Necessary and Sufficient Conditions of Solution Uniqueness in 1-Norm Minimization

Hui Zhang, Wotao Yin () and Lizhi Cheng
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Hui Zhang: National University of Defense Technology
Wotao Yin: University of California
Lizhi Cheng: National University of Defense Technology

Journal of Optimization Theory and Applications, 2015, vol. 164, issue 1, No 5, 109-122

Abstract: Abstract This paper shows that the solutions to various 1-norm minimization problems are unique if, and only if, a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as certain other 1-norm related models. This condition is previously known to be sufficient for the basis pursuit model to have a unique solution. Indeed, it is also necessary, and applies to a variety of 1-norm related models. The paper also discusses ways to recognize unique solutions and verify the uniqueness conditions numerically. The proof technique is based on linear programming strong duality and strict complementarity results.

Keywords: l1 minimization; Basis pursuit; Lasso; Solution uniqueness; Strict complementarity; 65K05; 90C25 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (8)

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DOI: 10.1007/s10957-014-0581-z

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