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The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045]. 

Abstract This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “$$\ell _0$$ ${\ell }_0$-path” where the discontinuous$$\ell _0$$ ${\ell }_0$-function provides the exact sparsity count; the “$$\ell _1$$ ${\ell }_1$-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$$\ell _0$$ ${\ell }_0$-path and the$$\ell _1$$ ${\ell }_1$-path. Indeed, while the$$\ell _0$$ ${\ell }_0$-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of$$\ell _1$$ ${\ell }_1$-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 pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function. 

Under the linear regression framework, we study the variable selection problem when the underlying model is assumed to have a small number of nonzero coefficients. Non-convex penalties in speci c forms are well-studied in the literature for sparse estimation. A recent work, Ahn, Pang, and Xin (2017), has pointed out that nearly all existing non-convex penalties can be represented as difference-of-convex (DC) functions, which are the difference of two convex functions, while itself may not be convex. There is a large existing literature on optimization problems when their objectives and/or constraints involve DC functions. Efficient numerical solutions have been proposed. Under the DC framework, directional-stationary (d-stationary) solutions are considered, and they are usually not unique. In this paper, we show that under some mild conditions, a certain subset of d-stationary solutions in an optimization problem (with a DC objective) has some ideal statistical properties: namely, asymptotic estimation consistency, asymptotic model selection consistency, asymptotic efficiency. Our assumptions are either weaker than or comparable with those conditions that have been adopted in other existing works. This work shows that DC is a nice framework to offer a uni ed approach to these existing works where non-convex penalties are involved. Our work bridges the communities of optimization and statistics.