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On the solution uniqueness characterization in the L1 norm and polyhedral gauge recovery

Jean Charles Gilbert (Jean-Charles.Gilbert***at***inria.fr)

Abstract: This paper first proposes another proof of the \textit{necessary and sufficient conditions of solution uniqueness in 1-norm minimization} given recently by H. Zhang, W. Yin, and L. Cheng. The analysis avoids the need of the surjectivity assumption made by these authors and should be mainly appealing by its short length (it can therefore be proposed to students exercising in convex optimization). In the second part of the paper, the previous existence and uniqueness characterization is extended to the recovery problem where the $\ell_1$ norm is substituted by a polyhedral gauge. In addition to present interest for a number of practical problems, this extension clarifies the geometrical aspect of the previous uniqueness characterization. Numerical techniques are proposed to compute a solution to the polyhedral gauge recovery problem in polynomial time and to check its possible uniqueness by a simple linear algebra~test.

Keywords: basis pursuit - convex polyhedral function - gauge recovery - $\ell_1$ minimization - Minkowski function - optimality conditions - sharp minimum - solution existence and uniqueness

Category 1: Convex and Nonsmooth Optimization

Citation:

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Entry Submitted: 08/25/2015
Entry Accepted: 08/25/2015
Entry Last Modified: 08/10/2016

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