Applied and Computational Mathematics 2019; 8(1): 21-28 http://www.sciencepublishinggroup.com/j/acm doi: 10.11648/j.acm.20190801.14 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online) Search Directions in Infeasible Newton’s Method for Computing Weighted Analytic Center for Linear Matrix Inequalities Shafiu Jibrin * , Ibrahim Abdullahi Department of Mathematics, Faculty of Science, Federal University, Dutse, Nigeria Email address: * Corresponding author To cite this article: Shafiu Jibrin, Ibrahim Abdullahi. Search Directions in Infeasible Newton’s Method for Computing Weighted Analytic Center for Linear Matrix Inequalities. Applied and Computational Mathematics. Vol. 8, No. 1, 2019, pp. 21-28. doi: 10.11648/j.acm.20190801.14 Received: February 11, 2019; Accepted: March 22, 2019; Published: April 22, 2019 Abstract: Four different search directions for Infeasible Newton’s method for computing the weighted analytic center defined by a system of linear matrix inequality constraints are studied. Newton’s method is applied to find the weighted analytic center and the starting point can be infeasible, that is, outside the feasible region determined by the linear matrix inequality constraints. More precisely, Newton’s method is used to solve system of equations given by the KKT optimality conditions for the weighted analytic center. The search directions for the Newton’s method considered are the ZY, ZY+YZ, Z −1 and NT methods that have been used in semidefinite programming. Backtracking line search is used for the Newton’s method. Numerical experiments are used to compare these search direction methods on randomly generated test problems by looking at the iterations and time taken to compute the weighted analytic center. The starting points are picked randomly outside the feasible region. Our numerical results indicate that ZY+YZ and ZY are the best methods. The ZY+YZ method took the least number of iterations on average while ZY took the least time on average and they handle weights better than the other methods when some of the weights are very large relative to the other weights. These are followed by NT method and then Z −1 method. Keywords: Linear Matrix Inequalities, Weighted Analytic Center, Newton’s Method, Semidefinite Programming 1. Introduction Consider a system of linear matrix inequality (LMI) constraints given below: () () () 0 1 ( ): 0, ( 1, 2, , ) n j j j i i i A x A xA j q = = + = … ∑  , (1) where x ∈ IR n is a variable and each () j i A is a m j x m j symmetric matrix for i=0,1,...,n. LMI constraints have applications in a variety of areas including engineering, geometry and statistics [1, 9]. Assume that feasible determined by the constraints is bounded and has a nonempty interior. Let R denote the feasible region defined by the inequalities (1). Given ω > 0, define the barrier function φ ω (x) : IR n −→ IR by: () 1 1 logdet[( ( )) ] ( ) () q j j j A x if x int x otherwise ω ω φ - =  ∈   =   ∞   ∑ R The weighted analytic center for the system (1) was introduced by Pressman and Jibrin [7], and discussed in a paper by Jibrin and Swift [5]. It is defined by: x ac (ω) = argmin{φ ω (x) | x ∈ IR n } This is a more general form of the determinant maximization problem considered by Vandenberghe et al. [11]. In the special case of linear constraints, weighted analytic center has been studied extensively in the past [3]. This definition of weighted analytic center for LMIs extends that of linear constraints studied by Atkinson and Vaidya [3]. Jibrin presents Infeasible Newton’s method for finding