Nonlinear Complementarity Problem Research Articles

An index detecting algorithm for a class of TCP([formula omitted]) equipped with nonsingular [formula omitted]-tensors

As a generalization of the well-known linear complementarity problem, tensor complementarity problem (TCP) has been studied extensively in the literature from theoretical perspective. In this paper, we consider a class of TCPs equipped with nonsingular (not necessarily symmetric) M-tensors. The considered TCPs can be regarded as a special class of nonlinear complementarity problems (NCPs), but the underlying mappings in TCPs are not necessarily P0-functions, which preclude the possibility of applying existing Newton-type methods for NCPs to TCPs directly. By introducing a pivot minimum function, we propose an index detecting algorithm, which efficiently exploits the beneficial properties of nonsingular M-tensors and goes beyond algorithmic frameworks designed for general NCPs. Moreover, the proposed algorithm is well-defined in the sense that it generates a nonnegative element-wise nonincreasing sequence converging to a solution of the problem under consideration. Finally, numerical results further support the efficiency and reliability of the proposed algorithm.

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.

Neural network approaches based on new NCP-functions for solving tensor complementarity problem

Two new NCP-functions are constructed firstly in this paper. The main purpose is to accelerate the process of solution-finding for tensor complementarity problem, which is implemented by neural network methods based on the promising NCP-functions. Moreover, the stability properties of the proposed neural networks are achieved via some theoretics and properties of generalization for linear and nonlinear complementarity problems. Plentiful numerical simulations demonstrate that the presented neural networks possess significantly better stability and comparable convergence rates than neural networks based on some existing NCP-functions.

A modulus-based multigrid method for nonlinear complementarity problems with application to free boundary problems with nonlinear source terms

To overcome the dependence of the convergence rate on the grid size in the existing modulus-based method, we present a modulus-based multigrid method to efficiently solve the nonlinear complementarity problems. In this paper, the nonlinear complementarity problems under consideration arise from free boundary problems with nonlinear source terms. The two-grid local Fourier analysis is given to predict the asymptotic convergence factor and the optimal relaxation parameter of the presented modulus-based multigrid method, and the predictions are agreement with the experimental results. Numerical results also show that both W- and F-cycles significantly outperform the existing modulus-based method and achieve asymptotic optimality in terms of grid-independent convergence rate and linear CPU time when the grid is refined.

An Evolutionary Based Method for Solving the Nonlinear Gripper Problem

ABSTRACT This paper presents an Evolutionary Programming based (EP) solution for the nonlinear frictional gripper problem for both the isotropic and orthotropic cases. The presented method is compared with other methods that involve piecewise linearization of the friction law, and methods that are based on the solution of Nonlinear Complementarity Problem (NCP). The EP method could generate more optimal solutions than those of other methods. Numerical examples that illustrate the proposed method are presented.

A Monotone Semismooth Newton Method for a Kind of Tensor Complementarity Problem

Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the system of nonsmooth equations, we develop a semismooth Newton method for the tensor complementarity problem. We prove the monotone convergence theorem for the proposed method under proper conditions.

Modeling Post-Liberalized European Gas Market Concentration—A Game Theory Perspective

The question of whether the liberalization of the gas industry has led to less concentrated markets has attracted much interest among the scientific community. Classical mathematical regression tools, statistical tests, and optimization equilibrium problems, more precisely non-linear complementarity problems, were used to model European gas markets and their effect on prices. In this research, the parametric and nonparametric game theory methods are employed to study the effect of the market concentration on gas prices. The parametric method takes into account the classical Cournot equilibrium test, with assumptions on cost and demand functions. However, the non-parametric method does not make any prior assumptions, a factor that allows greater freedom in modeling. The results of the parametric method demonstrate that the gas suppliers’ behavior in Austria and The Netherlands gas markets follows the Nash–Cournot equilibrium, where companies act rationally to maximize their payoffs. The non-parametric approach validates the fact that suppliers in both markets follow the same behavior even though one market is more liquid than the other. Interestingly, our findings also suggest that some of the gas suppliers maximize their ‘utility function’ not by only relying on profit, but also on some type of non-profit objective, and possibly collusive behavior.

A Double Nonmonotone Quasi-Newton Method for Nonlinear Complementarity Problem Based on Piecewise NCP Functions

In this paper, a double nonmonotone quasi-Newton method is proposed for the nonlinear complementarity problem. By using 3-1 piecewise and 4-1 piecewise nonlinear complementarity functions, the nonlinear complementarity problem is reformulated into a smooth equation. By a double nonmonotone line search, a smooth Broyden-like algorithm is proposed, where a single solution of a smooth equation at each iteration is required with the reduction in the scale of the calculation. Under suitable conditions, the global convergence of the algorithm is proved, and numerical results with some practical applications are given to show the efficiency of the algorithm.

Robust Secrecy Competition With Aggregate Interference Constraint in Small-Cell Networks

In this article, we address the security issue in a tiered small-cell network aiming at security optimization for small-cell users (SUEs) to defend against eavesdropping. Meanwhile, the transmissions from small-cell base stations (SBSs) are subject to the aggregate interference constraints of macro-cell users (MUEs). In particular, we consider two-fold information uncertainties in small cells, i.e., the uncertainties regarding the eavesdroppers and interference channels to the MUEs. As such, the SBSs compete for robust secrecy rate with robust protection for the MUEs. We adopt the generalized robust Nash equilibrium problem (GRNEP) formulation, for which we confirm the existence of equilibrium and analyze the condition for the uniqueness with variational inequality-assisted analysis. Furthermore, to solve for the equilibrium, we introduce the pricing mechanism and decompose the original GRNEP as a nonlinear complementarity problem with a priced NEP, where the former provides solution of price coefficients and the latter for resource allocation strategies based on given prices. Finally, extensive simulation results are provided to demonstrate the impacts of the interference constraint and uncertainties upon the security performance of an individual SUE and the overall network, which also corroborate the effectiveness of our proposal in security provisioning for the SUEs and interference protection for the MUEs.

Procedure for non-smooth contact for planar flexible beams with cone complementarity problem

Contact description plays an important role in modeling of applications involving flexible multibody dynamics. Example of such applications include contact between a belt and pulley, crash-worthiness analysis in aerospace and automotive engineering. Approaches such as the linear complementarity problem (LCP), nonlinear complementarity problem (NCP) and penalty method have been proposed for contact detection and imposition of contact constraints. Contact description within multibody dynamics, however, continues to be a challenging topic, particularly in the case of flexible bodies. This paper describes and compares the use of two contact descriptions in the framework of flexible multibody dynamics; (1) the use of nonlinear cone complementarity approach (CCP) and (2) the penalty method. Both contact models are presented together with a master-slave detection algorithm. The modified form of node-to-node approach presented facilitates creation of pseudo-nodes where gap function can be calculated. This reduces the cumbersome effort of contact search. Since large deformations can be an important phenomenon in flexible multibody applications, beam elements based on the absolute nodal coordinate formulation (ANCF) are implemented in this study. To make a comparison of two approaches, the damping component is included in the penalty method by using the continuous contact model introduced by Hunt and Crossley. Numerical results are based on the simulation of ANCF beam contact with rigid ground, rigid body with an arbitrary shape and pendulum contact. Although kinematic results show a good agreement between both approaches when the coefficient of restitution is zero, the unphysical interpenetration appears in the penalty method. Nonlinear minimization problem solved by CCP approach helps to prevent the penetration during contact event. Furthermore, the proposed contact detection algorithm is proved to be capable of being used for multiple contact between beam and arbitrary shape rigid body; different contact types, such as side-by-side and corner-by-side, can be performed without prediction.

Smoothing inexact Newton method based on a new derivative-free nonmonotone line search for the NCP over circular cones

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.

A New Algorithm to Solve the Generalized Nash Equilibrium Problem

We try a new algorithm to solve the generalized Nash equilibrium problem (GNEP) in the paper. First, the GNEP is turned into the nonlinear complementarity problem by using the Karush–Kuhn–Tucker (KKT) condition. Then, the nonlinear complementarity problem is converted into the nonlinear equation problem by using the complementarity function method. For the nonlinear equation equilibrium problem, we design a coevolutionary immune quantum particle swarm optimization algorithm (CIQPSO) by involving the immune memory function and the antibody density inhibition mechanism into the quantum particle swarm optimization algorithm. Therefore, this algorithm has not only the properties of the immune particle swarm optimization algorithm, but also improves the abilities of iterative optimization and convergence speed. With the probability density selection and quantum uncertainty principle, the convergence of the CIQPSO algorithm is analyzed. Finally, some numerical experiment results indicate that the CIQPSO algorithm is superior to the immune particle swarm algorithm, the Newton method for normalized equilibrium, or the quasivariational inequalities penalty method. Furthermore, this algorithm also has faster convergence and better off-line performance.

A 6M digital twin for modeling and simulation in subsurface reservoirs

A 6M digital twin for modeling and simulation in subsurface reservoirs

On a Nonsmooth Gauss–Newton Algorithms for Solving Nonlinear Complementarity Problems

In this paper, we propose a new version of the generalized damped Gauss–Newton method for solving nonlinear complementarity problems based on the transformation to the nonsmooth equation, which is equivalent to some unconstrained optimization problem. The B-differential plays the role of the derivative. We present two types of algorithms (usual and inexact), which have superlinear and global convergence for semismooth cases. These results can be applied to efficiently find all solutions of the nonlinear complementarity problems under some mild assumptions. The results of the numerical tests are attached as a complement of the theoretical considerations.

Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems

In this paper, we generalize modulus-based matrix splitting methods to a class of horizontal nonlinear complementarity problems (HNCPs). First, we write the HNCP as an implicit fixed-point equation and we introduce the proposed solution procedures. We then prove the convergence of the methods under some assumptions. We also comment on how the proposed methods and convergence theorems generalize existing results on (standard) linear and nonlinear complementarity problems and on horizontal linear complementarity problems. Finally, numerical experiments are solved to demonstrate the efficiency of the procedures. In this context, the effects of the splitting, of the dimension of the matrices, and of the nonlinear term of the problem are analyzed.

A novel generalization of the natural residual function and a neural network approach for the NCP

The natural residual (NR) function is a mapping often used to solve nonlinear complementarity problems (NCPs). Recently, three discrete-type families of complementarity functions with parameter p⩾3 (where p is odd) based on the NR function were proposed. Using a neural network approach based on these families, it was observed from some preliminary numerical experiments that lower values of p provide better convergence rates. Moreover, higher values of p require larger computational time for the test problems considered. Hence, the value p=3 is recommended for numerical simulations, which is rather unfortunate since we cannot exploit the wide range of values for the parameter p of the family of NCP functions. This paper is a follow-up study on the aforementioned results. Motivated by previously reported numerical results, we formulate a continuous-type generalization of the NR function and two corresponding symmetrizations. The new families admit a continuous parameter p>0, giving us a wider range of choices for p and smooth NCP functions when p>1. Moreover, the generalization subsumes the discrete-type generalization initially proposed. The numerical simulations show that in general, increased stability and better numerical performance can be achieved by taking values of p in the interval (1,3). This is indeed a significant improvement of preceding studies.

Convergence Analysis of a Trust-Region Multidimensional Filter Method for Nonlinear Complementarity Problems

For solving nonlinear complementarity problems, a new algorithm is proposed by using multidimensional filter techniques and a trust-region method. The algorithm is shown to be globally convergent under the reasonable assumptions and does not depend on any extra restoration procedure. In particular, it shows that the subproblem is a convex quadratic programming problem, which is easier to be solved. The results of numerical experiments show its efficiency.

On nonlinear complementarity problems with applications

In this paper, it is shown that nonlinear complementarity problems has a unique solution if function is continuous and strongly monotone of higher order on nonnegative orthant of finite dimensional Euclidean space. The uniqueness of solution is discussed for both primal and dual for a symmetric dual nonlinear program. A unique nonnegative saddle point for a differentiable scalar function is obtained. Further, we obtained a unique equilibrium point of an n-person game.

A market equilibrium model for electricity, gas and district heating operations

Abstract With increasing penetration of renewable energy, multi-energy systems constitute an effective mechanism to optimize energy distribution and improve social welfare. However, a centralized operation of the multi-energy system might not be appropriate under the existing energy markets. Therefore, this paper proposes an equilibrium model for improving the operation of the electricity, gas and district heating subsystems of a district or urban area. The proposed model allows each energy subsystem to pursue its own objective (i.e., maximum social welfare), while considering the interconnection with other subsystems. More specifically, this model represents the behavior of each subsystem and reflects the interactions of the multi-energy system in a practical way. This equilibrium problem is formulated as a nonlinear complementarity problem. An illustrative case study is analyzed to show the relevance of the proposed approach.

Modulus-Based Multisplitting Iteration Method for a Class of Weakly Nonlinear Complementarity Problem

In this paper, we present a modulus-based multisplitting iteration method based on multisplitting of the system matrix for a class of weakly nonlinear complementarity problem. And we prove the convergence of the method when the system matrix is an $$H_+$$-matrix. Finally, we give two numerical examples.