A RIEMANNIAN-GEOMETRICAL APPROACH TO STRICTLY CONVEX QUADRATIC PROGRAMMING WITH CONVEXITY-PRESERVING METRIC PARAMETERIZATION

Abstract

In this study, we propose a new approach to strictly convex quadratic programming based on differential geometry. Broadly, our approach is an interior-point method. However, it can also be viewed as Newton's method on a Riemannian manifold on a set of interior points. In contrast to existing works on Newton's method on Riemannian manifolds, we introduce a parameterized metric and a retraction on the manifold, which are required to find a descent direction on the tangent space and update the solution on the manifold, respectively. The parameter of the metric is chosen at each iteration to preserve the local geodesic convexity of the objective function, while the retraction is designed to guarantee local convergence of the algorithm. The convergence rate is proven to be quadratic. Furthermore, we propose a modified algorithm emphasizing effective performance, which is numerically illustrated to be computationally as efficient as the primal-dual interior-point method, which has been widely used in practice. Our approach is also capable of warm start, which are preferable for model predictive control.