Low-Rank Matrix Optimization Over Affine Set

TL;DR

This paper develops first- and second-order optimality conditions for low-rank matrix optimization over affine sets, providing theoretical insights and applications in system identification and signal processing.

Contribution

It introduces the first comprehensive optimality conditions for rank-MOA, enriching the theoretical framework and guiding algorithm design for low-rank matrix problems.

Findings

Established intersection rule of Fréchet normal cone

Proposed first-order optimality conditions in terms of F- and α-stationary points

Provided second-order necessary and sufficient optimality conditions

Abstract

The low-rank matrix optimization with affine set (rank-MOA) is to minimize a continuously differentiable function over a low-rank set intersecting with an affine set. Under some suitable assumptions, the intersection rule of the Fr\'{e}chet normal cone to the feasible set of the rank-MOA is established. This allows us to propose the first-order optimality conditions for the rank-MOA in terms of the so-called F-stationary point and the $\alpha$-stationary point. Furthermore, the second-order optimality analysis, including the necessary condition and the sufficient one, is proposed as well. All these results will enrich the theory of low-rank matrix optimization and give potential clues to designing efficient numerical algorithms for seeking low-rank solutions. Meanwhile, we illustrate our proposed optimality analysis for several specific applications of the rank-MOA including the Hankel matrix approximation problem in system identification and the low-rank representation on linear manifold in signal processing.