fact, this reduction may be remove some SDP variables which correspond to variables in the original POP. ** This reduction works in both MATLAB and C++ when ** ** param.aggressiveSW = 1 ** ** Combining param.reduceMomentMatSW = 1 or 2 with this, a much ** ** smaller SDP relaxation may be obatined. ** (5) Implemented an SDP relaxation described in the following paper: ``A Perturbed Sums of Squares Theorem for Polynomial Optimization and its Applications by H. Waki, M. Muramatsu and L. Tuncel'' ** This method works in only MATLAB by setting param.sparseSW = 2 ** ** and 3. ** ** param.sparseSW = 2 for a smaller SDP relax. based on Lasserre ** ** param.sparseSW = 3 for a smaller SDP relax. based on Sparse SDP ** ** relaxation. ** (6) Fix bugs reported by users Version 2.99 (February 15, 2012) (1) SparsePOP can be compiled by mex on Windows 7 with Visual Stadio 2010. (2) Some elements of the structure SDPinfo returned from SDPrelaxation.m and SDPrelaxationMex.m are changed: [Add] SDPinfo.SeDuMiA: the coefficient matrix A in the SDP with the SeDuMi format SDPinfo.SeDuMib: the coefficient vector b in the SDP with the SeDuMi format SDPinfo.SeDuMic: the objective vector c in the SDP with the SeDuMi format SDPinfo.SeDuMiK: the cone K in the SDP with the SeDuMi format SDPinfo.objeConstant: trans.objVonstant SDPinfo.objValScale: trans.objValScale SDPinfo.Amat: trans.Amat SDPinfo.bVect: trans.bVect [Delete] SDPinfo.K, SDPinfo.A, SDPinfo.b, SDPinfo.c, SDPinfo.K.f, SDPinfo.J. (3) We checked that SparsePOP can be compiled under a Windows machine with Microsoft Visual C++ 7.0 or later. (4) The previous version of readGMS.m recognized the expression `+ - 0.5' as 0.5, not -0.5. We fixed this bug. The current version read it as `-0.5'. (5) symamd in SDPrelaxationMex.m returns a segmentation fault for a POP with unused variables. We fixed this bug. Version 2.98 (December 15, 2011) (1) Fixed a bug in readGMS.m. The previous version could not read the monomials which are located after the keyword ``objvar". (2) Fixed a bug in deleteVar.m. The previous version might substitute some fixed values incorrectly for POPs whose lower and upper bounds are identical. (3) Some functions for scaling POPs with polynomial sdp constraints do not work in the current version. (4) SparsePOP with param.mex=1 returns the same format as param.mex=0 when users set param.detailedInfFile. Version 2.97 (September 30, 2011) (1) Fixed a bug in substituteEq.m. (2) Improved output of information on SDPNAL in solveBySDPNAL.m and SDPT3 in solveBySDPT3.m. Version 2.96 (August 31, 2011) (1) Fixed bugs in compileSparsePOP.m and gen_basisindecies in conversion.cpp. (2) Improved the part of checking the linear independency of the coefficient matrix A. Version 2.95 (July 30, 2011) (1) For POPs which have constraints "x in {0,1}" and/or "x in {-1, +1}", SparsePOP can reduce the size of the resulting SDP relaxation problem by exploiting their constraints. This reduction is based on the following paper: J.B.Lasserre, "An explicit equivalent positive semidefinite program for nonlinear 0-1 programs", SIAM J.Optim., 12, pp. 756--769. * For the reduction for a POP which has constraint "x in {0, 1}", set param.binarySW = 1. * For the reduction for a POP which has constraint "x in {-1, +1}", set param.SquareOneSW = 1. However, these reductions do not work when user computes error bounds of an approximated solution obtained by SparsePOP. (2) The reductions executed by reduceMomentMatSW = 1 and/or reduceAMatSW = 1 work when user computes error bounds of an approximated solution obtained by SparsePOP. (3) We move "class pop_params" from conversion.h into Parameters.h. (4) Some bugs are fixed. Version 2.90 (June 21, 2011) (1) SparsePOP can read the reserved keyword "maximizing objvar" and "binary variables". If an input file contains the keyword "maximizing objvar", then SparsePOP deals with the POP as the maximization problem. In addition, if an input file contains the keyword "binary variables", then SparsePOP adds the polynomial equality x(i)^2 -x(i) = 0 in the original POP for such variables x(i). (2) Some gams files in GLOBAL Library are added in the directory example/GMSformat. (3) In the case where some variables in SDP relaxation problem corresponding to variables in a given POP are removed by setting param.reduceMomentMatSW = 1, SparsePOP returns only the optimal value of the resulting SDP relaxation problem. (In the previous version, for such a POP, SparsePOP did not solve the SDP relaxation problem.) (4) Some bugs are fixed. Version 2.85 (May 23, 2011) (1) Some bugs in C++ files are fixed. The latest sparsepop can handle POPs which contain polynomial matrix inequalities by setting param.mex = 1. Version 2.80 (Feburary 7, 2011) (1) Users can use SDPA, SeDuMi, SDPT3, CSDP and SDPNAL for solving SDP relaxation problems. Users can tune up paramters of the SDP solvers by editing files in the subdirectory SparsePOP280/subPrograms/Mfiles/Solvers. (2) Small bugs in C++ files are fixed. Version 2.60 (May 9, 2010) (1) Add some functions for computing error bounds based on the paper M. Kojima and M. Yamashita, "Enclosing Ellipsoids and Elliptic Cylinders of Semialgebraic Sets and Their Application to Error Bounds in Polynomial Optimization", November 2009. Version 2.50 (March 30, 2010) (1) readGMS.m can print lucid error messages for some incorrect gms files. (2) The directory example/SDPAformat is removed. (3) Some bugs are fixed. Version 2.20 (August 20, 2009) (1) Users can use SDPA instead of SeDuMi for solving an SDP relaxation problem. See `readmeRevision.txt' for more detail. Version 2.15 (June 22, 2009) (1) The function for writing an SDP as the sdpa sparse format is modified and included into the function solveBySeDuMi.m (2) SparsePOP outputs a file which contains information on scaling for a given POP. See `readmeRevision.txt' for more detail. Version 2.10 (April 3, 2009) (1) SparsePOP can accept a new input format for a constrained nonlinear least square problem. (2) To refine approximated solution obtained by SparsePOP, users can call some functions in Optimization Toolbox. See `readmeRevision.txt' for more detail. Version 2.00 (June, 2007) (1) For speedup, C++ subroutines are developed. (2) The second version (B) of SparsePOP are added in SparsePOP. (3) Improve user interface. (4) Some bugs are fixed. Version 1.20 (September, 2005)
Source: readme.txt, updated 2014-09-28