A Full-Newton step infeasible-interior-point algorithm for P*(k)-horizontal linear complementarity problems

Abstract

In this paper we generalize an infeasible interior-point method for linear optimization to horizontal linear complementarity problem (HLCP). This algorithm starts from strictly feasible iterates on the central path of a perturbed problem that is produced by suitable perturbation in HLCP problem. Then, we use so-called feasibility steps that serves to generate strictly feasible iterates for the next perturbed problem. After accomplishing a few centering steps for the new perturbed problem, we obtain strictly feasible iterates close enough to the central path of the new perturbed problem. The complexity of the algorithm coincides with the best known iteration complexity for infeasible interior-point methods.

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

“…The following theorem states a condition on the proximity measure, that ensures a quadratic convergence of the Newton step. (5) and…”

“…Furthermore in [4,29] the authors extend the result of worst-case polynomial complexity to the wider class of sucient LCPs for FN-IPM. Finally Input: an accuracy parameter > 0 ; a sequence of update parameters {θ k }, 0 < θ k < 1 ∀k ∈ N ; initial values (z 0 , s 0 ) ∈ F + , µ 0 = z 0 s 0 ; 1 z := z 0 , s := s 0 , µ := µ 0 , θ := θ 0 , k := 0 ; 2 while z T s ≥ n do Algorithm 1: Full Newton step IPM (FN-IPM) a method independent of the choice of the initial iterate can be found in [5]. In all these dierent contexts, this new technique with ϕ(t) = √ t gives the best known complexity upper bound.…”

Funciones no lineales para la determinación de un punto interior en la región factible de problemas de programación lineal

“…Sin importar en cuál de los grupos se clasifique un método de solución, tienen en común la necesidad de un punto de partida. En general, obtener un punto de inicio para un algoritmo de punto interior no es un trabajo fácil, tanto es así que se han diseñado procedimientos que parten de puntos no factibles o puntos arbitrarios, entre los que se cuentan el método de Newton basado en una función de Kernel (Liu et al, 2009), otra variante del método de Newton (Roos y Mansouri, 2009) y el de punto interior no factible (Asadi y Mansouri, 2015).…”

Interior-point algorithm for sufficient LCPs based on the technique of algebraically equivalent transformation

“…In [13,15] the transformation is achieved by the square root function. Since then, this technique has been extended to different types of optimization problems, for LCP [1,3,5,6,34,39,44,46], for convex quadratic programming (CQP) [2], for semidefinite programing (SDP) [7,45], for second-order cone programming (SOCP) [46] and symmetric optimization (SO) [33,47].…”