Eigenvalue techniques for proving bounds for convex objective, nonconvex programs

• F (x) is a convex quadratic, i.e. F (x) = xTMx + vTx (with M 0 and v ∈ Rn). • P ⊆ Rn is a convex set over which we can efficiently optimize F , • K ⊆ Rn is a non-convex set with “special structure”. • Typically, n could be quite large. A standard approach to solving this problem would start by solving a convex relaxation to F , thereby obtaining a lower bound on F z. However, when K is complex, such a lower bound is likely to be weak. In this paper we present efficient techniques that enable us to tighten the lower bound. Our techniques are backed by theory and also prove computationally effective – our approach yields bounds comparable to or better than those produced by sophisticated formulations, but at a very small fraction of the computational cost. To illustrate the situation we have in mind, we focus next on the example where P = {x ∈ Rn : Ax ≥ b} for a given matrix A and vector b, and K is given by a cardinality constraint, i.e.