Generalized Complementarity Problem Research Articles

Coercive and Noncoercive Mixed Generalized Complementarity Problems

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces.

Error Bounds and Merit Functions for Exponentially General Variational Inequalities

In this paper, some new classes of classes of exponentially general variational inequalities are introduced. It is shown that the odd-order and nonsymmetric exponentially boundary value problems can be studied in the framework of exponentially general variational inequalities. We consider some classes of merit functions for exponentially general variational inequalities. Using these functions, we derive error bounds for the solution of exponentially general variational inequalities under some mild conditions. Since the exponentially general variational inequalities include general variational inequalities, quasi-variational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. Results obtained in this paper represent a refinement of previously known results for several classes of variational inequalities and related optimization problems.

A nonsmooth Newton method for solving the generalized complementarity problem

In this paper, we present a new nonsmooth Newton-type algorithm for solving the generalized complementarity problem based on its reformulation as a system of nonlinear equations using a one-parametric family of complementarity functions. We demonstrate, under suitable hypotheses, that this algorithm converges locally and q-quadratically. In addition, we show numerical experiments that allow us to see the good performance of the proposed algorithm.

Predictor-Corrector Interior-Point Algorithm for the General Linear Complementarity Problem

Abstract We study a predictor-corrector interior-point algorithm for solving general linear complementarity problems from the implementation point of view. We analyze the method proposed by Illés, Nagy and Terlaky [1] that extends the algorithm published by Potra and Liu [2] to general linear complementarity problems. A new method for determining the step size of the corrector direction is presented. Using the code implemented in the C++ programming language, we can solve large-scale problems based on sufficient matrices.

A Nonmonotone Smoothing Newton Algorithm for Weighted Complementarity Problem

The weighted complementarity problem (denoted by WCP) significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this paper, by introducing a one-parametric class of smoothing functions which includes the weight vector, we propose a smoothing Newton algorithm with nonmonotone line search to solve WCP. We show that any accumulation point of the iterates generated by this algorithm, if exists, is a solution of the considered WCP. Moreover, when the solution set of WCP is nonempty, under assumptions weaker than the Jacobian nonsingularity assumption, we prove that the iteration sequence generated by our algorithm is bounded and converges to one solution of WCP with local superlinear or quadratic convergence rate. Promising numerical results are also reported.

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.

Generalized Complementarity Problem with Three Classes of Generalized Variational Inequalities Involving⊕Operation

In this study, we introduce and study a generalized complementarity problem involving XOR operation and three classes of generalized variational inequalities involving XOR operation. Under certain appropriate conditions, we establish equivalence between them. An iterative algorithm is defined for solving one of the three generalized variational inequalities involving XOR operation. Finally, an existence and convergence result is proved, supported by an example.

A smoothing quasi-Newton method for solving general second-order cone complementarity problems

Recently, there are much interests in studying smoothing Newton method for solving montone second-order cone complementarity problem (SOCCP) or SOCCPs with Cartesian $$P/P_0$$ -property. In this paper, we propose a smoothing quasi-Newon method for solving general SOCCP. We show that the proposed method is well-defined without any additional assumption and has global convergence under standard conditions. Moreover, under the Jacobian nonsingularity assumption, the method is shown to have local superlinear or quadratic convergence rate. Our preliminary numerical experiments show the method could be very effective for solving SOCCPs.

Existence of solutions for extended generalized complementarity problems

In this paper we introduce extended generalized complementarity problems in Hausdorff topological vector spaces in duality and study the existence of their solutions. We use a different method than those in literature on the existence of solutions of complementarity problems, which are usually based on arguments from generalized monotonicity. This leads us to obtain new results and improve many existing results in literature. We also prove some existence results for extended generalized complementarity problems in reflexive Banach spaces by means of a Tikhonov regularization procedure under a copositivity assumption and arguments from the recession analysis.

Nonemptiness and Compactness of Solution Sets to Generalized Polynomial Complementarity Problems

In this paper, we investigate the generalized polynomial complementarity problem, which is a subclass of generalized complementarity problems with the involved map pairs being two polynomials. Based on the analysis on two structured tensor pairs located in the heading items of polynomials involved, and by using the degree theory, we achieve several results on the nonemptiness and compactness of solution sets. When generalized polynomial complementarity problems reduce to polynomial complementarity problems (or tensor complementarity problems), our results reduce to the existing ones. In particular, one of our results broadens the one proposed in a very recent paper to guarantee the nonemptiness and compactness of solution sets to generalized polynomial complementarity problems. Furthermore, we establish several existence and uniqueness results, which enrich the theory of generalized complementarity problems with the observation that some known conditions to guarantee the existence and uniqueness of solutions may not hold for a lot of generalized polynomial complementarity 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.

Numerical Results for the General Linear Complementarity Problem

Abstract In this article we discuss the interior-point algorithm for the general complementarity problems (LCP) introduced by Tibor Illés, Marianna Nagy and Tamás Terlaky. Moreover, we present a various set of numerical results with the help of a code implemented in the C++ programming language. These results support the efficiency of the algorithm for both monotone and sufficient LCPs.

Iteration complexity of generalized complementarity problems

The goal of this paper is to discuss iteration complexity of two projection type methods for solving generalized complementarity problems and establish their worst-case sub-linear convergence rates measured by the iteration complexity in both the ergodic and the nonergodic sense.

A sub-additive DC approach to the complementarity problem

In this article, we propose a new merit function based on sub-additive functions for solving a general complementarity problem. This leads to consider an optimization problem that is equivalent to the NCP. In the case of a concave NCP this optimization problem is a Difference of Convex (DC) program and we can therefore use DC Algorithm to locally solve it. We prove that in the case of a monotone NCP, it is sufficient to compute a stationary point of the optimization problem to get a solution of the complementarity problems. In the case of a general NCP, assuming that a DC decomposition of the complementarity problem is known, we propose a penalization technique to reformulate the optimization problem as a DC program. Numerical results on linear complementarity problems and absolute value equations show that our method is promising.

Verifiable sufficient conditions for the error bound property of second-order cone complementarity problems

The error bound property for a solution set defined by a set-valued mapping refers to an inequality that bounds the distance between vectors closed to a solution of the given set by a residual function. The error bound property is a Lipschitz-like/calmness property of the perturbed solution mapping, or equivalently the metric subregularity of the underlining set-valued mapping. It has been proved to be extremely useful in analyzing the convergence of many algorithms for solving optimization problems, as well as serving as a constraint qualification for optimality conditions. In this paper, we study the error bound property for the solution set of a very general second-order cone complementarity problem (SOCCP). We derive some sufficient conditions for error bounds of SOCCP which is verifiable based on the initial problem data.

The deterministic ERM and CVaR reformulation for the stochastic generalized complementarity problem

In this paper, we consider the stochastic generalized complementarity problem (SGCP), which has a variety of important applications and has the stochastic complementarity problem as a special case. In order to give reasonable solutions of SGCP, we propose a deterministic formulation that minimizes the expected value of the residual function as the objective function and uses CVaR constraints to ensure the probabilistic feasibility of its solutions. The resulting model containsnon-smooth constraints and expected value functions both in the objective function and constraints. We present approximation problems for solving this model using smoothing and Monte-Carlo techniques. We prove two convergence results of the proposed approximation problems. One is to increase the sample size only, the other is to let smoothing parameter tend to zero as sample increase together. The effectiveness of the proposed approach was demonstrated using numerical examples.

A descent algorithm for generalized complementarity problems based on generalized Fischer-Burmeister functions

We study an unconstrained minimization approach to the generalized complementarity problem GCP(f, g) based on the generalized Fischer-Burmeister function and its generalizations when the underlying functions are $$C^1$$ . Also, we show how, under appropriate regularity conditions, minimizing the merit function corresponding to f and g leads to a solution of the generalized complementarity problem. Moreover, we propose a descent algorithm for GCP(f, g) and show a result on the global convergence of a descent algorithm for solving generalized complementarity problem. Finally, we present some preliminary numerical results. Our results further give a unified/generalization treatment of such results for the nonlinear complementarity problem based on generalized Fischer-Burmeister function and its generalizations.

Direct solution of piecewise linear systems

Let S be a real n×n matrix, z,cˆ∈Rn, and |z| the componentwise modulus of z. Then the piecewise linear equation systemz−S|z|=cˆ is called an absolute value equation (AVE). It has been proven to be equivalent to the general linear complementarity problem, which means that it is NP-hard in general.We will show that for several system classes (in the sense of structural impositions on S) the AVE essentially retains the good-natured solvability properties of regular linear systems. I.e., it can be solved directly by a slightly modified Gaussian elimination that we call the signed Gaussian elimination. For dense matrices S this algorithm has, up to a term in O(n), the same operations count as the classical Gaussian elimination with column pivoting. For tridiagonal systems in n variables its computational cost is roughly that of sorting n floating point numbers. The sharpness of the proposed restrictions on S will be established.

Global Error for Generalized Complementarity Problems based on Generalized Fisher-Burmeister Function

Global Error for Generalized Complementarity Problems based on Generalized Fisher-Burmeister Function

Differentiability v.s. convexity for complementarity functions

It is known that complementarity functions play an important role in dealing with complementarity problems. The most well known complementarity problem is the symmetric cone complementarity problem (SCCP) which includes nonlinear complementarity problem (NCP), semidefinite complementarity problem (SDCP), and second-order cone complementarity problem (SOCCP) as special cases. Moreover, there is also so-called generalized complementarity problem (GCP) in infinite dimensional space. Among the existing NCP-functions, it was observed that there are no differentiable and convex NCP-functions. In particular, Miri and Effati (J Optim Theory Appl 164:723–730, 2015) show that convexity and differentiability cannot hold simultaneously for an NCP-function. In this paper, we further establish that such result also holds for general complementarity functions associated with the GCP.