On sparsity of the solution to a random quadratic optimization problem

TL;DR

This paper investigates the sparsity of solutions to large random quadratic optimization problems on the simplex, showing that solutions are typically sparse with support size polylogarithmic in the problem dimension under broad distributional assumptions.

Contribution

The paper extends previous results on solution sparsity to a wider class of distributions, including those with super/sub-exponentially narrow tails, and for non-symmetric matrices, establishing polylogarithmic support size.

Findings

Solutions are sparse with support size polylogarithmic in dimension.

Sparsity results hold for broader distribution classes, including heavy-tailed distributions.

Support size estimates are derived using optimality conditions and tail distribution bounds.

Abstract

The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form $\bold x^TQ\bold x$ on the standard simplex $\{\bold x\ge\bold 0: \bold x^T\bold e=1\}$, is studied. The StQP arises in numerous applications, and it is known to be NP-hard. The first author, Peng and Zhang~\cite{int:Peng-StQP} showed that almost certainly the StQP with a large random matrix $Q=Q^T$, whose upper-triangular entries are i. i. concave-distributed, attains its minimum at a point with few positive components. In this paper we establish sparsity of the solution for a considerably broader class of the distributions, including those supported by $(-\infty,\infty)$, provided that the distribution tail is (super/sub)-exponentially narrow, and also for the matrices $Q=(M+M^T)/2$, when $M$ is not symmetric. {The likely support size in those cases is shown to be polylogarithmic in $n$, the problem dimension.} Following~\cite{int:Peng-StQP} and Chen and Peng ~\cite{ChenPeng2015}, the key ingredients are the first and second order optimality conditions, and the integral bound for the tail distribution of the solution support size. To make these results work for our goal, we obtain a series of estimates involving, in particular, the random interval partitions induced by the order statistics of the elements $Q_{i,j}$.