An Error Analysis for Polynomial Optimization over the Simplex Based on the Multivariate Hypergeometric Distribution

Abstract

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. [9] and improves on previously known O(1/r) bounds in the quadratic case.

Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization

“…This concludes the proof for relation (8), and relation (9) follows from (19) in an analogous way. This finishes the proof of Theorem 3.…”

“…This approach has been investigated, in particular, for minimization over the standard simplex and when selecting some discrete distributions over the grid points in the simplex. The multinomial distribution is used in [7,24] to show convergence in O(1/r ) and the multivariate hypergeometric distribution is used in [8] to show convergence in O(1/r 2 ) for quadratic minimization over the simplex (and in the general case assuming a rational minimizer exists).…”

A Survey of Semidefinite Programming Approaches to the Generalized Problem of Moments and Their Error Analysis

“…A first observation made in [36] is that this semidefinite program (19) can in fact be reformulated as a generalized eigenvalue problem. Indeed, its dual semidefinite program reads max{λ : A 0 − λA 1 0}, whose optimal value gives again the parameter val (r) inner (since strong duality holds).…”

“…2r that is orthonormal with respect to the reference measure µ 0 (i.e., such that ∫ K b α b β dµ 0 = 1 if α = β and 0 otherwise), then in the above semidefinite program (19) we may set A 1 = I to be the identity matrix and…”

On the convergence rate of grid search for polynomial optimization over the simplex

“…On the other hand if, as opposed to the PTAS property, one is only interested in the dependence of the accuracy f min,∆(n,r) − f min,∆n on r, then one may obtain O(1/r 2 ) bounds, as shown in [9]. Here the constant in the big-O notation may depend on the polynomial f .…”