Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso

Yunier Bello-Cruz (), Guoyin Li () and Tran Thai An Nghia ()
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Yunier Bello-Cruz: Northern Illinois University
Guoyin Li: University of New South Wales
Tran Thai An Nghia: Oakland University

Journal of Optimization Theory and Applications, 2022, vol. 194, issue 1, No 7, 167-190

Abstract: Abstract In the previous paper Bello-Cruz et al. (J Optim Theory Appl 188:378–401, 2021), we showed that the quadratic growth condition plays a key role in obtaining Q-linear convergence of the widely used forward–backward splitting method with Beck–Teboulle’s line search. In this paper, we analyze the property of quadratic growth condition via second-order variational analysis for various structured optimization problems that arise in machine learning and signal processing. This includes, for example, the Poisson linear inverse problem as well as the $$\ell _1$$ ℓ 1 -regularized optimization problems. As a by-product of this approach, we also obtain several full characterizations for the uniqueness of optimal solution to Lasso problem, which complements and extends recent important results in this direction.

Keywords: Nonsmooth and convex optimization problems; Forward–backward splitting method; Linear convergence; Uniqueness; Lasso; Quadratic growth condition; Variational analysis; 65K05; 90C25; 90C30 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-022-02013-2

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