{"title":"用渐进式 MIP 方法改进带互补约束的不定二次程序和线性程序的求解方法","authors":"Xinyao Zhang, Shaoning Han, Jong-Shi Pang","doi":"arxiv-2409.09964","DOIUrl":null,"url":null,"abstract":"Indefinite quadratic programs (QPs) are known to be very difficult to be\nsolved to global optimality, so are linear programs with linear complementarity\nconstraints. Treating the former as a subclass of the latter, this paper\npresents a progressive mixed integer linear programming method for solving a\ngeneral linear program with linear complementarity constraints (LPCC). Instead\nof solving the LPCC with a full set of integer variables expressing the\ncomplementarity conditions, the presented method solves a finite number of\nmixed integer subprograms by starting with a small fraction of integer\nvariables and progressively increasing this fraction. After describing the PIP\n(for progressive integer programming) method and its various implementations,\nwe demonstrate, via an extensive set of computational experiments, the superior\nperformance of the progressive approach over the direct solution of the\nfull-integer formulation of the LPCCs. It is also shown that the solution\nobtained at the termination of the PIP method is a local minimizer of the LPCC,\na property that cannot be claimed by any known non-enumerative method for\nsolving this nonconvex program. In all the experiments, the PIP method is\ninitiated at a feasible solution of the LPCC obtained from a nonlinear\nprogramming solver, and with high likelihood, can successfully improve it.\nThus, the PIP method can improve a stationary solution of an indefinite QP,\nsomething that is not likely to be achievable by a nonlinear programming\nmethod. Finally, some analysis is presented that provides a better\nunderstanding of the roles of the LPCC suboptimal solutions in the local\noptimality of the indefinite QP.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":"6 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Improving the Solution of Indefinite Quadratic Programs and Linear Programs with Complementarity Constraints by a Progressive MIP Method\",\"authors\":\"Xinyao Zhang, Shaoning Han, Jong-Shi Pang\",\"doi\":\"arxiv-2409.09964\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Indefinite quadratic programs (QPs) are known to be very difficult to be\\nsolved to global optimality, so are linear programs with linear complementarity\\nconstraints. Treating the former as a subclass of the latter, this paper\\npresents a progressive mixed integer linear programming method for solving a\\ngeneral linear program with linear complementarity constraints (LPCC). Instead\\nof solving the LPCC with a full set of integer variables expressing the\\ncomplementarity conditions, the presented method solves a finite number of\\nmixed integer subprograms by starting with a small fraction of integer\\nvariables and progressively increasing this fraction. After describing the PIP\\n(for progressive integer programming) method and its various implementations,\\nwe demonstrate, via an extensive set of computational experiments, the superior\\nperformance of the progressive approach over the direct solution of the\\nfull-integer formulation of the LPCCs. It is also shown that the solution\\nobtained at the termination of the PIP method is a local minimizer of the LPCC,\\na property that cannot be claimed by any known non-enumerative method for\\nsolving this nonconvex program. In all the experiments, the PIP method is\\ninitiated at a feasible solution of the LPCC obtained from a nonlinear\\nprogramming solver, and with high likelihood, can successfully improve it.\\nThus, the PIP method can improve a stationary solution of an indefinite QP,\\nsomething that is not likely to be achievable by a nonlinear programming\\nmethod. Finally, some analysis is presented that provides a better\\nunderstanding of the roles of the LPCC suboptimal solutions in the local\\noptimality of the indefinite QP.\",\"PeriodicalId\":501286,\"journal\":{\"name\":\"arXiv - MATH - Optimization and Control\",\"volume\":\"6 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Optimization and Control\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09964\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Optimization and Control","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09964","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Improving the Solution of Indefinite Quadratic Programs and Linear Programs with Complementarity Constraints by a Progressive MIP Method
Indefinite quadratic programs (QPs) are known to be very difficult to be
solved to global optimality, so are linear programs with linear complementarity
constraints. Treating the former as a subclass of the latter, this paper
presents a progressive mixed integer linear programming method for solving a
general linear program with linear complementarity constraints (LPCC). Instead
of solving the LPCC with a full set of integer variables expressing the
complementarity conditions, the presented method solves a finite number of
mixed integer subprograms by starting with a small fraction of integer
variables and progressively increasing this fraction. After describing the PIP
(for progressive integer programming) method and its various implementations,
we demonstrate, via an extensive set of computational experiments, the superior
performance of the progressive approach over the direct solution of the
full-integer formulation of the LPCCs. It is also shown that the solution
obtained at the termination of the PIP method is a local minimizer of the LPCC,
a property that cannot be claimed by any known non-enumerative method for
solving this nonconvex program. In all the experiments, the PIP method is
initiated at a feasible solution of the LPCC obtained from a nonlinear
programming solver, and with high likelihood, can successfully improve it.
Thus, the PIP method can improve a stationary solution of an indefinite QP,
something that is not likely to be achievable by a nonlinear programming
method. Finally, some analysis is presented that provides a better
understanding of the roles of the LPCC suboptimal solutions in the local
optimality of the indefinite QP.