On sparsity of the solution to a random quadratic optimization problem

The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form x^TQx on the standard simplex { x≥ 0: x^Te= 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, Qi,j,(i≤j) being independent and identically concave-distributed, attains its minimum at a point 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 - ∞, provided that the distribution of Qi,j,(i≤j) (of M_i,j, if Q= (M+ M^T) / 2 resp.) has a (super/sub) exponentially narrow left tail.