Closing the duality gap of the generalized trace ratio problem

TL;DR

This paper demonstrates that the generalized trace ratio problem (GTRP) can have zero Lagrangian duality gap when a redundant constraint is added and the problem is well scaled, providing new insights into its duality properties.

Contribution

The paper introduces a matrix S-lemma and shows how adding a redundant constraint and proper scaling can eliminate the duality gap in GTRP.

Findings

GTRP has zero duality gap with added redundant constraint and proper scaling

The matrix S-lemma is key to establishing duality results

Scaling and constraints are crucial for duality in GTRP

Abstract

The generalized trace ratio problem {\rm (GTRP)} is to maximize a quadratic fractional objective function in trace formulation over the Stiefel manifold. In this paper, based on a newly developed matrix S-lemma, we show that {\rm (GTRP)}, if a redundant constraint is added and well scaled, has zero Lagrangian duality gap. However, this is not always true without the technique of scaling or adding the redundant constraint.