{"title":"On parametric semidefinite programming with unknown boundaries","authors":"Jonathan D. Hauenstein , Tingting Tang","doi":"10.1016/j.jsc.2024.102324","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study parametric semidefinite programs (SDPs) where the solution space of both the primal and dual problems change simultaneously. Given a bounded set, we aim to find the <em>a priori</em> unknown maximal permissible perturbation set within it where the semidefinite program problem has a unique optimum and is analytic with respect to the parameters. Our approach reformulates the parametric SDP as a system of partial differential equations (PDEs) where this maximal analytical permissible set (MAPS) is the set on which the system of PDEs is well-posed. A sweeping Euler scheme is developed to approximate this <em>a priori</em> unknown perturbation set. We prove local and global error bounds for this second-order sweeping Euler scheme and demonstrate the method in comparison to existing SDP solvers and its performance on several two-parameter and three-parameter SDPs for which the MAPS can be visualized.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Symbolic Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0747717124000282","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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Abstract
In this paper, we study parametric semidefinite programs (SDPs) where the solution space of both the primal and dual problems change simultaneously. Given a bounded set, we aim to find the a priori unknown maximal permissible perturbation set within it where the semidefinite program problem has a unique optimum and is analytic with respect to the parameters. Our approach reformulates the parametric SDP as a system of partial differential equations (PDEs) where this maximal analytical permissible set (MAPS) is the set on which the system of PDEs is well-posed. A sweeping Euler scheme is developed to approximate this a priori unknown perturbation set. We prove local and global error bounds for this second-order sweeping Euler scheme and demonstrate the method in comparison to existing SDP solvers and its performance on several two-parameter and three-parameter SDPs for which the MAPS can be visualized.
期刊介绍：
An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects.
It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.
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