Parabolic Relaxation for Quadratically-constrained Quadratic Programming -- Part II: Theoretical & Computational Results
This work provides a theoretical guarantee for an optimization algorithm, which is incremental but important for researchers and practitioners dealing with non-convex QCQPs in fields like system identification.
The authors tackled the problem of solving non-convex quadratically-constrained quadratic programs by proving that their sequential penalized parabolic relaxation algorithm converges to points satisfying Karush-Kuhn-Tucker optimality conditions under certain regularity conditions, and demonstrated its efficiency through numerical experiments on benchmark and large-scale system identification problems.
In the first part of this work [32], we introduce a convex parabolic relaxation for quadratically-constrained quadratic programs, along with a sequential penalized parabolic relaxation algorithm to recover near-optimal feasible solutions. In this second part, we show that starting from a feasible solution or a near-feasible solution satisfying certain regularity conditions, the sequential penalized parabolic relaxation algorithm convergences to a point which satisfies Karush-Kuhn-Tucker optimality conditions. Next, we present numerical experiments on benchmark non-convex QCQP problems as well as large-scale instances of system identification problem demonstrating the efficiency of the proposed approach.