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Mathematics > Optimization and Control

arXiv:1407.2108 (math)
[Submitted on 8 Jul 2014]

Title:An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution

Authors:Etienne de Klerk, Monique Laurent, Zhao Sun
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Abstract:We study the minimization of fixed-degree polynomials over the simplex. This problem is well-known to be NP-hard, as it contains the maximum stable set problem in graph theory as a special case. In this paper, we consider a rational approximation by taking the minimum over the regular grid, which consists of rational points with denominator $r$ (for given $r$). We show that the associated convergence rate is $O(1/r^2)$ for quadratic polynomials. For general polynomials, if there exists a rational global minimizer over the simplex, we show that the convergence rate is also of the order $O(1/r^2)$. Our results answer a question posed by De Klerk et al. (2013) and improves on previously known $O(1/r)$ bounds in the quadratic case.
Comments: 17 pages
Subjects: Optimization and Control (math.OC)
MSC classes: 90C26, 90C30
Cite as: arXiv:1407.2108 [math.OC]
  (or arXiv:1407.2108v1 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.1407.2108

Submission history

From: Zhao Sun [view email]
[v1] Tue, 8 Jul 2014 14:32:49 UTC (16 KB)
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