Rounding on the standard simplex: regular grids for global optimization

Abstract

Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all p-norms for p ≥ 1. We show that the minimal p-distance to the regular grid on the standard simplex can exceed one, even for very fine mesh sizes in high dimensions. Furthermore, for p = 1, the maximum minimal distance approaches the 1-diameter of the standard simplex. We also put our results into perspective with respect to the literature on approximating global optimization problems over the standard simplex by means of the regular grid.

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

“…The results in this note complement a growing literature on the complexity of polynomial optimization and interpolation on a simplex; see [3,4,5,7,8,9,10,11] and the references therein.…”

BGEN: a binary file format for imputed genotype and haplotype data

“…exactly encodable in b binary digits. An algorithm for finding q was given by Bomze et al 18 . Given q, we encode the probability data for the sample by storing q1,…,qS-1 in consecutive bits of the file, for a total storage of b(S-1) bits.…”

“…Thus, the minimum number of bits per probability required for a given level of accuracy ε is b=ceil(log2(1+1/ε)). For example, hard-called genotype data can be faithfully stored using b=1; while the values b=5, 8,11,15,18 give approximately one, two, three, four, and five decimal places of accuracy respectively. (Based on the results below, we recommend using b=8 or above for imputed datasets).…”

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

“…For quadratic f ∈ H n,2 , Vavasis [18] shows that problem (1) admits a rational global minimizer x * , whose bit-size is polynomial in the bit-size of the input data. On the other hand, when the degree of f is larger than 2, there exist polynomials f for which problem (1) does not have any rational global minimizer. This is the case, for instance, for the polynomial f (x) = 2x 1 3 − x 1 ( n i=1 x i ) 2 , whose global minimizer always has the irrational component x 1 = 1/ √ 6.…”