Optimality Conditions for Sparse Bilinear Least Squares Problems

TL;DR

This paper investigates various first-order optimality conditions for sparse bilinear least squares problems, characterizing different stationary points and their relationships to necessary optimality conditions.

Contribution

It introduces new stationarity concepts like the L-like stationary point and coordinate-wise minima, and analyzes their properties and interrelations.

Findings

All discussed stationary points satisfy necessary optimality conditions.

Characterization of T-type, N-type, and L-like stationary points.

Relationships between different stationary points are established.

Abstract

The first-order optimality conditions of sparse bilinear least squares problems are studied. The so-called T-type and N-type stationary points for this problem are characterized in terms of tangent cone and normal cone in Bouligand and Clarke senses, and another stationarity concept called the coordinate-wise minima is introduced and discussed. Moreover, the L-like stationary point for this problem is introduced and analyzed through the newly introduced concept of like-projection, and the M-stationary point is also investigated via a complementarity-type reformulation of the problem. The relationship between these stationary points is discussed as well. It turns out that all stationary points discussed in this work satisfy the necessary optimality conditions for the sparse bilinear least squares problem.