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We present a sufficient condition for monotonicity of the nonlinear nonsmooth system generated by Fischer–Burmeister function associated with nonlinear complementarity problem. Based on the presented condition, the nonlinear complementarity problem considered in this paper is equivalently formulated to a nonsmooth monotone system. We then propose a modified multivariate spectral gradient projection method for the resulting system, and establish the global convergence without smoothness and Lipschitz condition. Preliminary numerical experiments show that, compared to some existing methods, the proposed method is effective. [ABSTRACT FROM AUTHOR]

By equivalently transforming a class of weakly nonlinear complementarity problems into a modulus equation, and introducing a smoothing approximation of the absolute value function, a smoothing Newton method is established for solving the weakly nonlinear complementarity problem. Under some mild assumptions, the proposed method is shown to possess global convergence and locally quadratical convergence. Especially, the global convergence results do not need a priori existence of an accumulation point with some suitable conditions. Numerical results are given to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a geometrical approach to the optimisation problems motivated by transportation system management. Here, we provide a comprehensive account of geometric programming based on the elementary Finsler geometry in R n . Then, we present a Finslerian dynamical model for the nonlinear complementarity problem of traffic equilibrium that can be applied to a variety of equilibrium problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a one-layer recurrent neural network (NN) for solving convex optimization problems by using the Mangasarian and Solodov (MS) implicit Lagrangian function. In this paper by using Krush–Kuhn–Tucker conditions and MS function the NN model was derived from an unconstrained minimization problem. The proposed NN model is one layer and compared to the available NNs for solving convex optimization problems, which has a better performance in convergence time. The proposed NN model is stable in the sense of Lyapunov and globally convergent to optimal solution of the original problem. Finally, simulation results on several numerical examples are presented and the validity of the proposed NN model is demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we establish a two-step modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems with the system matrix being an H+-matrix. The convergence analysis of the proposed method is given. Numerical examples show that the proposed method is efficient. [ABSTRACT FROM AUTHOR]

Recently, the tensor complementarity problem has been investigated in the literature. An important question involving the property of global uniqueness and solvability for a class of tensor complementarity problems was proposed by Song and Qi (J Optim Theory Appl, 165:854-873, ). In the present paper, we give an answer to this question by constructing two counterexamples. We also show that the solution set of this class of tensor complementarity problems is nonempty and compact. In particular, we introduce a class of related structured tensors and show that the corresponding tensor complementarity problem has the property of global uniqueness and solvability. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of quasiconvex nonlinear complementarity functions. However, we prove that a nonlinear complementarity function cannot be pseudoconvex. As a consequence of this, we show that every convex nonlinear complementarity function is nondifferentiable. Furthermore, some properties of homogeneous nonlinear complementarity functions are proved. [ABSTRACT FROM AUTHOR]

The nonlinear complementarity or NCP functions were introduced by Mangasarian and these functions are proved to be useful in constrained optimization and elsewhere. Interestingly enough there are only two general methods to derive such functions, while the known or used NCP functions are either individual constructions or modifications of the few individual NCP functions such as the Fischer-Burmeister function. In the paper we analyze the elementary properties of NCP functions and the various techniques used to obtain such functions from old ones. We also prove some new nonexistence results on the possible forms of NCP functions. Then we develop and analyze several new methods for the construction of nonlinear complementarity functions that are based on various geometric arguments or monotone transformations. The appendix of the paper contains the list and source of the known NCP functions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new generalized penalized Fischer-Burmeister merit function, and show that the function possesses a system of favorite properties. Moreover, for the merit function, we establish the boundedness of level set under a weaker condition. We also propose a derivative-free algorithm for nonlinear complementarity problems with a nonmonotone line search. More specifically, we show that the proposed algorithm is globally convergent and has a locally linear convergence rate. Numerical comparisons are also made with the merit function used by Chen (J Comput Appl Math 232:455-471, ), which confirm the superior behaviour of the new merit function. [ABSTRACT FROM AUTHOR]

In this paper we propose a class of new large-update primal-dual interior-point algorithms for P( κ) nonlinear complementarity problem (NCP), which are based on a class of kernel functions investigated by Bai et al. in their recent work for linear optimization (LO). The arguments for the algorithms are followed as Peng et al.'s for P( κ) complementarity problem based on the self-regular functions [Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, Princeton, 2002]. It is worth mentioning that since this class of kernel functions includes a class of non-self-regular functions as special case, so our algorithms are different from Peng et al.'s and the corresponding analysis is simpler than theirs. The ultimate goal of the paper is to show that the algorithms based on these functions have favorable polynomial complexity. [ABSTRACT FROM AUTHOR]

The nonlinear complementarity problem (denoted by NCP( F)) can be reformulated as the solution of a nonsmooth system of equations. In this paper, we propose a new smoothing and regularization Newton method for solving nonlinear complementarity problem with P-function ( P-NCP). Without requiring strict complementarity assumption at the P-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions. Numerical experiments indicate that the proposed method is quite effective. In addition, in this paper, the regularization parameter ε in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods. [ABSTRACT FROM AUTHOR]

In this paper we propose a non-interior-point smoothing algorithm for solving the monotone nonlinear complementarity problem (NCP). The proposed algorithm is simpler than many existing non-interior-point smoothing algorithms in the sense that it only needs to solve one system of linear equations and to perform one line search at each iteration. We show that the proposed algorithm is globally convergent under the assumption that the NCP concerned has a nonempty solution set. Such assumption is weaker than those required by most other non-interior-point smoothing algorithms. In particular, we prove that the solution obtained by the proposed algorithm is a maximally complementary solution of the NCP concerned. Preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

In this paper we focus on the problem of identifying the index sets P(x):={i|x[sub i]>0}, N(x):={i|F[sub i](x)>0} and C(x):={i|x[sub i]=F[sub i](x)=0} for a solution x of the monotone nonlinear complementarity problem NCP(F). The correct identification of these sets is important from both theoretical and practical points of view. Such an identification enables us to remove complementarity conditions from the NCP and locally reduce the NCP to a system which can be dealt with more easily. We present a new technique that utilizes a sequence generated by the proximal point algorithm (PPA). Using the superlinear convergence property of PPA, we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution. [ABSTRACT FROM AUTHOR]

It is shown that McCormick's second order sufficient optimality conditions are also necessary for a solution to a quadratic program to be locally unique and hence these conditions completely characterize a locally unique solution of any quadratic program. This result is then used to give characterizations of a locally unique solution to the linear complementarity problem. Sufficient conditions are also given for local uniqueness of solutions of the nonlinear complementarity problem. [ABSTRACT FROM AUTHOR]