Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

Shaoning Han (), Ying Cui () and Jong-Shi Pang ()
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Shaoning Han: Department of Mathematics, National University of Singapore, Singapore 119076
Ying Cui: Department of Industrial Engineering and Operations Research, University of California, Berkeley, California 94720
Jong-Shi Pang: Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, California 90089

Mathematics of Operations Research, 2025, vol. 50, issue 3, 2175-2198

Abstract: 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.

Keywords: Primary: 90C26; 90C30; nonsmooth analysis; lower semicontinuity; Heaviside functions; local convexity-like property (search for similar items in EconPapers)
Date: 2025
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