Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization
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arXiv
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| Format: | Preprint |
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2026
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| author | García-Castaño, Fernando Melguizo-Padial, Miguel Ángel |
| author_facet | García-Castaño, Fernando Melguizo-Padial, Miguel Ángel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06186 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization García-Castaño, Fernando Melguizo-Padial, Miguel Ángel Optimization and Control Functional Analysis 2020: 46N10, 90C26, 90C29, 90C33, 90C46, 90C48 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. |
| title | Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization |
| topic | Optimization and Control Functional Analysis 2020: 46N10, 90C26, 90C29, 90C33, 90C46, 90C48 |
| url | https://arxiv.org/abs/2602.06186 |