Nonlinear Complementarity Problem Function Research Articles

On construction of new NCP functions

We report a new method to construct complementarity functions for the nonlinear complementarity problem (NCP). Basic properties related to growth behavior, convexity and semismoothness of the newly discovered NCP functions are proved. We also present some variants, generalizations and other transformations of these NCP functions. Finally, we propose some interesting research directions that can be explored in the NCP research.

An efficient neural network for solving convex optimization problems with a nonlinear complementarity problem function

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.

Generalized Nash Equilibrium Problem Involving Uncertain Random Variable

This paper investigates Nash equilibrium problem with uncertain random variable which is a tool to deal with a mixture of uncertainty and randomness. By using nonlinear complementarity problem function (denoted by NCP function), quasi-Monte Carlo method and inverse uncertain distribution method, the expected minimization model is put forward,the method of solution and convergence result of the corresponding solution are given.

Investigations on enhanced Fischer–Burmeister NCP functions: application to a rate-dependent model for ferroelectrics

This contribution deals with investigations on enhanced Fischer–Burmeister nonlinear complementarity problem (NCP) functions applied to a rate-dependent laminate-based material model for ferroelectrics. The framework is based on the modelling and parametrisation of the material’s microstructure via laminates together with the respective volume fractions. These volume fractions are treated as internal-state variables and are subject to several inequality constraints which can be treated in terms of Karush–Kuhn–Tucker conditions. The Fischer–Burmeister NCP function provides a sophisticated scheme to incorporate Karush–Kuhn–Tucker-type conditions into calculations of internal-state variables. However, these functions are prone to numerical instabilities in their original form. Therefore, some enhanced formulations of the Fischer–Burmeister ansatz are discussed and compared to each other in this contribution.

A Continuous Approach to Binary Quadratic Problems

This paper presents a continuous method for solving binary quadratic programming problems. First, the original problem is converted into an equivalent continuous optimization problem by using NCP (Nonlinear Complementarity Problem) function, which can be further carry on the smoothing processing by aggregate function. Therefore, the original combinatorial optimization problem could be transformed into a general differential nonlinear programming problem, which can be solved by mature optimization technique. Through some numerical experiments, the applicability, robustness, and solution quality of the approach are proved, which could be applied to large scale problems.

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 Modified Levenberg-Marquardt (L-M) Algorithm for Traffic Equilibrium Problem with Nonadditive Route Costs

The traditional traffic equilibrium problem (TEP) is mainly based on the additivity assumption that the route cost is simply the sum of the link costs on that route. However, there are many situations where this assumption on the route costs is inappropriate, and thus we have to explicitly formulate and solve the TEP in the route space instead of link space. In this paper, we firstly reformulate the TEP with nonadditive route cost function to a nonlinear complementarity problem (NCP), and then the NCP is converted to an equivalent least square problem (LSP) with a new NCP function; then we propose a modified Levenberg- Marquardt algorithm to solve the LSP, and also, the quadratic convergence and the equivalent condition of the proposed L-M algorithm are proved under some assumptions. Finally, a numerical example is presented in the paper. As the results shown, the proposed method has the capability to converge to a high level accuracy with reasonable computational efforts.

Comparison of Levenberg-Marquardt Method and Path Following Interior Point Method for the Solution of Optimal Power Flow Problem

Abstract The Levenberg-Marquardt (L-M) method and the path following interior point (PFIP) method are two most popular methods to solve the optimal power flow (OPF) problem. The performance comparison between the L-M method and the PFIP method applied to solve the OPF problem has not been made so far. The main objective of this paper provides a more comprehensive reference for researcher by analysing, comparing and evaluating the performances of the two methods. The L-M method is employed to solve the OPF problem formulated by the nonlinear complementarity problem (NCP). After the Karush-Kuhn-Tucker (KKT) conditions of OPF model are represented in term of a set of semismooth equations by introducing the NCP function, the semismooth equations are solved by the L-M method. The PFIP method, combined logarithmic barrier function with Newton-Raphson method, has been then applied to solve the OPF problem. Finally, the two methods are tested on the 5-bus, the IEEE 30-bus and the IEEE 118-bus system, respectively. Simulation results show that the two methods get the same optimal value of the objective function under the convergent criterion, while the L-M method has higher computation precision and faster convergence speed than the PFIP method.

A gradient-based approach for discrete optimum design

In this paper, structural optimization with discrete variables in engineering design is modeled and investigated as a zero-one programming problem. The zero-one programming problem is first reformulated as an equivalent continuous problem through replacing the zero-one constraints by complementarity constraints, then as an equivalent ordinary nonlinear programming problem with the help of the NCP(nonlinear complementarity problem) function. Furthermore, an aggregate function method is introduced with the aim to simplify computation in the following augmented Lagrangian method. Numerical experiments on discrete optimum design showed the proposed method is promising.

The Semismooth Approach for Semi-Infinite Programming without Strict Complementarity

As a complement of our recent article [O. Stein and A. Tezel, J. Global Optim., 41 (2008), pp. 245–266], we study convergence of a semismooth Newton method for generalized semi-infinite programming problems with convex lower level problems. The semismooth Newton method is applied to a semismooth reformulation of the upper and lower level Karush–Kuhn–Tucker conditions by nonlinear complementarity problem functions into a semismooth system of equations. In the present paper we assume strict complementary slackness neither in the upper nor in the lower level. The auxiliary functions of the locally reduced problem then are not necessarily twice continuously differentiable. Still, we can show that a standard regularity condition for quadratic convergence of the semismooth Newton method holds under a natural assumption for semi-infinite programs.

Filter-sequence of quadratic programming method with nonlinear complementarity problem function

A mechanism for proving global convergence in filter-SQP (sequence of quadratic programming) method with the nonlinear complementarity problem (NCP) function is described for constrained nonlinear optimization problem. We introduce an NCP function into the filter and construct a new SQP-filter algorithm. Such methods are characterized by their use of the dominance concept of multi-objective optimization, instead of a penalty parameter whose adjustment can be problematic. We prove that the algorithm has global convergence and superlinear convergence rates under some mild conditions.

A family of NCP functions and a descent method for the nonlinear complementarity problem

In last decades, there has been much effort on the solution and the analysis of the nonlinear complementarity problem (NCP) by reformulating NCP as an unconstrained minimization involving an NCP function. In this paper, we propose a family of new NCP functions, which include the Fischer-Burmeister function as a special case, based on a p-norm with p being any fixed real number in the interval (1,+?), and show several favorable properties of the proposed functions. In addition, we also propose a descent algorithm that is indeed derivative-free for solving the unconstrained minimization based on the merit functions from the proposed NCP functions. Numerical results for the test problems from MCPLIB indicate that the descent algorithm has better performance when the parameter p decreases in (1,+?). This implies that the merit functions associated with p?(1,2), for example p=1.5, are more effective in numerical computations than the Fischer-Burmeister merit function, which exactly corresponds to p=2.

Boundedness and Regularity Properties of Semismooth Reformulations of Variational Inequalities

The Karush-Kuhn-Tucker (KKT) system of the variational inequality problem over a set defined by inequality and equality constraints can be reformulated as a system of semismooth equations via an nonlinear complementarity problem (NCP) function. We give a sufficient condition for boundedness of the level sets of the norm function of this system of semismooth equations when the NCP function is metrically equivalent to the minimum function; and a sufficient and necessary condition when the NCP function is the minimum function. Nonsingularity properties identified by Facchinei, Fischer and Kanzow, 1998, SIAM J. Optim. 8, 850–869, for the semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function, which is an irrational regular pseudo-smooth NCP function, hold for the reformulation based on other regular pseudo-smooth NCP functions. We propose a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent to the minimum NCP function. When it is used to the generalized Newton method for solving the variational inequality problem, an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers only.

Regular Pseudo-Smooth NCP and BVIP Functions and Globally and Quadratically Convergent Generalized Newton Methods for Complementarity and Variational Inequality Problems

The nonlinear complementarity problem (NCP) can be reformulated as a system of semismooth equations by some NCP functions. A well-known NCP function is the Fischer-Burmeister function, which is a strongly semismooth function. It is smooth everywhere except at the origin. The generalized Newton direction of the system of semismooth equations formulated with the Fischer-Burmeister function is always a descent direction at a nonsolution point. The generalized Jacobian of the system is nonsingular under mild conditions. Efficient algorithms have been developed based upon these nice properties. In this paper, we define a class of NCP functions, called regular pseudo-smooth NCP functions, and show that they have these nice properties. Regular pseudo-smooth NCP functions can be easily identified. They include the Fischer-Burmeister function, the Tseng-Luo NCP function family, and the Kanzow-Kleinmichel NCP function family. We give two new regular pseudo-smooth NCP function families: the ratio generated NCP function family and the C curve generated NCP function family. We then discuss the box constrained variational inequality problem (BVIP). We define a class of BVIP functions, called regular pseudo-smooth BVIP functions, and show that they have these nice properties too. We present three different approaches to generate regular pseudo-smooth BVIP functions from regular pseudo-smooth NCP functions. Globally and quadratically convergent generalized Newton methods are established for solving the NCP and the BVIP, based upon regular pseudo-smooth NCP and BVIP functions.