Feasible Point Pursuit and Successive Approximation of Non-Convex QCQPs

Abstract

Quadratically constrained quadratic programs (QC-QPs) have a wide range of applications in signal processing and wireless communications. Non-convex QCQPs are NP-hard in general. Existing approaches relax the non-convexity using semidefinite relaxation (SDR) or linearize the non-convex part and solve the resulting convex problem. However, these techniques are seldom successful in even obtaining a feasible solution when the QCQP matrices are indefinite. In this paper, a new feasible point pursuit successive convex approximation (FPP-SCA) algorithm is proposed for non-convex QCQPs. FPP-SCA linearizes the non-convex parts of the problem as conventional SCA does, but adds slack variables to sustain feasibility, and a penalty to ensure slacks are sparingly used. When FPP-SCA is successful in identifying a feasible point of the non-convex QCQP, convergence to a Karush-Kuhn-Tucker (KKT) point is thereafter ensured. Simulations show the effectiveness of our proposed algorithm in obtaining feasible and near-optimal solutions, significantly outperforming existing approaches. Index Terms-Non-convex QCQP, feasible point pursuit, successive convex approximation, semi-definite relaxation, linearization, multicast beamforming.

Multicast multigroup beamforming for per-antenna power constrained large-scale arrays

“…Inspired by the recent development of the feasible point pursuit (FPP) successive convex approximation (SCA) of non-convex quadratically constrained quadratic problems (QCQPs), as developed in [12], the present work aims at improving the max min fair solutions of [11]. The FPP − SCA tool has been preferred over other existing approximations (for instance [13]) due to its guaranteed feasibility regardless of the initial state of the iterative optimization [12].…”

“…The FPP − SCA tool has been preferred over other existing approximations (for instance [13]) due to its guaranteed feasibility regardless of the initial state of the iterative optimization [12].…”

“…However, the FPP − SCA, a recently proposed alternative to SDR, is herein considered [12]. By defining…”

An iterative approach to nonconvex QCQP with applications in signal processing

“…Interestingly, we show that in fact it is not necessary to numerically tackle such an optimization problem to find optimal {Q i }. Recall the optimality condition of Q i in (20), which may be written as,…”

“…Recently a variant of SCA known as feasible point pursuit-successive convex approximation (FPP-SCA) has also been proposed in [20]. To the best of our knowledge, SDR is yet the most prominent and widely used technique employed for tackling nonconvex QCQP.…”