Constrained Least Squares, SDP, and QCQP Perspectives on Joint Biconvex Radar Receiver and Waveform design

TL;DR

This paper presents new optimization perspectives on joint radar receiver and waveform design, demonstrating that the problem can be formulated as constrained least squares, SDP, and QCQP, with proven convergence of alternating minimization.

Contribution

It introduces novel formulations of the biconvex radar design problem using classical optimization methods and proves convergence for the proposed alternating minimization approach.

Findings

Waveform optimization can be cast as constrained least squares, SDP, and QCQP.

Convergence of alternating minimization is proven for biconvex problems with biconvex constraints.

The approach handles biconvex constraints by relating to convex hulls of small diameter.

Abstract

Joint radar receive filter and waveform design is non-convex, but is individually convex for a fixed receiver filter while optimizing the waveform, and vice versa. Such classes of problems are fre- quently encountered in optimization, and are referred to biconvex programs. Alternating minimization (AM) is perhaps the most popu- lar, effective, and simplest algorithm that can deal with bi-convexity. In this paper we consider new perspectives on this problem via older, well established problems in the optimization literature. It is shown here specifically that the radar waveform optimization may be cast as constrained least squares, semi-definite programs (SDP), and quadratically constrained quadratic programs (QCQP). The bi-convex constraint introduces sets which vary for each iteration in the alternat- ing minimization. We prove convergence of alternating minimization for biconvex problems with biconvex constraints by showing the equivalence of this to a biconvex problem with constrained Cartesian product convex sets but for convex hulls of small diameter.