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In this paper, we give a new penalized semidefinite programming approach fornon-convex quadratically-constrained quadratic programs (QCQPs). We incorporatepenalty terms into the objective of convex relaxations in order to retrievefeasible and near-optimal solutions for non-convex QCQPs. We introduce ageneralized linear independence constraint qualification (GLICQ) criterion andprove that any GLICQ regular point that is sufficiently close to the feasibleset can be used to construct an appropriate penalty term and recover a feasiblesolution. Inspired by these results, we develop a heuristic sequentialprocedure that preserves feasibility and aims to improve the objective value ateach iteration. Numerical experiments on large-scale system identificationproblems as well as benchmark instances from the library of quadraticprogramming (QPLIB) demonstrate the ability of the proposed penalizedsemidefinite programs in finding near-optimal solutions for non-convex QCQP.

Penalized Semidefinite Programming for Quadratically-Constrained Quadratic Optimization