New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability

Abstract

A standard quadratic optimization problem (StQP) consists in minimizing a quadratic form over a simplex. Among the problems which can be transformed into a StQP are the general quadratic problem over a polytope, and the maximum clique problem in a graph. In this paper we present several new polynomial-time bounds for StQP ranging from very simple and cheap ones to more complex and tight constructions. The main tools employed in the conception and analysis of most bounds are Semidefinite Programming and decomposition of the objective function into a sum of two quadratic functions, each of which is easy to minimize. We provide a complete diagram of the dominance, incomparability, or equivalence relations among the bounds proposed in this and in previous works. In particular, we show that one of our new bounds dominates all the others. Furthermore, a specialization of such bound dominates Schrijver's improvement of Lovasz's theta function bound for the maximum size of a clique in a graph

Copositivity and constrained fractional quadratic problems

“…is feasible to (14), then Z − αI n+1 ∈ C * n+1 for a small α > 0, and in particular this matrix is psd. But…”

“…Theorem 2 Under the model assumptions (10), the dual problem (17) is strictly feasible (i.e., Slater's condition is satisfied). Hence the duality gap is zero, and the primal problem (14) has always an optimal solution, that is, ψ = λ * = C • Z * for some Z * feasible to (14).…”

“…For instance, a study with the standard quadratic problem reveals the dominance of copositivity based bounds over alternative techniques [14]. Although copositive matrices have been studied for long in linear algebra [29], direct applications in optimization are relatively recent.…”

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

“…The problem appears in numerous applications such as resource allocation [26], portfolio selection [30] and machine learning [32]. It also covers several other important problems such as the maximal clique problem in discrete optimization [20] and the determination of co-positivity of a matrix in linear algebra [9]. In addition to its broad range of applications, the StQP provides a prototype for numerous classes of quadratic optimization problems.…”

“…As such, the study of StQP has caught the attention of experts in many different fields and various algorithms have been proposed in the literature. For details, we refer to recent papers [9,36,41] and the references therein.…”