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Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization

Fernando García-Castaño, Miguel Ángel Melguizo-Padial

TL;DR

This work develops a general Lagrange multiplier theory for nonconvex set-valued optimization by replacing classical linear functionals with closed convex processes as multipliers. The theory proves the existence of multiplier processes under Lipschitz-type regularity and cone-nondegeneracy, and establishes a one-to-one correspondence between multiplier processes and lower semicontinuous sublinear functions in the scalar case, yielding exact penalization without additional constraint qualifications. The framework extends separation principles beyond convexity and applies naturally to set-valued vector equilibrium problems, including infinite-dimensional examples that relax interiority assumptions. Overall, the paper provides a geometric, process-based duality that informs theory and potential algorithmic approaches for nonconvex set-valued optimization.

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.

Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization

TL;DR

This work develops a general Lagrange multiplier theory for nonconvex set-valued optimization by replacing classical linear functionals with closed convex processes as multipliers. The theory proves the existence of multiplier processes under Lipschitz-type regularity and cone-nondegeneracy, and establishes a one-to-one correspondence between multiplier processes and lower semicontinuous sublinear functions in the scalar case, yielding exact penalization without additional constraint qualifications. The framework extends separation principles beyond convexity and applies naturally to set-valued vector equilibrium problems, including infinite-dimensional examples that relax interiority assumptions. Overall, the paper provides a geometric, process-based duality that informs theory and potential algorithmic approaches for nonconvex set-valued optimization.

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.
Paper Structure (8 sections, 17 theorems, 185 equations)

This paper contains 8 sections, 17 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Motivation and main contributions
  3. Structure of the paper
  4. Relation to prior work
  5. Preliminaries and Notation
  6. The Optimization Problem and the General Lagrange Duality Framework
  7. Verification of the General Assumptions under Lipschitz and Cone-Regularity Conditions
  8. An Application to Set-Valued Vector Equilibrium Problems

Key Result

Theorem 3.5

Let $y_0 \in \mathrm{ND}(P(0_Z))$ and assume that $\Gamma_{y_0}\not = \varnothing$. Then, every $\Delta \in \Gamma_{y_0}$ is a Lagrange multiplier of $(P(0_Z))$ at $y_0$; that is, $y_0$ is a nondominated point of the program Furthermore, if $y_0$ is a minimal point of $(P(0_Z))$ (that is, if $y_0 \in F(x_0)$ for some feasible solution $x_0$), then $y_0$ is also a minimal point of $(P[\Delta])$ (a

Theorems & Definitions (49)

  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5
  • Remark 3.6
  • Theorem 3.7
  • proof
  • proof : Proof of Theorem \ref{['ThLagMult11']}
  • Example 3.8
  • ...and 39 more