Computing Weighted Analytic Center for Linear Matrix Inequalities Using Infeasible Newton’s Method

Shafiu Jibrin

doi:10.1155/2015/456392

Abstract

We study the problem of computing weighted analytic center for system of linear matrix inequality constraints. The problem can be solved using Standard Newton’s method. However, this approach requires that a starting point in the interior point of the feasible region be given or a Phase I problem be solved. We address the problem by using Infeasible Newton’s method applied to the KKT system of equations which can be started from any point. We implement the method using backtracking line search technique and also study the effect of large weights on the method. We use numerical experiments to compare Infeasible Newton’s method with Standard Newton’s method. The results show that Infeasible Newton’s method moves in the interior of the feasible regions often very quickly, starting from any point. We recommend it as a method for finding an interior point by setting each weight to be 1. It appears to work better than Standard Newton’s method in finding the weighted analytic center when none of weights is very large relative to the other weights. However, we find that Infeasible Newton’s method is more sensitive than Standard Newton’s method to large variation in the weights.

Highlights

We consider a system of linear matrix inequality constraints given as follows: n subject to A(j) (x) := A(0j) + ∑xiA(ij) ⪰ 0, i=1 (1)
Weighted analytic center has been used in interior point methods for linear programs and semidefinite programs [2, 3, 7, 8, 10]
We presented Infeasible Newton’s method for computing weighted analytic center for system of linear matrix inequalities and compared it with Standard Newton’s method

Summary

Introduction

We consider a system of linear matrix inequality constraints given as follows:. j = 1, 2, . . . , q, where x ∈ Rn is a variable and each A(ij) is an mj × mj symmetric matrix. Weighted analytic center for linear matrix inequalities can be found using Standard Newton’s method by minimizing the barrier function. This approach has the drawback that a starting in the interior of the feasible region must be given. We find that Infeasible Newton’s method moves very quickly into the interior of the feasible regions for most of our test problems It seems to be a suitable method for finding an interior point for the system by setting each weight to be 1. In the case of very large variation in the weights, we recommend using Infeasible Newton’s method to get into the interior with each weight set to 1 and switching to Standard Newton’s method for convergence to the weighted analytic center using the original weights and starting from the interior

Weighted Analytic Center for Linear Matrix Inequalities

Infeasible Newton’s Method for Computing Weighted Analytic Center

Numerical Experiments

Conclusion