Solution Uniqueness of Convex Piecewise Affine Functions Based Optimization with Applications to Constrained $\ell_1$ Minimization

TL;DR

This paper investigates the conditions for solution uniqueness in convex optimization problems involving convex piecewise affine functions with polyhedral constraints, focusing on $ ext{l}_1$ minimization problems like LASSO and basis pursuit.

Contribution

It develops unified dual-based necessary and sufficient conditions for solution uniqueness, extending and generalizing existing results for constrained $ ext{l}_1$ minimization problems.

Findings

Provides a linear program scheme to verify solution uniqueness.

Recovers known conditions without restrictive assumptions.

Introduces new uniqueness conditions for broader $ ext{l}_1$ problems.

Abstract

In this paper, we study the solution uniqueness of an individual feasible vector of a class of convex optimization problems involving convex piecewise affine functions and subject to general polyhedral constraints. This class of problems incorporates many important polyhedral constrained $\ell_1$ recovery problems arising from sparse optimization, such as basis pursuit, LASSO, and basis pursuit denoising, as well as polyhedral gauge recovery. By leveraging the max-formulation of convex piecewise affine functions and convex analysis tools, we develop dual variables based necessary and sufficient uniqueness conditions via simple and yet unifying approaches; these conditions are applied to a wide range of $\ell_1$ minimization problems under possible polyhedral constraints. An effective linear program based scheme is proposed to verify solution uniqueness conditions. The results obtained in this paper not only recover the known solution uniqueness conditions in the literature by removing restrictive assumptions but also yield new uniqueness conditions for much broader constrained $\ell_1$-minimization problems.