[Paper Review] The Fermat Rule for Set Optimization Problems with Lipschitzian Set-Valued Mappings

Gemayqzel Bouza, Ernest Quintana+2 more|arXiv (Cornell University)|Jul 26, 2021

Optimization and Variational Analysis51 references4 citations

TL;DR

This paper establishes a Fermat rule for set optimization problems using the lower and upper less set relations, under Lipschitzianity assumptions on the set-valued objective mapping. By analyzing scalarizing functionals and their limiting subdifferentials, the authors derive necessary optimality conditions without requiring convexity, compactness, or strongly minimal elements in the image set.

ABSTRACT

In this paper, we consider set optimization problems where the solution concept is given by the set approach. Specifically, we deal with the lower less and the upper less set relations. First, we derive the convexity and Lipschitzianity of suitable scalarizing functionals under the assumption that the set-valued objective mapping has certain convexity and Lipschitzianity properties. Then, we obtain upper estimates of the limiting subdifferential of these functionals. These results, together with the properties of the scalarization functionals, allow us to obtain a Fermat rule for set optimization problems with Lipschitzian data.

Motivation & Objective

• To develop necessary optimality conditions for set optimization problems under the set approach, specifically using the lower and upper less set relations.
• To address limitations in existing approaches that require convexity, compactness, or the existence of strongly minimal elements in the image set.
• To establish optimality conditions using limiting subdifferentials of scalarizing functionals derived from Lipschitzian set-valued mappings.
• To extend the applicability of Fermat-type rules to set optimization by leveraging generalized differentiation in primal and dual spaces.
• To provide a foundation for future algorithmic development converging to stationary points in set optimization.

Proposed method

• Derives convexity and Lipschitzianity properties of scalarizing functionals constructed from the set-valued objective mapping under assumed convexity and Lipschitz conditions.
• Computes upper estimates of the limiting subdifferential of these scalarizing functionals using generalized differential tools.
• Applies the set approach via the lower less (≤ₗ) and upper less (≤ᵤ) relations to define minimal solutions in the power set of the image space.
• Uses the contingent derivative and coderivative structures to characterize stationarity in terms of subdifferential inclusions.
• Employs the embedding of set-valued mappings into a normed space to relate set optimization to standard vector optimization problems.
• Applies the concept of directional derivatives based on distance-type functionals and set differences to analyze optimality.

Experimental results

Research questions

• RQ1 How can necessary optimality conditions be derived for set optimization problems under the lower and upper less set relations without assuming convexity or compactness of the image set?
• RQ2 What are the limiting subdifferential properties of scalarizing functionals induced by Lipschitzian set-valued mappings in set optimization?
• RQ3 In what way do the proposed optimality conditions generalize or improve upon existing results that require strongly minimal elements or compact images?
• RQ4 Can the Fermat rule be extended to other set relations beyond the lower and upper less relations using similar scalarization and subdifferential techniques?
• RQ5 To what extent can the proposed framework support the development of convergent algorithms for set optimization problems?

Key findings

• The paper establishes a Fermat rule for set optimization problems under the lower less (≤ₗ) and upper less (≤ᵤ) set relations, valid under Lipschitzianity of the set-valued objective mapping.
• The limiting subdifferential of the scalarizing functional is upper estimated using the generalized coderivative and normal cone structures, enabling stationarity characterization.
• The proposed optimality conditions do not require the image set to be convex or compact, nor the existence of a strongly minimal element, overcoming key limitations of prior approaches.
• For the example case, the point ̄x is shown to be both ≤ₗ- and ≤ᵤ-stationary, verified through explicit computation of subdifferential and coderivative inclusions.
• The results are robust to isolated image sets, as demonstrated by the normal cone computation N(f(̄x), F(̄x)) = ℝ² in the example.
• The framework is extendable to other set relations such as those introduced by Jahn, Ha, and Karaman et al., suggesting broad applicability beyond the current scope.