Yang and Xia prove the generalized trace ratio problem needs both a redundant constraint and scaling to close its Lagrangian duality gap

The paper establishes a matrix S-lemma: for H∈S^{p×p} and Q∈S^{n×n}, the set {X∈R^{n×p}: tr(HX^TQX)<0, X^TX=I_p} is empty if and only if there exist M∈S^{p×p} and W∈S^{n×n} with W⪰0 such that tr(HX^TQX)+tr(M(X^TX−I_p))+tr(W(XX^T−I_n))≥0 for all X. The lemma extends the classical and convex Farkas lemmas to a matrix setting with a quadratic equality constraint on the Stiefel manifold, a form not previously available in the cited literature. A complete proof is given: the convex Farkas lemma yields a nonhomogeneous Farkas lemma (Lemma 2), and Lemma 3 (X^TX=I_p implies XX^T⪯I_n) together with Lemma 4 (S1⊆S2 and S1=∅⇔S2=∅, using the Birkhoff–von Neumann theorem) completes both directions.

Building on the matrix S-lemma, the paper gives a semidefinite programming characterization of the optimal value of (GTRP), v(GTRP)=inf{μ: μG⊗A−G⊗B+M⊗I_n+I_p⊗W⪰0, tr(M)+tr(W)≤0, W⪰0}, and proves that the Lagrangian dual of (GRS) attains this value, so (GRS) has no duality gap. While the hidden convexity of (GTRP) was known, explicitly closing the duality gap through the equivalent reformulation (GRS) and exhibiting its semidefinite programming form is new here. Theorems 2 and 3 provide full derivations, with the final dual form (12) matching (10) term by term.

The paper proves that the optimal values of the Lagrangian duals of (GTRP) and (GR) are both λ_max(A^{-1}B), and gives a necessary and sufficient condition for zero duality gap: every unit eigenvector vec(X̂) corresponding to λ_max(I_p⊗(A^{-1}B)) satisfies X̂^TX̂=(1/p)I_p. This shows that adding the seemingly redundant constraint XX^T⪯I_n alone does nothing to reduce the duality gap, in contrast to the known result for (GTP) where adding that constraint yields strong duality. Theorems 4 and 5 derive the two dual values in full, and Theorem 6 states the necessary and sufficient condition; the p=1 case (GRQ1) is used to illustrate that strong duality does hold in that special case.

Through an explicit example with n=p=2, the paper shows that (GS) can have a positive duality gap: with A=I_2, G=diag(1,2), and B=diag(1,3), the primal optimal value is 7/3 while the dual optimal value is 3, giving a gap of 2/3. This rules out the idea that scaling alone can close the duality gap of (GTRP), showing that the redundant constraint and the scaling must be combined to obtain (GRS). The example gives explicit matrices and a step-by-step derivation, with the primal value obtained via Lemma 6 and the dual value from the semidefinite constraints (20) and (21).

The conclusions are set for (GTRP) with G and A positive definite, X∈R^{n×p} with n≥p, and the constraint X^TX=I_p; in this setting, if the redundant constraint XX^T⪯I_n is added and a scaling is applied to obtain (GRS), one can safely turn to solving its Lagrangian dual, since Theorem 3 guarantees the two values coincide. For readers working on trace-ratio discriminant analysis, orthogonally constrained quadratic programming, or Brockett cost function optimization, this means the nonconvex primal can be replaced by a semidefinite-programming-form dual. The paper also notes that the nonhomogeneous form (NGTRP) can be equivalently converted into a (GTRP)-type problem by folding β/(p tr(G))I_n and α/(p tr(G))I_n into B and A, so the theory applies to that nonhomogeneous case as well.

The paper gives the dual optimal values and a necessary and sufficient condition for the duality gaps of (GTRP), (GR), and (GS), but does not provide a closed-form expression for the primal optimal value, so the concrete size of the gap still depends on the data; the example covers only one choice with n=p=2, and the behavior of the gap at other scales and parameters remains to be observed. The authors state in the conclusion that they will study the more general problem max tr(G_2X^TBX)/tr(G_1X^TAX) and its properties and global algorithms, indicating that extending the present results to that form is still open. In addition, the paper is theoretical and reports no numerical experiments or algorithm implementations, so the computational cost and scalability of actually solving the semidefinite program corresponding to (GRS) are outside its scope.