Quadratic Optimization with Orthogonality Constraints: Explicit Łojasiewicz Exponent and Linear Convergence of Line-Search Methods

A fundamental class of matrix optimization problems that arise in many areas of science and engineering is that of quadratic optimization with orthogonality constraints. Such problems can be solved using line-search methods on the Stiefel manifold, which are known to converge globally under mild conditions. To determine the convergence rates of these methods, we give an explicit estimate of the exponent in a Łojasiewicz inequality for the (non-convex) set of critical points of the aforementioned class of problems. This not only allows us to establish the linear convergence of a large class of line-search methods but also answers an important and intriguing problem in mathematical analysis and numerical optimization. A key step in our proof is to establish a local error bound for the set of critical points, which may be of independent interest.