二元多项式优化问题的紧平方和下界

IF 0.8 Q3 COMPUTER SCIENCE, THEORY & METHODS ACM Transactions on Computation Theory Pub Date : 2023-10-17 DOI:10.1145/3626106
Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
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引用次数: 16

摘要

Sakaue, Takeda, Kim和Ito [j] .最优化。[j], 2017]证明了\(\lceil \frac{n+2d-1}{2}\rceil \)在SDP松弛的SoS/Lasserre层次中,半确定(SDP)松弛提供了精确的最优值。当n是奇数时,我们证明他们的分析是紧密的,即我们证明SoS/Lasserre层次的\(\frac{n+2d-1}{2} \)级别也是必要的。劳伦特[数学。哦。Res., 2003]表明Sherali-Adams层次需要n个层次来检测没有积分点的集合的线性表示的空整数壳。她推测同样问题的SoS/Lasserre秩是n−1。本文证明了这一猜想,并推导出秩的下界和上界。
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Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
For binary polynomial optimization problems of degree 2 d with n variables Sakaue, Takeda, Kim and Ito [SIAM J. Optim., 2017] proved that the \(\lceil \frac{n+2d-1}{2}\rceil \) th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e. we prove that \(\frac{n+2d-1}{2} \) levels of the SoS/Lasserre hierarchy are also necessary. Laurent [Math. Oper. Res., 2003] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n − 1. In this paper we disprove this conjecture and derive lower and upper bounds for the rank.
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来源期刊
ACM Transactions on Computation Theory
ACM Transactions on Computation Theory COMPUTER SCIENCE, THEORY & METHODS-
CiteScore
2.30
自引率
0.00%
发文量
10
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