Interplay of non-convex quadratically constrained problems with adjustable robust optimization

Abstract

In this paper we explore convex reformulation strategies for non-convex quadratically constrained optimization problems (QCQPs). First we investigate such reformulations using Pataki’s rank theorem iteratively. We show that the result can be used in conjunction with conic optimization duality in order to obtain a geometric condition for the S-procedure to be exact. Based upon known results on the S-procedure, this approach allows for some insight into the geometry of the joint numerical range of the quadratic forms. Then we investigate a reformulation strategy introduced in recent literature for bilinear optimization problems which is based on adjustable robust optimization theory. We show that, via a similar strategy, one can leverage exact reformulation results of QCQPs in order to derive lower bounds for more complicated quadratic optimization problems. Finally, we investigate the use of reformulation strategies in order to derive characterizations of set-copositive matrix cones. Empirical evidence based upon first numerical experiments shows encouraging results.

Sparse Conic Reformulation of Structured QCQPs based on Copositive Optimization with Applications in Stochastic Optimization

“…1 so that we have a feasible solution for (4) that gives the same objective function value so that in total we have val(4) ≥ val(5) ≥ val(8) ≥ val(4), implying that val(4) = val (5).…”

“…What remains to be shown is that, in case we implemented changes in the objective function mentioned at the beginning, we can undo these changes in the objective function of the reformulation so that we truly arrive at the reformulation stated in the theorem. So consider a feasible solution to (5). We can rearrange the transformed objective function…”

“…A well established framework of achieving such reformulation is based on the set G(F ) := clconv x, xx T : x ∈ F where F is the feasible set of a QCQP and clconv(•) is the is the closure of the convex hull of a set. For examples of such reformulations see [1,5,7,9,10,13,21], where [13, Lemma 6] contains a detailed description of the use of G(F ). These reformulations typically involve set-completely positive matrix cones given by…”