Quadratic Optimization with Orthogonality Constraints: Explicit Lojasiewicz Exponent and Linear Convergence of Line-Search Methods
This work addresses a fundamental issue in matrix optimization relevant to science and engineering, offering incremental improvements in convergence analysis.
The paper tackles the problem of determining the convergence rate of line-search methods for quadratic optimization with orthogonality constraints by providing an explicit estimate of the Lojasiewicz exponent for critical points, establishing linear convergence for a broad class of these 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 rate of these methods, we give an explicit estimate of the exponent in a Lojasiewicz inequality for the (non-convex) set of critical points of the aforementioned class of problems. By combining such an estimate with known arguments, we are able to establish the linear convergence of a large class of line-search methods. 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.