Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization

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.

💡 Research Summary

The paper introduces a novel Lagrange multiplier framework for nonconvex, nondifferentiable set‑valued optimization problems. Classical multiplier theory relies on continuous linear functionals (hyperplanes) to separate the feasible set from the objective level sets. When convexity or differentiability fails, such linear functionals may not exist, and the traditional theory collapses. To overcome this limitation, the authors replace linear functionals with closed convex processes—set‑valued mappings whose graphs are closed convex cones. This geometric object requires far fewer structural restrictions than a hyperplane and can be employed in any normed space, regardless of whether the space is ordered.

The main problem studied is: minimize a set‑valued map (F:\Omega\to Y) subject to (x\in\Omega) and (z\in G(x)) for a parameter (z) in an open neighbourhood (U\subset Z). The value map (V(z)=F\bigl(G^{-1}(z)\bigr)) captures all objective values attainable under the constraint (z). A point (y_0) is assumed to be nondominated for the unperturbed problem ((z=0)).

A Lagrange multiplier process (\Delta:Z\rightrightarrows Y) is defined as a proper closed convex process such that (y_0) remains nondominated for the penalized problem (\min{F(x)+\Delta(G(x))}). Three geometric assumptions guarantee the existence and usefulness of such a process:

1. Richness – the algebraic interior (core) of the graph of (\Delta) is non‑empty, ensuring the process has enough directional freedom.
2. Compatibility with the ordering cone – (\Delta(0)) contains the whole ordering cone (Y^+) and its intersection with the opposite cone is confined to (Y^+). This prevents the penalty from destroying the problem’s order structure.
3. Non‑degeneracy – the graph of (-\Delta) intersects the shifted graph of (V) only at the origin, which guarantees that the multiplier does not annihilate feasible directions.

Under these assumptions, Theorem 3.5 proves that any process satisfying them acts as a Lagrange multiplier: the nondominated (or minimal) solutions of the original problem remain nondominated (or minimal) in the augmented problem. Hence the multiplier process preserves global optimality, a stronger result than the local optimality conditions typical of KKT theory.

In the scalar case ((Y=\mathbb{R})), the authors establish a one‑to‑one correspondence between multiplier processes and lower semicontinuous sublinear penalty functions. Consequently, exact penalty formulations can be derived without any additional constraint qualifications, a notable improvement over existing results that usually require Slater‑type conditions.

Section 4 provides verifiable sufficient conditions. If the composite mapping (V) is Lipschitz continuous at the origin, the ordering cone (Y^+) admits a bounded base, and the non‑degeneracy condition holds, then the family of admissible multiplier processes is non‑empty (Theorem 4.9). The proof relies on conic dilations and separation arguments adapted to the process setting. Specialized results for the scalar and single‑valued cases (Theorems 4.15 and 4.17) give explicit constructions of exact penalty functions.

An infinite‑dimensional example in the Banach space (C