This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the biobjective of estimation loss versus solution sparsity. Three such paths are considered: the “
We study
 Award ID(s):
 2208386
 NSFPAR ID:
 10521854
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Journal of Scientific Computing
 Volume:
 100
 Issue:
 2
 ISSN:
 08857474
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract path” where the discontinuous$$\ell _0$$ ${\ell}_{0}$ function provides the exact sparsity count; the “$$\ell _0$$ ${\ell}_{0}$ path” where the$$\ell _1$$ ${\ell}_{1}$ function provides a convex surrogate of sparsity count; and the “capped$$\ell _1$$ ${\ell}_{1}$ path” where the nonconvex nondifferentiable capped$$\ell _1$$ ${\ell}_{1}$ function aims to enhance the$$\ell _1$$ ${\ell}_{1}$ approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped$$\ell _1$$ ${\ell}_{1}$ path. Our study of the capped$$\ell _1$$ ${\ell}_{1}$ path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped$$\ell _1$$ ${\ell}_{1}$ regularized problem offers interesting theoretical properties and practical compromise between the$$\ell _1$$ ${\ell}_{1}$ path and the$$\ell _0$$ ${\ell}_{0}$ path. Indeed, while the$$\ell _1$$ ${\ell}_{1}$ path is computationally prohibitive and greatly handicapped by the repeated solution of mixedinteger nonlinear programs, the quality of$$\ell _0$$ ${\ell}_{0}$ path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped$$\ell _1$$ ${\ell}_{1}$ path; the latter can be obtained efficiently by a combination of a parametric pivotinglike scheme supplemented by an algorithm that takes advantage of the Zmatrix structure of the loss function.$$\ell _1$$ ${\ell}_{1}$ 
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