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In this paper, we study the solution uniqueness of an individual feasiblevector of a class of convex optimization problems involving convex piecewiseaffine functions and subject to general polyhedral constraints. This class ofproblems incorporates many important polyhedral constrained $\ell_1$ recoveryproblems arising from sparse optimization, such as basis pursuit, LASSO, andbasis pursuit denoising, as well as polyhedral gauge recovery. By leveragingthe max-formulation of convex piecewise affine functions and convex analysistools, we develop dual variables based necessary and sufficient uniquenessconditions via simple and yet unifying approaches; these conditions are appliedto a wide range of $\ell_1$ minimization problems under possible polyhedralconstraints. An effective linear program based scheme is proposed to verifysolution uniqueness conditions. The results obtained in this paper not onlyrecover the known solution uniqueness conditions in the literature by removingrestrictive assumptions but also yield new uniqueness conditions for muchbroader constrained $\ell_1$-minimization problems.

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