A generalized direction in interior point method for monotone linear complementarity problems

Abstract

In this paper, we present a new interior point method with full Newton step for monotone linear complementarity problems. The specicity of our method is to compute the Newton step using a modied system similar to that introduced by Darvay in [11]. We prove that this new method possesses the best known upper bound complexity for these methods. Moreover, we extend results known in the literature since we consider a general family of smooth concave functions in the Newton system instead of the square root. Some computational results are included to illustrate the validity of the proposed algorithm.

A new smoothing method for nonlinear complementarity problems involving P0-function

“…To obtain such a family, we follow the methodology developed by Haddou and his coauthors [17,2], the key ingredient of which is a smoothing function. This notion turned out to be a versatile tool in a wide variety of pure and applied mathematical problems [3,18,19,34]. We begin with a "father" function, from which all other regularized functions will be generated.…”

“…Replacing ℑ by θ r in (18) is logical. Replacing "≤" by "=" in (18) and the (19) seems to be a bold move, but this is motivated by the fact that we want an equality to be mounted into the system of equations. Some times an additional assumption (strict complementarity x + z > 0) is made to get such equations.…”

Feasible Corrector-Predictor Interior-Point Algorithm for $P_{*} (\kappa)$-Linear Complementarity Problems Based on a New Search Direction

“…Another important question would be how to find a general class of functions ϕ for AET which gives IPAs with the best known complexity results for solving sufficient LCPs. Haddou, Migot, and Omer [34] proposed a family of smooth concave functions which yields IPAs with the best known iteration complexity bound for monotone LCPs (i.e., their matrix is either skew-symmetric or positive semidefinite). Note that our function ϕ(t) = t − √ t for the AET does not belong to the family defined by Haddou, Migot, and Omer.…”

“…An interesting question regarding the AET method is whether a general class of functions ϕ can be given in order to define a polynomialtime IPA. Related to this problem, Haddou, Migot, and Omer [34] have recently proposed a family of smooth concave functions which yields to IPAs with the best known iteration bound. The AET technique has been extended to other areas, too, such as LCPs [1,5,6,41,46], semidefinite programming [47,48,69], second-order cone programming [71], and symmetric optimization [40,72].…”

“…That CP IPA in some aspects differs from our CP IPA even for LPs or P * (0)matrices. We point out that the function ϕ(t) = t − √ t does not belong to the family introduced by Haddou, Migot, and Omer [34]. We apply Newton's method to the transformed system to find new search directions.…”