In this paper, we give a new penalized semidefinite programming approach for non-convex quadratically-constrained quadratic programs (QCQPs). We incorporate penalty terms into the objective of convex relaxations in order to retrieve feasible and near-optimal solutions for non-convex QCQPs. We introduce a generalized linear independence constraint qualification (GLICQ) criterion and prove that any GLICQ regular point that is sufficiently close to the feasible set can be used to construct an appropriate penalty term and recover a feasible solution. Inspired by these results, we develop a heuristic sequential procedure that preserves feasibility and aims to improve the objective value at each iteration. Numerical experiments on large-scale system identification problems as well as benchmark instances from the library of quadratic programming demonstrate the ability of the proposed penalized semidefinite programs in finding near-optimal solutions for non-convex QCQP.

在本文中，我们为非凸二次约束二次程序（QCQP）提供了一种新的惩罚半定规划方法。我们将惩罚项纳入凸松弛的目标中，以检索非凸QCQP的可行和接近最优的解决方案。我们引入了广义线性独立约束限定（GLICQ）准则，并证明了足够接近可行集的任何GLICQ正则点都可用于构造适当的惩罚项并恢复可行解。受这些结果的启发，我们开发了一种启发式顺序过程，该过程保留了可行性，并旨在提高每次迭代的目标价值。