{"title":"T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization","authors":"Hiroki Marumo, Sunyoung Kim, Makoto Yamashita","doi":"10.1007/s10589-024-00582-8","DOIUrl":null,"url":null,"abstract":"<p>We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng et al. (JGO 84:415–440, 2022). In this work, we propose a T-SDP relaxation for POPs with polynomial inequality constraints and show that the resulting T-SDP relaxation formulated with third-order tensors can be transformed into the standard SDP relaxation with block-diagonal structures. The convergence of the T-SDP relaxation to the optimal value of a given constrained POP is established under moderate assumptions as the relaxation level increases. Additionally, the feasibility and optimality of the T-SDP relaxation are discussed. Numerical results illustrate that the proposed T-SDP relaxation enhances numerical efficiency.</p>","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":null,"pages":null},"PeriodicalIF":1.6000,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Optimization and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10589-024-00582-8","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng et al. (JGO 84:415–440, 2022). In this work, we propose a T-SDP relaxation for POPs with polynomial inequality constraints and show that the resulting T-SDP relaxation formulated with third-order tensors can be transformed into the standard SDP relaxation with block-diagonal structures. The convergence of the T-SDP relaxation to the optimal value of a given constrained POP is established under moderate assumptions as the relaxation level increases. Additionally, the feasibility and optimality of the T-SDP relaxation are discussed. Numerical results illustrate that the proposed T-SDP relaxation enhances numerical efficiency.
期刊介绍：
Computational Optimization and Applications is a peer reviewed journal that is committed to timely publication of research and tutorial papers on the analysis and development of computational algorithms and modeling technology for optimization. Algorithms either for general classes of optimization problems or for more specific applied problems are of interest. Stochastic algorithms as well as deterministic algorithms will be considered. Papers that can provide both theoretical analysis, along with carefully designed computational experiments, are particularly welcome.
Topics of interest include, but are not limited to the following:
Large Scale Optimization,
Unconstrained Optimization,
Linear Programming,
Quadratic Programming Complementarity Problems, and Variational Inequalities,
Constrained Optimization,
Nondifferentiable Optimization,
Integer Programming,
Combinatorial Optimization,
Stochastic Optimization,
Multiobjective Optimization,
Network Optimization,
Complexity Theory,
Approximations and Error Analysis,
Parametric Programming and Sensitivity Analysis,
Parallel Computing, Distributed Computing, and Vector Processing,
Software, Benchmarks, Numerical Experimentation and Comparisons,
Modelling Languages and Systems for Optimization,
Automatic Differentiation,
Applications in Engineering, Finance, Optimal Control, Optimal Design, Operations Research,
Transportation, Economics, Communications, Manufacturing, and Management Science.
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