Non-interior continuation algorithm for solving system of inequalities over symmetric cones

Abstract

As a basic mathematical structure, the system of inequalities over symmetric cones and its solution can provide an effective method for solving the startup problem of interior point method which is used to solve many optimization problems. In this paper, a non-interior continuation algorithm is proposed for solving the system of inequalities under the order induced by a symmetric cone. It is shown that the proposed algorithm is globally convergent and well-defined. Moreover, it can start from any point and only needs to solve one system of linear equations at most at each iteration. Under suitable assumptions, global linear and local quadratic convergence is established with Euclidean Jordan algebras. Numerical results indicate that the algorithm is efficient. The systems of random linear inequalities were tested over the second-order cones with sizes of 10,100, ...,1 000 respectively and the problems of each size were generated randomly for 10 times. The average iterative numbers show that the proposed algorithm can generate a solution at one step for solving the given linear class of problems with random initializations. It seems possible that the continuation algorithm can solve larger scale systems of linear inequalities over the second-order cones quickly. Moreover, a system of nonlinear inequalities was also tested over Cartesian product of two simple second-order cones, and numerical results indicate that the proposed algorithm can deal with the nonlinear cases.