A Weighted Analytic Center for Second-Order Cone Constraints

Abstract

This paper introduces a weighted analytic center for a system of second order cone constraints. The associated barrier function is shown to be convex and conjugate gradient (CG) methods are used to compute the weighted analytic center. In contrast with Newton’s-like methods, CG methods use only the gradient and not the Hessian to minimize a function. The methods considered are the HPRP and ZA with exact and inexact line searches. The exact line search uses Newton’s method and quadratic interpolation is used for the inexact line search. The performance of each method on random test problems was evaluated by observing the number of iterations and time required to find the weighted analytic center. Our numerical methods indicate that ZA is better than HPRP with any of the two line searches, in terms of the number of iterations and time to find the weighted analytic center. Quadratic interpolation inexact line search gives the best success rate and fewest number of iterations for the CG methods considered. On the other hand, the fastest time for the CG methods is found with the Newton’s exact line search. In addition, these results indicate that for each of the methods, our Quadratic interpolation inexact line search has a higher cost per iteration than that of the Newton’s exact line search.