Improving the convergence of non-interior point algorithms for nonlinear complementarity problems

Abstract

Abstract. Recently, based upon the Chen-Harker-Kanzow-Smale smoothing function and the trajectory and the neighbourhood techniques, Hotta and Yoshise proposed a noninterior point algorithm for solving the nonlinear complementarity problem. Their algorithm is globally convergent under a relatively mild condition. In this paper, we modify their algorithm and combine it with the superlinear convergence theory for nonlinear equations. We provide a globally linearly convergent result for a slightly updated version of the HottaYoshise algorithm and show that a further modified Hotta-Yoshise algorithm is globally and superlinearly convergent, with a convergence Q-order 1 + t, under suitable conditions, where t ∈ (0, 1) is an additional parameter.

A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

“…Since A has full row rank, (21) implies y = 0. Thus the linear system of equations (19) has only zero solution, and hence G (z) is nonsingular. Thus, the proof is completed.…”

“…Conventional IPMs usually apply a Newton-type method to the equations in (4) with a suitable line search dealing with constraints x ∈ Q and s ∈ Q explicitly. Recently smoothing Newton methods [2,14,25,19,20,7,15,8,12] have attracted a lot of attention partially due to their superior numerical performances. However, some algorithms [2,19] depend on the assumptions of uniform nonsingularity and strict complementarity.…”

“…Recently smoothing Newton methods [2,14,25,19,20,7,15,8,12] have attracted a lot of attention partially due to their superior numerical performances. However, some algorithms [2,19] depend on the assumptions of uniform nonsingularity and strict complementarity. Without the uniform nonsingularity assumption, the algorithm given in [27] usually needs to solve two linear systems of equations and to perform at least two line searches at each iteration.…”

“…The method in [20] was shown to be locally superlinearly/quadratically convergent without strict complementarity. Moreover, the smoothing methods available are mostly for solving the complementarity problems [2,3,19,20,7,8], but there is little work on smoothing methods for the SOCP.…”

A Globally and Locally Superlinearly Convergent Non--Interior-Point Algorithm for P0LCPs

“…The first global linear convergence result for the LCP with a P 0 and R 0 matrix was obtained by Burke and Xu [3], who also proposed in [4] a non-interior-point predictor-corrector algorithm for monotone LCPs which was both globally linearly and locally quadratically convergent under certain assumption. Further development of non-interior-point methods can be found in [35,5,40,33,8,7,21]. It is worth mentioning that Chen and Xiu [8] and Chen and Chen [7] proposed a class of noninterior-point methods using the Chen-Mangasarian smoothing function family [9] that includes the CHKS smoothing function as a special case.…”