Meijia Yang

This paper studies the generalized trace ratio problem (GTRP), which maximizes a trace-form quadratic fractional objective over the Stiefel manifold, and, using a newly established matrix S-lemma, proves that adding the redundant constraint XX^T⪯I_n together with a well-chosen scaling yields an equivalent problem (GRS) with zero Lagrangian duality gap, whereas the original (GTRP), the redundant-constraint-only version (GR), and the scaling-only version (GS) can all exhibit a positive Lagrangian duality gap.

This paper studies the generalized trace ratio problem (GTRP), which maximizes a trace-form quadratic fractional objective over the Stiefel manifold, and, using a newly established matrix S-lemma, proves that adding the redundant constraint XX^T⪯I_n together with a well-chosen scaling yields an equivalent problem (GRS) with zero Lagrangian duality gap, whereas the original (GTRP), the redundant-constraint-only version (GR), and the scaling-only version (GS) can all exhibit a positive Lagrangian duality gap.