New Analysis on Sparse Solutions to Random Standard Quadratic Optimization Problems and Extensions

Abstract

The standard quadratic optimization problem (StQP) refers to the problem of minimizing a quadratic form over the standard simplex. Such a problem arises from numerous applications and is known to be NPhard. In a recent paper [15], we showed that with a high probability close to 1, StQPs with random data have sparse optimal solutions when the associated data matrix is randomly generated from a certain distribution such as uniform and exponential distributions. In this paper, we present a new analysis for random StQPs combining probability inequalities derived from both the first-order and second-order optimality conditions. The new analysis allows us to significantly improve the probability bounds. More important, it allows us to handle normal distributions which is left open in [15]. The existence of sparse approximate solutions to convex StQPs and extensions to other classes of QPs are discussed as well.

On sparsity of the solution to a random quadratic optimization problem

“…Thus, a good understanding of the behavior of the optimal solutions under randomly generated instances may shed light on the behaviors of various algorithms tested on these instances. Indeed, our results, together with those in [10] and [11], establishing the sparsity of the optimal solutions of randomly generated StQPs under quite general distribution assumptions, indicate that the performance of algorithms tested on these instances must be carefully analyzed before any general statement can be made. Interestingly, motivated by the sparsity of the optimal solutions, Bomze et al [6] construct StQP instances with a rich solution structure.…”

“…The surprisingly short proof of this general claim follows from Theorem 3 in [11]. With additional, mildly restrictive, constraints, we are able to reduce, significantly, the likely range of the support size.…”

“…a = −1, b = 2, both the minimum point and the maximum point are sparse, with the likely support size of order (log n) 3/2 in each case. As we mentioned, Chen and Peng [11] proved that for the GOE matrix Q = (M +M T )/2, M i,j being i.i.d. exactly normal, whp the support size of X * is 2, at most.…”

“…The core analysis in [10] is based on the estimates coming from the firstorder optimality condition on X * and some probabillistic bounds on the order statistics for the attendant random variables. The further analysis in [11] relied in addition on the second-order optimality condition.…”

“…The likely support size in those cases is shown to be polylogarithmic in n, the problem dimension. Following [10] and Chen and Peng [11], 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 Qi,j.where Q = [Q ij ] ∈ ℜ n×n is a symmetric matrix, and e ∈ ℜ n is the all 1-vector.…”