Optimization Problems with Difference of Tangentially Convex Functions under Uncertainty

TL;DR

This paper studies robust optimization problems with objectives expressed as differences of tangentially convex functions under data uncertainty, developing calculus rules and optimality conditions for such nonsmooth, nonconvex problems.

Contribution

It introduces new nonsmooth calculus rules and optimality conditions for DTC functions in robust optimization, advancing understanding of nonsmooth, nonconvex problem analysis.

Findings

Established relationships between subdifferentials for maximum functions.

Derived optimality conditions using tangential subdifferentials.

Provided illustrative examples demonstrating the theoretical results.

Abstract

This paper investigates a specific class of nonsmooth nonconvex optimization problems in the face of data uncertainty, namely, robust optimization problems, where the given objective function can be expressed as a difference of two tangentially convex (DTC) functions. More precisely, we develop a range of nonsmooth calculus rules to establish relationships between Frechet and limiting subdifferentials for a particular maximum function and the tangential subdifferential of its constituent functions. Subsequently, we derive optimality conditions for problems involving DTC functions, employing generalized constraint qualifications within the framework of the tangential subdifferential concept. Several illustrative examples are presented to demonstrate the obtained results.