Primal-dual optimization conditions for the robust sum of functions with applications

TL;DR

This paper develops primal-dual optimization conditions for the robust sum of functions, providing existence theorems, characterizations, and formulas that enhance understanding and solution methods for robust optimization problems.

Contribution

It introduces a dual problem framework for the robust sum function, offering new primal-dual optimality conditions and subdifferential formulas with applications to subaffine functions and inconsistent systems.

Findings

Established existence of primal optimal solutions.

Derived characterizations of primal-dual optimal sets.

Provided formulas for subdifferential of the robust sum function.

Abstract

This paper associates a dual problem to the minimization of an arbitrary linear perturbation of the robust sum function introduced in DOI 10.1007/s11228-019-00515-2. It provides an existence theorem for primal optimal solutions and, under suitable duality assumptions, characterizations of the primal-dual optimal set, the primal optimal set, and the dual optimal set, as well as a formula for the subdiffential of the robust sum function. The mentioned results are applied to get simple formulas for the robust sums of subaffine functions (a class of functions which contains the affine ones) and to obtain conditions guaranteeing the existence of best approximate solutions to inconsistent convex inequality systems.