Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

Han, Shaoning & Cui, Ying & Pang, Jong-Shi. (2024). Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity. Mathematics of Operations Research. 10.1287/moor.2023.0295.

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 …