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

Abstract

We consider the approximate minimization of a given polynomial on the standard simplex, obtained by taking the minimum value over all rational grid points with given denominator r ∈ N. It was shown in [De Klerk, E., Laurent, M., Sun, Z.: An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution. SIAM J. Optim. 25 (3) 1498-1514 (2015)] that the accuracy of this approximation depends on r as O(1/r 2 ) if there exists a rational global minimizer. In this note we show that the rational minimizer condition is not necessary to obtain the O(1/r 2 ) bound.

Two methods for the maximization of homogeneous polynomials over the simplex

“…There is a unique (global) maximizer x = ( 1 3 , 1 3 , 1 3 ) which trivially satisfies the SC condition, cf., (12). Thus according to Theorem 2 the replicator iteration (20) should always show a linear convergence when starting with x (0) > 0.…”

“…Then the sequence x (k) converges to x , i.e., the point x is the only limit point. (b) Suppose x is a KKT point of P satisfying the second order condition SSOC, as well as the SC condition (12). Then there is some 1 > 0 such that the following is true: The point x is the only fixed point x ∈ Δ n of y(x) with ‖x − x‖ < 1 , and for any starting point x (0) ∈ Δ n , with ‖x (0) − x‖ < 1 the replicator iterates (20) converge to x.…”

“…It is well-known that for d ≥ 2 the global maximization problem P is NP-hard (cf., [19]). Approximation methods for computing a global maximizer have been investigated in [2] for d = 2 , and in [12,13] for d > 2 . However, in the present article we are less interested in the solution of the global maximization problem but we are looking for an algorithm to compute local maximizers of P. For this purpose we focus our attention to the so-called replicator dynamics (cf., (20) below).…”

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

“…where C d is a constant that depends only on the degree d of p; see [17] for K = ∆ n and [13] for K = [0, 1] n . A faster regime in O(1/r 2 ) can be shown when allowing a constant that depends on the polynomial p (see [20] for ∆ n and [12] for [0, 1] n ). Note that the number of rational points with denominator r in the simplex ∆ n is n+r−1 r = O(n r ) and thus the computation time for these upper bounds is polynomial in the dimension n for any fixed order r. On the other hand, there are (r +1) n = O(r n ) such grid points in the hypercube [0, 1] n and thus the computation time of the upper bounds grows exponentially with the dimension n.…”

“…where C d is a constant that depends only on the degree d of p; see [17] for K = ∆ n and [13] for K = [0, 1] n . A faster regime in O(1/r 2 ) can be shown when allowing a constant that depends on the polynomial p (see [20] for ∆ n and [12] for [0, 1] n ).…”