On Subadditive Duality for Conic Mixed-Integer Programs

TL;DR

This paper establishes conditions under which the subadditive dual of a feasible conic mixed-integer program is strong and feasible, linking it to the dual of the continuous relaxation and extending key convexity properties.

Contribution

It proves the strong duality of subadditive duals for conic MIPs, connecting dual feasibility to the continuous relaxation and extending the finiteness property to complex convex intersections.

Findings

Subadditive dual is strong and feasible under certain conditions.

Dual feasibility is equivalent to the dual of the continuous relaxation.

Extended the finiteness property to intersections of convex sets.

Abstract

In this paper, we show that the subadditive dual of a feasible conic mixed-integer program (MIP) is a strong dual whenever it is feasible. Moreover, we show that this dual feasibility condition is equivalent to feasibility of the conic dual of the continuous relaxation of the conic MIP. In addition, we prove that all known conditions and other 'natural' conditions for strong duality, such as strict mixed-integer feasibility, boundedness of the feasible set or essentially strict feasibility imply that the subadditive dual is feasible. As an intermediate result, we extend the so-called 'finiteness property' from full-dimensional convex sets to intersections of full-dimensional convex sets and Dirichlet convex sets.