Journal
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
Volume 164, Issue 1, Pages 109-122Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10957-014-0581-z
Keywords
l1 minimization; Basis pursuit; Lasso; Solution uniqueness; Strict complementarity
Funding
- NUDT Funding of Innovation [B110202]
- NSF [DMS-0748839, ECCS-1028790]
- NSFC [61271014, 61072118]
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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.
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