On the local convergence of quasi-Newton methods for nonlinear complementarity problems

Lopes, Véra Lucia Rocha; Martínez, José Mário; Perez, R.B.

doi:10.1016/s0168-9274(98)00080-4

Cited by 13 publications

(7 citation statements)

References 12 publications

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“…On the other hand, some recent papers (Refs. 19,28) show that NCP-functions different from the Fischer-Burmeister function lead to semismooth reformulations of nonlinear complementarity and variational inequality problems with appealing theoretical and computational properties. The practical advantages of such reformulations should be particularly interesting for large problems.…”

Section: Discussionmentioning

confidence: 99%

Hybrid Newton-Type Method for a Class of Semismooth Equations

Pieraccini

2002

Journal of Optimization Theory and Applications

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In this paper, we present a hybrid method for the solution of a class of composite semismooth equations encountered frequently in applications. The method is obtained by combining a generalized finite difference Newton method to a inexpensive direct search method. We prove that, under standard assumptions, the method is globally convergent with a local rate of convergence which is superlinear or quadratic. We also report several numerical results obtained applying the method to suitable reformulations of well known nonlinear complementarity problems.

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Section: Discussionmentioning

confidence: 99%

Hybrid Newton-Type Method for a Class of Semismooth Equations

Pieraccini

2002

Journal of Optimization Theory and Applications

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“…From (1), a vector x * is a solution of (2), if, and only if, x * it is a solution of the NCP. To solve (2) and thus, to solve the NCP, a nonsmooth algorithms type Newton [27], [29] and quasi-Newton [20], [21], among others [1], [6], [22], [26], [31] have been proposed. The natural merit function [24] Ψ : R n → R, defined by Ψ(x) = 1 2 ∥Φ(x)∥ 2 2 , is used in the globalization of these methods.…”

Section: Introductionmentioning

confidence: 99%

Un algoritmo global con jacobiano suavizado para problemas de complementariedad no lineal

Sánchez¹,

Perez²,

Martínez³

2021

Rev Integr temas mat

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In this paper, we use the smoothing Jacobian strategy to propose a new algorithm for solving complementarity problems based on its reformulation as a nonsmooth system of equations. This algorithm can be seen as a generalization of the one proposed in [18]. We develop its global convergence theory and under certain assumptions, we demonstrate that the proposed algorithm converges locally and, q-superlinearly or q-quadratically to a solution of the problem. Some numerical experiments show a good performance of this algorithm.

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“…The nonsmooth system of nonlinear equations (2), equivalently the NCP, has been solved using nonsmooth methods of Newton [8] and quasi-Newton [9,10,11,12] type, and smooth methods [13,14,15]. These methods are based on Clarke's generalized Jacobian [16] defined by a Lipschitz continuous function G : n → n as follows…”

Section: Introductionmentioning

confidence: 99%

A local Jacobian smoothing method for solving Nonlinear Complementarity Problems

2020

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In this paper, we present a smoothing of a family of nonlinear complementarity functions and use its properties in combination with the smooth Jacobian strategy to present a new generalized Newton-type algorithm to solve a nonsmooth system of equations equivalent to the Nonlinear Complementarity Problem. In addition, we prove that the algorithm converges locally and q-quadratically, and analyze its numerical performance.

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On the local convergence of quasi-Newton methods for nonlinear complementarity problems

Cited by 13 publications

References 12 publications

Hybrid Newton-Type Method for a Class of Semismooth Equations

Hybrid Newton-Type Method for a Class of Semismooth Equations

Un algoritmo global con jacobiano suavizado para problemas de complementariedad no lineal

A local Jacobian smoothing method for solving Nonlinear Complementarity Problems

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