On $C$-Pareto dominance in decomposably $C$-antichain-convex sets

TL;DR

This paper investigates the structure of $C$-Pareto dominance within decomposably $C$-antichain-convex sets, establishing key properties of their convex hulls and implications for Pareto optima and solution invariance.

Contribution

It proves that elements in the convex hull are $C$-Pareto dominated by set elements and shows disjointness of convex hulls under certain conditions, advancing understanding of $C$-antichain-convex set structures.

Findings

Convex hulls of disjoint $C$-antichain-convex sets are disjoint if one is $C$-upward.

Any element in the convex hull of a decomposably $C$-antichain-convex set is $C$-Pareto dominated by some set element.

The set of $C$-Pareto optima and maximizers exhibits convexity and invariance under convexification.

Abstract

This paper shows that---under suitable conditions on a cone $C$---any element in the convex hull of a decomposably $C$-antichain-convex set $Y$ is $C$-Pareto dominated by some element of $Y$. Building on this, the paper proves the disjointness of the convex hulls of two disjoint decomposably $C$-antichain-convex sets whenever one of latter is $C$-upward. These findings are used to obtain several consequences on the structure of the $C$-Pareto optima of decomposably $C$-antichain-convex sets, on the separation of decomposably $C$-antichain-convex sets and on the convexity of the set of maximals of $C$-antichain-convex relations and of the set of maximizers of $C$-antichain-quasiconcave functions. Special emphasis is placed on the invariance of the solution set of a problem after its "convexification".