Globally maximizing the ratio of two generalized quadratic matrix form functions over the Stiefel manifold

We consider the problem of maximizing the ratio of two generalized quadratic matrix form functions over the Stiefel manifold, i.e., \(\max \limits _{X^{T}X=I} \frac{\text {tr}(GX^{T}AX)}{\text {tr}(GX^{T}BX)}\) $\underset{X^{T}X=I}{max}\frac{\text{tr}(G{X}^{T}AX)}{\text{tr}(G{X}^{T}BX)}$ (RQMP). We utilize the Dinkelbach algorithm to globally solve RQMP, where each subproblem is evaluated by the closed-form solution. For a special case of RQMP with \(AB=BA\) $AB=BA$ , we propose an equivalent linear programming problem. Numerical experiments demonstrate that it is more efficient than the recent SDP-based algorithm.