Variational Theory and Algorithms for a Class of Asymptotically Approachable Nonconvex Problems

TL;DR

This paper introduces a new variational framework for a class of challenging nonconvex functions, providing theoretical optimality conditions and a practical algorithm with convergence guarantees for problems like inverse optimal value optimization.

Contribution

It develops an asymptotic decomposition method for non-Lipschitz composite functions, extending the theory of amenable functions and designing algorithms with verifiable termination criteria.

Findings

Algorithm generates sequences whose accumulation points satisfy optimality conditions.

The decomposition guarantees epi-convergence to the original nonconvex function.

Preliminary numerical results demonstrate practical implementability.

Abstract

We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of this class is that the inner function may fail to be locally Lipschitz continuous. It covers a range of important yet challenging applications, including inverse optimal value optimization and problems under value-at-risk constraints. We propose an asymptotic decomposition of the composite function that guarantees epi-convergence to the original function, leading to necessary optimality conditions for the corresponding minimization problem. The proposed decomposition also enables us to design a numerical algorithm such that any accumulation point of the generated sequence, if exists, satisfies the newly introduced optimality conditions. These results expand on the study of so-called amenable functions introduced by Poliquin and Rockafellar in 1992, which are compositions of convex functions with smooth maps, and the prox-linear methods for their minimization. To demonstrate that our algorithmic framework is practically implementable, we further present verifiable termination criteria and preliminary numerical results.