Improved Penalty Function Approaches for Optimization Problems with General Orthogonality

Abstract

In this paper, we consider a class of generalized orthogonal optimization constraint problems (GOOCP) over $\mathbb{R}^{n \times p}$, where the variable $X$ is restricted within the intersection of a certain subspace $\mathcal{F}$ and satisfies the quadratic constraint $\{X \in \mathbb{R}^{n \times p}: X^{\top} \phi(X) = I_p\}$. Such constraints generalize a wide range of structured matrix manifolds, such as the Stiefel manifold, the symplectic Stiefel manifold, the indefinite Stiefel manifold, the third-order tensor Stiefel manifold, etc. We show that the feasible region of GOOCP is a closed embedded submanifold of $\mathbb{R}^{n \times p}$ and characterize the necessary geometric materials for the existing Riemannian optimization frameworks. Based on the constraint dissolving approach for Riemannian optimization problems, we propose the constraint dissolving penalty function (GOCDF) for the constrained optimization problem GOOCP with easy-to-compute formulations. We further establish the equivalence between GOCDF and GOOCP in the aspects of first-order and second-order stationary points. We also analyze the computational complexity of applying first-order methods to minimize GOOCP, which could be significantly lower than those of first-order Riemannian optimization methods. Numerical experiments demonstrate that solving GOOCP through applying unconstrained optimization methods to minimize constraint dissolving function demonstrates superior efficiency to existing Riemannian optimization methods.