The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's). In particular, we show that our approach leads to a polynomial-time approximation scheme for the standard quadratic optimization problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3-approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.
I. Bomze, E. de Klerk. Solving standard quadratic optimization problems via semidefinite and copositive programming. J. Global Optimization, Volume 24, Issue 2, pp. 163-185, 2002.