Randomized Algorithms for Optimal Solutions of Double-Sided QCQP With Applications in Signal Processing

• Published in: IEEE transactions on signal processing Vol. 62; no. 5; pp. 1093 - 1108
• Main Authors: Yongwei Huang, Palomar, Daniel P.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Approximation; Array signal processing; Beamforming; Coding, codes; Decomposition; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; Heuristic; homogeneous quadratically constrained quadratic program (QCQP); Information, signal and communications theory; Interference; multicast downlink beamforming; optimal radar waveform; Optimization; Radar; Reduction; Robust receive beamforming; Robustness; semidefinite programming (SDP) relaxation; Signal and communications theory; Signal processing; Signal processing algorithms; Signal, noise; Telecommunications and information theory; Theorems; Vectors; Multiple access; Downlink; Matrix decomposition; Quadratic programming; Algorithm; Data broadcast; Information dissemination; Beam forming; Optimal design; Constrained optimization; Robust receive beamforming; Multicast; Relaxation; homogeneous quadratically constrained quadratic program (QCQP); Randomization; multicast downlink beamforming; Optimal solution; Radar; Signal processing; Optimal code; Matrix method; semidefinite programming (SDP) relaxation; Approximate solution; optimal radar waveform
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2013.2297683

Quadratically constrained quadratic programming (QCQP) with double-sided constraints has plenty of applications in signal processing as have been addressed in recent years. QCQP problems are hard to solve, in general, and they are typically approached by solving a semidefinite programming (SDP) relaxation followed by a postprocessing procedure. Existing postprocessing schemes include Gaussian randomization to generate an approximate solution, rank reduction procedure (the so-called purification), and some specific rank-one matrix decomposition techniques to yield a globally optimal solution. In this paper, we propose several randomized postprocessing methods to output not an approximate solution but a globally optimal solution for some solvable instances of the double-sided QCQP (i.e., instances with a small number of constraints). We illustrate their applicability in robust receive beamforming, radar optimal code design, and broadcast beamforming for multiuser communications. As a byproduct, we derive an alternative (shorter) proof for the Sturm-Zhang rank-one matrix decomposition theorem.