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.
Highlights
Consider a system of linear matrix inequality (LMI) constraints given below: ∑nA( j) (x) := A0( j) + xi Ai( j) 0, ( j = 1, 2,..., q), (1)i =1 where x ∈ IRn is a variable and each Ai( j) is a mj x mj symmetric matrix for i=0,1,...,n
Four search directions for the method, namely: ZY, ZY+YZ, Z−1 and NT methods are compared in computing the weighted analytic center
The results agree with what is known in semidefinite programming, where ZY+YZ and ZY are found to be more efficient among the four methods
Summary
Consider a system of linear matrix inequality (LMI) constraints given below:. i =1 where x ∈ IRn is a variable and each Ai( j) is a mj x mj symmetric matrix for i=0,1,...,n. 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: xac(ω) = argmin{φω(x) | x ∈ IRn}. In the special case of linear constraints, weighted analytic center has been studied extensively in the past [3]. Four search directions for the method, namely: ZY, ZY+YZ, Z−1 and NT methods are compared in computing the weighted analytic center. These search directions have been used in the past in the problem of semidefinite programming [2]. The results agree with what is known in semidefinite programming, where ZY+YZ and ZY are found to be more efficient among the four methods
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