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Abstract: The smoothing-type algorithm has been successfully applied to solve various optimization problems. In general, the smoothing-type algorithm is designed based on some monotone line search. However, in order to achieve better numerical results, the non-monotone line search technique has been used in the numerical computations of some smoothing-type algorithms. In this paper, we propose a smoothing-type algorithm for solving the nonlinear complementarity problem with a non-monotone line search. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are also reported. [Copyright &y& Elsevier]

Abstract: Recently, filter methods are extensively studied to handle nonlinear programming problems. Because of good numerical results, filter techniques are attached importance to. The nonlinear complementarity problem can be reformulated as the least -norm solution of an optimization problem. In this paper, basing on the filter technique and the new smoothing function, we present a new Filter-Levenberg–Marquardt method for solving the equation system with the disturbance . Under the assumption that the lever set of the problem is compact, we prove its global convergence. [Copyright &y& Elsevier]

Abstract: The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a possibly inconsistent nonsmooth system of equations. Based on the ideas developed in smoothing Newton methods, we approximated the problem of the least l 2-norm solution of the equivalent nonsmooth equations of NCP(F) with a family of parameterized optimization problem with twice continuously differentiable objective functions by making use of a new smoothing function. Then we presented a smoothing Levenberg–Marquardt method to solve the parameterized smooth optimization problem. By using the smooth and semismooth technique, the local quadratic convergence of the proposed method is proved under some suitable assumptions. [Copyright &y& Elsevier]

Abstract: The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the least l 2-norm solution of a optimization problem. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a smoothing Levenberg–Marquardt method is applied to solve the parameterized optimization problems. The global convergence of the proposed method is proved under an assumption that the level set of the problem is compact. [Copyright &y& Elsevier]

Abstract: In this paper, we describe H-differential of the implicit Lagrangian function. We show how, under appropriate regularity conditions on an H-differential of f, minimizing the implicit Lagrangian function corresponding to f leads to a solution of the nonlinear complementarity problem. Our results give a unified treatment of such results for C 1-functions and for locally Lipschitzian functions. [Copyright &y& Elsevier]