relaxation

Relaxations and Randomized Methods for Nonconvex QCQPs Alexandre d’Aspremont, Stephen Boyd EE392o, Stanford University Autumn, 2003 1 Introduction While some special classes of nonconvex problems can be efficiently solved, most nonconvex problems are very... More

Relaxations and Randomized Methods for Nonconvex QCQPs Alexandre d’Aspremont, Stephen Boyd EE392o, Stanford University Autumn, 2003 1 Introduction While some special classes of nonconvex problems can be efficiently solved, most nonconvex problems are very difficult to solve (at least, globally). In this set of notes we show how convex optimization can be used to find bounds on the optimal value of a hard problem, and can also be used to find good (but not necessarily optimal) feasible points. We first focus on Lagrangian relaxations, i. e. , using weak duality and the convexity of duals to get bounds on the optimal value of nonconvex problems. In a second section, we show how randomization techniques provide near optimal feasible points with, in some cases, bounds on their suboptimality. 1. 1 Nonconvex QCQPs In this note, we will focus on a specific class of problems: nonconvex quadratically constrained quadratic programs, or nonconvex QCQP (see also §4. 4 in [BV03]). We will see that the Less