On strong duality, theorems of the alternative, and projections in conic optimization

TL;DR

This paper investigates the conditions under which strong duality holds in conic optimization, explores the relationships among various constraint qualifications, and derives results on infeasibility and projections in conic problems.

Contribution

It establishes the implications among three key constraint qualifications for strong duality and generalizes the Clark-Duffin theorem, providing new insights into conic optimization theory.

Findings

Hierarchy of constraint qualifications for strong duality

Generalized form of the Clark-Duffin theorem

Explicit description of conic set projections

Abstract

A conic program is the problem of optimizing a linear function over a closed convex cone intersected with an affine preimage of another cone. We analyse three constraint qualifications, namely a Closedness CQ, Slater CQ, and Boundedness CQ (also called Clark-Duffin theorem), that are sufficient for achieving strong duality and show that the first implies the second which implies the third, and also give a more general form of the third CQ for conic problems. Furthermore, two consequences of strong duality are presented, the first being a theorem of the alternative on almost feasibility (also called weak infeasibility), and the second being an explicit description of the projection of conic sets onto linear subspaces, akin to using projection cones for polyhedral sets.