On sparsity of the solution to a random quadratic optimization problem

Abstract

The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form $$\mathbf{x}^TQ\mathbf{x}$$ on the standard simplex $$\{\mathbf{x}\ge \mathbf{0}: \mathbf{x}^T\mathbf{e}=1\}$$ , is studied. The StQP arises in numerous applications, and it is known to be NP-hard. Chen, Peng and Zhang showed that almost certainly the StQP with a large random matrix $$Q=Q^T$$ , $$Q_{i,j},\, (i\le j)$$ being independent and identically concave-distributed, attains its minimum at a point $$\mathbf{x}$$ with support size $$|\{j: x_j>0\}|$$ bounded in probability. Later Chen and Peng proved that for $$Q=(M+M^T)/2$$ , with $$M_{i,j}$$ i.i.d. normal, the likely support size is at most 2. In this paper we show that the likely support size is poly-logarithmic in n, the problem size, for a considerably broader class of the distributions. Unlike the cited papers, the mild constraints are put on the asymptotic behavior of the distribution at a single left endpoint of its support, rather than on the distribution’s shape elsewhere. It also covers the distributions with the left endpoint $$-\infty $$ , provided that the distribution of $$Q_{i,j},\, (i\le j)$$ (of $$M_{i,j}$$ , if $$Q=(M+M^T)/2$$ resp.) has a (super/sub) exponentially narrow left tail.