A Complementarity Partition Theorem for Multifold Conic Systems

TL;DR

This paper extends the Goldman-Tucker Theorem to multifold convex conic systems, establishing a canonical partition of the index set based on complementarity sets linked to interior solutions.

Contribution

It introduces a complementarity partition theorem for multifold convex conic systems, generalizing classical linear programming results to a broader conic setting.

Findings

Established a canonical partition of index sets for conic systems.

Extended Goldman-Tucker Theorem to multifold convex conic systems.

Connected interior solutions to complementarity sets in conic systems.

Abstract

Consider a homogeneous multifold convex conic system $$ Ax = 0, \; x\in K_1\times...\times K_r $$ and its alternative system $$ A\transp y \in K_1^*\times...\times K_r^*, $$ where $K_1,..., K_r$ are regular closed convex cones. We show that there is canonical partition of the index set ${1,...,r}$ determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming.