Journal of the Operations Research Society of Japan

Show abstractHide 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.

View full abstract

Download PDF (237K)

This paper considers the level-increment (LI) truncation approximation of M/G/1-type Markov chains. The LI truncation approximation is usually used to implement Ramaswami's recursion for the stationary distribution in M/G/1-type Markov chains. The main result of this paper is a subgeometric convergence formula for the total-variation distance between the stationary distribution and its LI truncation approximation.

View full abstract

Register with J-STAGE for free!