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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]

We enrich the theory of variational inequalities in the case of an aggregative structure by implementing recent results obtained by using the Selten–Szidarovszky technique. We derive existence, semi-uniqueness and uniqueness results for solutions and provide a computational method. As an application we derive very powerful practical equilibrium results for Nash equilibria of sum-aggregative games and illustrate with Cournot oligopolies. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems under a weakened condition, and present the general convergence conditions for the method in terms of spectral radius and matrix norm, respectively. Moreover, for some special cases of the method, we propose the concrete convergence conditions and optimal parameters. These convergence theories improve the existing results to some extent. The numerical experiments verify the validity and practicality of the presented results. [ABSTRACT FROM AUTHOR]

For constrained equations with nonisolated solutions and a certain family of Newton-type methods, it was previously shown that if the equation mapping is 2-regular at a given solution with respect to a direction which is interior feasible and which is in the null space of the Jacobian, then there is an associated large (not asymptotically thin) domain of starting points from which the iterates are well defined and converge to the specific solution in question. Under these assumptions, the constrained local Lipschitzian error bound does not hold, unlike the common settings of convergence and rate of convergence analyses. In this work, we complement those previous results by considering the case when the equation mapping is 2-regular with respect to a direction in the null space of the Jacobian which is in the tangent cone to the set, but need not be interior feasible. Under some further conditions, we still show linear convergence of order 1/2 from a large domain around the solution (despite degeneracy, and despite that there may exist other solutions nearby). Our results apply to constrained variants of the Gauss–Newton and Levenberg–Marquardt methods, and to the LP-Newton method. An illustration for a smooth constrained reformulation of the nonlinear complementarity problem is also provided. [ABSTRACT FROM AUTHOR]

For a class of second-order cone nonlinear complementarity problems, abbreviated as SOCNCPs, we establish the modulus-based matrix splitting relaxation iteration methods, which are obtained by reformulating equivalently SOCNCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The global convergence theorems are given under suitable choices of the involved splitting matrix and parameter matrices. When the splitting matrix is symmetric positive definite, the strategy choice of the parameters is discussed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SOCNCPs. [ABSTRACT FROM AUTHOR]

Having in mind singular solutions of smooth reformulations of complementarity problems, arising unavoidably when the solution in question violates strict complementarity, we study the behavior of Newton-type methods near singular solutions of nonlinear equations, assuming that the operator of the equation possesses a strongly semismooth derivative, but is not necessarily twice differentiable. These smoothness restrictions give rise to peculiarities of the analysis and results on local linear convergence and asymptotic acceptance of the full step, the issues addressed in this work. Moreover, we consider not only the basic Newton method, but also some stabilized versions of it intended for tackling singular (including nonisolated) solutions. Applications to nonlinear complementarity problems are also dealt with. [ABSTRACT FROM AUTHOR]

In this paper we consider the nonlinear complementarity problem over circular cones (CCNCP) which contains a lot of circular cone optimization problems. We study a one-parametric class of smoothing functions which can be used to reformulate the CCNCP as a system of smooth nonlinear equations. Based on the equivalent reformulation, we propose a smoothing inexact Newton method to solve the CCNCP. In each iteration, the proposed method solves the nonlinear equations only approximately. Since the inexact direction is not necessarily descent, a new derivative-free nonmonotone line search is developed to ensure that the proposed method has global and local superlinear and quadratical convergence. Some numerical results are also reported. [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, using the sign patterns of the solution of the equivalent modulus equation, the resolution of the nonlinear complementarity problem shrinks to find the zero of a differentiable nonlinear function. Then, a sign-based Newton's method is established by applying the Newton's iteration. The theoretical analysis for the sign patterns of the solution of the equivalent modulus equation is given under the assumption of strictly complementarity. Moreover, by using the known modulus-based matrix splitting iteration method to detect the sign patterns of the solution of the equivalent modulus equation, a practical sign-detection Newton's method is proposed. Numerical examples show that the new methods are efficient and accelerate the convergence performance with higher precision and less CPU time than the existing modulus-based matrix splitting iteration method and the projection-based matrix splitting iteration method, especially for the large sparse problems. [ABSTRACT FROM AUTHOR]

As is well-known, Jacobian smoothing method is a popular one to solve nonlinear complementarity problems, in which the Jacobian consistency is stressed. By investigating an element of related functions’ B-differential, a smoothing Levenberg-Marquardt(LM) method is proposed based on a Chen-Harker-Kanzow-Smale(CHKS) smoothing function, which satisfies a property called strongly Jacobian consistency. Finally, the numerical experiments illustrate the effectiveness of the given method. [ABSTRACT FROM AUTHOR]

This paper establishes a theoretical framework of infeasible Mehrotra-type predictor–corrector algorithm for nonmonotone nonlinear complementarity problems over symmetric cones which can be regarded as an extension the Mehrotra’s algorithm proposed by Salahi et al. (On Mehrotra-type predictor–corrector algorithms. SIAM J Optim 18(4):1377–1397, 2005) from nonnegative orthant to symmetric cone. The iteration complexity of the algorithm is estimated, and some numerical results are provided. The numerical results show that the algorithm is efficient and reliable. [ABSTRACT FROM AUTHOR]

In this paper, the convergence conditions of the modulus-based matrix splitting iteration method for nonlinear complementarity problem of H-matrices are weakened. The convergence domain given by the proposed theorems is larger than the existing ones. Numerical examples show the advantages of the new theorems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a generalized version of Caristi fixed point theorems in pseudo-metric spaces. Our results generalize and improve many of well-known theorems. As an application of our results, we give a new existence theorem to the generalized nonlinear complementarity problem and a solution of differential inclusion in the distributions setting. [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]

In this paper, we consider the splitting method and the two-stage splitting method for solving a class of nonlinear complementarity problems with the coefficient matrix being an $$H$$ -matrix. Convergence result for the splitting method is presented when the splitting is $$H$$ -splitting. Moreover, for the two-stage splitting method, we estimate weighted max-norm bounds for iteration errors, and thereby, we show that the sequence generated by the two-stage iteration scheme converges to the unique solution of the nonlinear complementarity problem without any restriction on the initial vector. Numerical results show that both methods are efficient for solving the class of nonlinear complementarity problems. [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 discuss the globalization of some kind of modified Levenberg-Marquardt methods for nonsmooth equations and their applications to nonlinear complementarity problems. In these modified Levenberg-Marquardt methods, only an approximate solution of a linear system at each iteration is required. Under some mild assumptions, the global convergence is shown. Finally, numerical results show that the present methods are promising. [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]

By using the Fischer-Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust region algorithm for solving the NCP withP0 functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict complementarity condition. [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]