Singleton Optimality in Standard Quadratic Programs with the GOE

TL;DR

This paper analyzes the behavior of the standard quadratic optimization problem over the simplex with GOE matrices, showing that the global optimizer is almost surely a single vertex as the dimension grows.

Contribution

It provides a probabilistic characterization of the support size of the optimizer in quadratic programs with GOE matrices, revealing that the optimizer is typically a single vertex.

Findings

Probability that the optimizer has support size greater than one diminishes as n increases.

The global optimizer is almost surely a single vertex for large n.

Explicit asymptotic probability for the optimizer being a single vertex.

Abstract

We study the standard quadratic optimization problem over the simplex when the objective matrix is drawn from the Gaussian Orthogonal Ensemble (GOE). Let \(\kappa_n\) denote the support size of the almost surely unique global optimizer. We prove \[ \Prob(\kappa_n>1)\sim 2\sqrt{2\pi}\,\frac{\sqrt{\log n}}{n}. \] The proof combines an exact two-coordinate condition for edge improvement with a product formula obtained by conditioning on the diagonal order statistics. Boundary-layer estimates identify the leading contribution and show that supports of size at least three are negligible. Consequently, the minimum-diagonal vertex is globally optimal with probability tending to one, with an explicit first-order correction.