General Nonlinear Complementarity Problems Research Articles

Binary NCP: a new approach for solving nonlinear complementarity problems

A general nonlinear complementarity problem (NCP) is formulated as a finite set optimization problem called binary NCP, where design variables are identified by n-bit binary numbers. The binary NCP corresponds to the problem of finding a vertex of an n-dimensional unit cube with zero cost value. Two geometrically intuitive algorithms based on the geometry of the unit cube and another algebraically intuitive one based on the distribution of negative variables are implemented. They are called VSA (vertex search algorithm), FSA (face search algorithm), and NSA (negative-entry search algorithm). FSA and NSA are robust as shown by test problems of small to medium size. FSA, which is like a coordinate search algorithm, is much superior to the other two. It is simple to program and most reliable not only for NCPs but also for a new category called joker search problems.

Connectedness of the solution set of the tensor complementarity problem

In this paper, we investigate the connectedness of the solution set of the tensor complementarity problem (TCP). By exploring the structure of underlying tensor in the TCP, we establish two results. The first is that if the solution set of the TCP is connected, then the underlying tensor must be semi-positive. The other is that the solution set of the TCP is connected under a newly proposed condition. The proposed condition is a tensor extension for the linear complementarity problem, and the result improves a direct adoption of the result for general nonlinear complementarity problem. Some examples are given which confirm our theoretical findings.

A smooth Newton method with 3-1 piecewise NCP function for generalized nonlinear complementarity problem

ABSTRACTIn this paper, based on the 3-1 piecewise nonlinear complementarity problem (NCP) function, we proposed a smoothing Newton-type method for the generalized nonlinear complementarity problem (GNCP) with a modified non-monotone line search. The algorithm for a GNCP is more difficult than that for an NCP, because two functions must be considered in the problem. We reformulate the (GNCP) to a smoothing system of equations by two independent variables, and then develop a smoothing Newton-type method for solving it. Under reasonable conditions, we obtain the global convergent properties. Also, the numerical experiments are reported in this paper.

A smoothing Broyden-like method with a nonmonotone derivative-free line search for nonlinear complementarity problems

By using the CHKS-function, we propose a smoothing Broyden-like method for general nonlinear complementarity problems (NCPs). The method is based on the smoothing equation for which we consider the smoothing parameter as an independent variable, and makes use of a new nonmonotone derivative-free line search rule. Under suitable assumptions, we show that the iteration sequence generated by the proposed algorithm converges globally and superlinearly. Furthermore, the algorithm has local quadratic convergence under mild assumptions. Some numerical results are reported, which show that the algorithm is quite effective.

Smoothing Methods for Nonlinear Complementarity Problems

In this paper, we present some new smoothing techniques to solve general nonlinear complementarity problems. Under a weaker condition than monotonicity as on the original problems, we prove convergence of our methods. We also present an error estimate under a general monotonicity condition. Some numerical tests confirm the efficiency of the proposed methods.

Generalized nonlinear complementarity problems with order $$P_0$$ P 0 and $$R_0$$ R 0 properties

We generalize the \(P, P_0, R_0\) properties for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of generalized nonlinear complementarity problem (GNCP). We prove that if a continuous function has the order \(P_0\) and \(R_0\) properties then all the associated GNCPs have solutions.

Newton-type interior-point methods for solving generalized complementarity problems in polyhedral cones

In this work the solution of the generalized nonlinear complementarity problem (GNCP) in polyhedral cones is addressed by two interior-point strategies: a perturbed Newton method and a predictor–corrector method. The latter may be considered as a member of the so-called Chebyshev–Halley family of methods for nonlinear systems, adapted to conform with the interior-point approach. Applied to a linear complementarity problem, the proposed method becomes the well-known Mehrotra's predictor–corrector method. Quadratic local convergence results are proved under the assumptions usually made for the GNCP. Numerical experiments validate the usage of both ideas for solving the GNCP in polyhedral cones. The proposed predictor–corrector method is implementable and competitive with Newton's method for some problems.

A Smoothing Trust Region Method for Ncps Based on the Smoothing Generalized Fischer-Burmeister Function

Based on a reformulation of the complementarity problem as a system of nonsmooth equations by using the generalized Fischer-Burmeister function, a smoothing trust region algorithm with line search is proposed for solving general (not necessarily monotone) nonlinear complementarity problems. Global convergence and, under a nonsingularity assumption, local Q-superlinear/Q-quadratic convergence of the algorithm are established. In particular, it is proved that a unit step size is always accepted after a finite number of iterations. Numerical results also confirm the good theoretical properties of our approach.

Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem

In this article, we first reformulate the generalized nonlinear complementarity problem (GNCP) over a polyhedral cone as a smoothing system of equations and then suggest a smoothing Broyden-like method for solving it. The proposed algorithm has to solve only one system of nonhomogeneous linear equations, perform only one line search and update only one matrix per iteration. We show that the iteration sequence generated by the proposed algorithm converges globally and superlinearly under suitable conditions. Furthermore, the algorithm has local quadratic convergence under mild assumptions. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.

Smoothing Levenberg–Marquardt method for general nonlinear complementarity problems under local error bound

By using the F–B function and smoothing technique to convert the nonlinear complementarity problems to smoothing nonlinear systems, and introducing perturbation parameter μ k into the smoothing Newton equation, we present a new smoothing Levenberg–Marquardt method for general nonlinear complementarity problems. For general mapping F, not necessarily a P 0 function, the algorithm has global convergence. Each accumulation point of the iterative sequence is at least a stationary point of the problem. Under the local error bound condition, which is much weaker than nonsingularity assumption or the strictly complementarity condition, we get the local superlinear convergence. Under some proper condition, quadratic convergence is also obtained.

Merit functions for general mixed quasi-variational inequalities

In this paper, we present some merit functions for general mixed quasi-variational inequalities, and we obtain the equivalent optimization problems to general mixed quasi-variational inequalities. Since the general mixed quasi-variational inequalities include general variational inequalities, quasi-variational inequalities and nonlinear (implicit) complementarity problems as special cases, our results continue to hold for these problems. In this respect, results obtained in this paper represent an extension of previously known results.

Numerical method for general mixed quasi-variational inequalities

In this paper, we use the resolvent operator to suggest and analyze a new numerical method for solving general mixed quasi-variational inequalities coupled with a new direction and a new step size α k . Under certain conditions, the global convergence of the proposed method is proved. Some preliminary computational results are given to illustrate the efficiency of the proposed method. Since the general mixed quasi-variational inequalities include general variational inequalities, quasi-variational inequalities and nonlinear (implicit) complementarity problems as special cases, results proved in this paper continue to hold for these problems. Results proved in this paper may be viewed as a refinement of the previous known results.

Convergence analysis of nonsmooth equations for the general nonlinear complementarity problem

This paper deals with a general nonlinear complementarity problem, where the underlying functions are assumed to be continuous. Based on a nonlinear complementarity function, it is transformed into a system of nonsmooth equations. Then, two kinds of approximate Newton methods for the nonsmooth equations are developed and their convergence are proved. Finally, numerical tests are also listed.

On Self‐Adaptive Method for General Mixed Variational Inequalities

We suggest and analyze a new self‐adaptive method for solving general mixed variational inequalities, which can be viewed as an improvement of the method of (Noor 2003). Global convergence of the new method is proved under the same assumptions as Noor′s method. Some preliminary computational results are given to illustrate the efficiency of the proposed method. Since the general mixed variational inequalities include general variational inequalities, quasivariational inequalities, and nonlinear (implicit) complementarity problems as special cases, results proved in this paper continue to hold for these problems.

Unconstrained optimization reformulation of the generalized nonlinear complementarity problem and related method

In this article, we first propose an unconstrained optimization reformulation of the generalized nonlinear complementarity problem (GNCP) over a polyhedral cone, and then discuss the conditions under which its any stationary point is a solution of the GNCP. The conditions which guarantee the nonsingularity and positive definiteness of the Hessian matrix of the objective function are also given. In the end, we design a Newton-type method to solve the GNCP and show the global and local quadratic convergence of the proposed method under certain assumptions.

A Nonsmooth L-M Method for Solving the Generalized Nonlinear Complementarity Problem over a Polyhedral Cone

In this paper the generalized nonlinear complementarity problem (GNCP) defined on a polyhedral cone is reformulated as a system of nonsmooth equations. Based on this reformulation, the famous Levenberg-Marquardt (L-M) algorithm is employed to obtain its solution. Theoretical results that relate the stationary points of the merit function to the solution of the GNCP are presented. Under mild assumptions, we show that the L-M algorithm is both globally and superlinearly convergent. Moreover, a method to calculate a generalized Jacobian is given and numerical experimental results are presented.

On the Resolution of the Generalized Nonlinear Complementarity Problem

Minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem (GNCP) defined on a polyhedral cone. It is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems. Theoretical results that relate stationary points of the function that is minimized to the solutions of the GNCP are presented. Perturbations of the GNCP are also considered, and results are obtained related to the resolution of GNCPs with very general assumptions on the data. These theoretical results show that local methods for box-constrained optimization applied to the associated problem are efficient tools for solving the GNCP. Numerical experiments are presented that encourage the use of this approach.

A SQP method for general nonlinear complementarity problems

In this paper, the nonlinear complementarity problem is transformed into the least squares problem with nonnegative constraints, and a SQP algorithm for this reformulation based on a damped Gauss-Newton type method is presented. It is shown that the algorithm is globally and locally superlinearly (quadratically) convergent without the assumption of monotonicity.

A smoothing Newton method for general nonlinear complementarity problems

Smoothing Newton methods for nonlinear complementarity problems NCP(F) often require F to be at least a P0-function in order to guarantee that the underlying Newton equation is solvable. Based on a special equation reformulation of NCP(F), we propose a new smoothing Newton method for general nonlinear complementarity problems. The introduction of Kanzow and Pieper's gradient step makes our algorithm to be globally convergent. Under certain conditions, our method achieves fast local convergence rate. Extensive numerical results are also reported for all complementarity problems in MCPLIB and GAMSLIB libraries with all available starting points.

QPCOMP: A quadratic programming based solver for mixed complementarity problems

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel [14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm.