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.
Highlights
Introduction and outlineIn this text we explore the connections between robust optimization and quadratically constrained quadratic optimization
An important strategy for achieving such reformulations is to lift the space of variables, thereby linearizing the quadratic terms and convexifying the feasible set, such that its extreme points correspond to rank-one matrices, which will decompose into vectors feasible to the original problem: Theorem 1 Let F := x ∈ K : xTAi x + aiTx ≤ bi, i ∈ [1 : m] ⊆ Rn be a feasible set of a quadratically constrained quadratic optimization problem (QCQP) and denote by
We show that a class of QCQPs can be reformulated as adjustable robust optimization problems which themselves allow for a bound based on the general reformulation strategy for semi-infinite constraints
Summary
In this text we explore the connections between robust optimization and quadratically constrained quadratic optimization. For the sake of this introduction we briefly review the most important concepts in these areas and give an outline on the aims of this text
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