Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization

TL;DR

This paper introduces a novel Lagrange multiplier framework for nonconvex set-valued optimization using convex processes, extending classical theories to broader settings and ensuring global optimality preservation.

Contribution

It develops a geometric, set-valued Lagrange multiplier theory that generalizes separation principles beyond convexity and differentiability, with applications to equilibrium problems.

Findings

Established existence of multiplier processes under verifiable conditions.

Preserved global optimality of solutions in the penalized problem.

Connected multiplier processes with sublinear functions in the scalar case.

Abstract

We develop a Lagrange multiplier theory for nonconvex set-valued optimization problems under Lipschitz-type regularity conditions. Instead of classical continuous linear functionals, we introduce closed convex processes -- set-valued mappings whose graphs are closed convex cones -- as generalized Lagrange multipliers. This geometric framework extends separation principles beyond convexity and differentiability. We establish the existence of multiplier processes under verifiable assumptions, including Lipschitz regularity at a reference point, the existence of a bounded base of the ordering cone, and a nondegeneracy condition ensuring proper isolation of optimal values. These processes preserve global optimality: nondominated (respectively, minimal) solutions of the primal problem remain nondominated (respectively, minimal) in the penalized problem. In the scalar case, we obtain a one-to-one correspondence between multiplier processes and lower semicontinuous sublinear functions, yielding exact penalty formulations without additional constraint qualifications. An infinite-dimensional example shows that interiority conditions on the ordering cone, while sufficient, are not necessary. Applications to set-valued vector equilibrium problems are also discussed.