二元多项式优化的伪布尔多面体和多项式大小扩展公式

IF 2.2 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Mathematical Programming Pub Date : 2024-07-22 DOI:10.1007/s10107-024-02122-y
Alberto Del Pia, Aida Khajavirad
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引用次数: 0

摘要

为了获得二元多项式优化问题的强放松,我们引入了伪布尔多面体,它被定义为满足一系列等式的二元点的集合(z in \{0,1\}^{V \cup S}\) (z_s = \prod _{v \in s} )。\通过用有符号的超图来表示伪布尔多面体,我们得到了该多面体具有多项式大小的扩展表述的充分条件。我们的新框架统一并扩展了之前关于至少三度二元多项式优化问题可行区域凸壳的多项式大小扩展公式存在性的所有结果。
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The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the set of binary points \(z \in \{0,1\}^{V \cup S}\) satisfying a collection of equalities of the form \(z_s = \prod _{v \in s} \sigma _s(z_v)\), for all \(s \in S\), where \(\sigma _s(z_v) \in \{z_v, 1-z_v\}\), and where S is a multiset of subsets of V. By representing the pseudo-Boolean polytope via a signed hypergraph, we obtain sufficient conditions under which this polytope has a polynomial-size extended formulation. Our new framework unifies and extends all prior results on the existence of polynomial-size extended formulations for the convex hull of the feasible region of binary polynomial optimization problems of degree at least three.

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来源期刊
Mathematical Programming
Mathematical Programming 数学-计算机:软件工程
CiteScore
5.70
自引率
11.10%
发文量
160
审稿时长
4-8 weeks
期刊介绍: Mathematical Programming publishes original articles dealing with every aspect of mathematical optimization; that is, everything of direct or indirect use concerning the problem of optimizing a function of many variables, often subject to a set of constraints. This involves theoretical and computational issues as well as application studies. Included, along with the standard topics of linear, nonlinear, integer, conic, stochastic and combinatorial optimization, are techniques for formulating and applying mathematical programming models, convex, nonsmooth and variational analysis, the theory of polyhedra, variational inequalities, and control and game theory viewed from the perspective of mathematical programming.
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