Complementarity Problem

We prove necessary and sufficient conditions for locally Lipschitz Löwner operators to be monotone, strictly monotone and strongly monotone. Utilizing our characterization of the strict monotonicity of Löwner operators, we generalize Mangasarian class of Nonlinear Complementarity Problem (NCP)-functions to the setting of symmetric cone complementarity problem. This affirmatively answers a question of Tseng [Math. Program. 83, 159-185(1998)].

We consider a regularization method for nonlinear complementarity problems with F being a P 0 -function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer-Burmeister (FB) NCP-functions p with p > 1. We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p, for example p ∈ [1.1, 2], usually has better numerical performance, and the generalized FB functions p with p ∈ [1.1, 2) can be used as the substitutions for the FB function 2 .

We consider a setting of a two settlement power market where firms compete in the forward market and an uncertain real-time market. A recourse-based framework is proposed where firms make simultaneous bids in the forward market and take recourse in the real-time market contingent on the realization of uncertainty. The market participants include both generation firms as well as the independent system operator (ISO), the latter of which is assumed to maximize wheeling revenue. The resulting stochastic game-theoretic problem is seen to be a Nash game with coupled strategy sets, often referred to as a generalized Nash game. In general, the primal variational conditions of such problems are given by quasi-variational inequality. Yet, the associated complementarity problem in a primal-dual space admits a monotonicity property that allows us to derive an appropriate existence statement. Computation of equilibria is complicated by the challenge arising from the size of the sample-space. We present a distributed iterative regularization technique that is shown to scale well with the size of the sample-space. Finally, the paper concludes with the application of this model on an electrical network and provides insights on market design and operation.

The modeling and control problem for a grid-connected photovoltaic (PV) power electronic system, which includes a dc/dc boost converter, an inverter and a filter are considered. A linear complementarity (LC) dynamic model of the PV system allows the design of a model predictive controller (MPC). Dynamic models of the subsystems are obtained and merged in order to represent the whole PV system in a compact and comprehensive LC model, which is valid for all operating modes of power converters and PV cells involved in the energy conversion process. A finite-control-set MPC problem is formulated as a mixed-integer quadratic program subject to the dynamic LC model and pulse width modulators. The minimization of an objective function aimed at tracking dc voltage and grid current references provides directly the commands for the switches of the boost converter and inverter. Numerical results show the effectiveness of the proposed strategy for maximum power point tracking and synchronization to the grid under dynamic scenarios characterized by variations of the solar irradiance.

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a

A fundamental mathematical problem is to find a solution to a square system of nonlinear equations. There are many methods to approach this problem, the most famous of which is Newton's method. In this paper, we describe a generalization of this problem, the complementarity problem. We show how such problems are modeled within the GAMS modeling language and provide details about the PATH solver, a generalization of Newton's method, for finding a solution. While the modeling format is applicable in many disciplines, we draw the examples in this paper from an economic background. Finally, some extensions of the modeling format and the solver are described.

In this paper, we describe an algorithm for the incremental state update of elasto-plastic systems with softening. The algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem. In developing the algorithm, we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance, and hence the ability to compute numerical solutions to practical size problems. For example, the notion of a tangent stiffness matrix arises. Numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm. The numerical examples also raise some interesting questions about multiplicity of the solutions.

In this paper it is shown that the implicit Euler time-discretization of some classes of switching systems with sliding modes, yields a very good stabilization of the trajectory and of its derivative on the sliding surface. Therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used, are avoided. Moreover the method (an event-capturing, or time-stepping algorithm) allows for multiple switching surfaces (i.e., a sliding surface of codimension 2). The details of the implementation are given, and numerical examples illustrate the developments. This method may be an alternative method for chattering suppression, keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces. Links with discrete-time sliding mode controllers are studied.

Several new interfaces have recently been developed requiring PATH to solve a mixed complementarity problem. To overcome the necessity of maintaining a different version of PATH for each interface, the code was reorganized using object-oriented design techniques. At the same time, robustness issues were considered and enhancements made to the algorithm. In this paper, we document the external interfaces to the PATH code and describe some of the new utilities using PATH. We then discuss the enhancements made and compare the results obtained from PATH 2.9 to the new version.

By using the topological degree we introduce the concept of ’’exceptionalfamily of elements‘‘ specifically for continuous functions. This has importantconsequences pertaining to the solvability of the explicit, the implicit andthe general order complementarity problems. In this way a new direction forresearch in the complementarity theory is now opened.

This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of nth-order linear time-invariant (LTI) systems with nu (respectively, n y ) independent inputs (respectively, outputs). The algorithm is based on a "cone complementarity" formulation of the problem and is guaranteed to produce a stabilizing controller of order m n 0 max(n u ; n y ), matching a generic stabilizability result of Davison and Chatterjee [7]. Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura's generic stabilizability result (m n0nu 0ny+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis.

In this paper, we characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.

In this work, we present some new existence theorems for solutions for a new class of generalized vector F-implicit complementarity problems and the corresponding generalized vector F-implicit variational inequality problems by using the Fan-KKM Theorem under some suitable assumptions without monotonicity, and establish the equivalence between those problems in Banach spaces.

General variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection, Wiener-Hopf equations, updating the solution, auxiliary principle, inertial proximal, penalty function, dynamical system and well-posedness. We also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection-type algorithms. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, results presented here continue to hold for these problems. Several open problems have been suggested for further research in these areas.

Vector general quasi-optimization problems are formulated and some sufficient conditions on the existence of solutions for these problems are shown. These concern the existence of solutions, the stability and the structure of solution set of general vector problems. As special case, we obtain results on the existence of solutions of vector quasi-optimization problem. Some relationships between these problems and others, as vector quasi-equilibrium problems, quasi-variational inequalities, complementarity problems,..., are shown. From these we extend some well-known results obtained by Blum and Oettli [4], Park [21], Chan and Pang [6], Parida and Sen [20], Browder and Minty [18], Fan [8], etc.

Diverse problems in optimization, engineering, and economics have natural formulations in terms of complementarity conditions, which state (in their simplest form) that either a certain nonnegative variable must be zero or a corresponding inequality must hold with equality, or both. A variety of algorithms have been devised for solving problems expressed in terms of complementarity conditions. It is thus attractive to consider extending algebraic modeling languages, which are widely used for sending ordinary equations and inequality constraints to solvers, so that they can express complementarity problems directly. We describe an extension to the AMPL modeling language that can express the most common complementarity conditions in a concise and flexible way, through the introduction of a single new "complements" operator. We present details of an efficient implementation that incorporates an augmented presolve phase to simplify complementarity problems, and that converts complementarity conditions to a canonical form convenient for solvers.

The present paper extends the classical limit analysis approach for steel structures to account for local strain softening behavior. As is wellknown, the bound theorems underpinning the collapse load estimation of ductile structures require an assumption of perfect plasticity. They are invalid for the class of structures, such as those being considered here, that exhibit local softening. We propose a method, based on the same conditions governing the path-independent state problem, for capturing the maximum load multiplier characterizing the one parameter proportionally applied loading regime. The problem is cast as a special instance of the notoriously difficult to solve class of problems known as a mathematical program with equilibrium constraints (MPEC). Various solution approaches, all based on solving the MPEC as a standard nonlinear programming problem, are briefly presented. One of them, namely the penalty scheme, is used to solve the three numerical examples provided to highlight application of the proposed limit analysis approach.

An approach is presented for dynamic analysis of tensegrity structures in the small displacement regime. Such analyses are characterized by cables switching between taut and slack states. The approach is based on casting the computation in each time increment as a complementarity problem. Numerical examples are presented to illustrate the approach. Despite the nonsmooth nature of cables switching between taut and slack states, the computed solutions exhibit remarkable long-term energy balance. Furthermore, by exploiting some features of the tensegrity model, significant computational efficiency can be gained in the solution of the complementarity problem in each time increment. Due to the use of linearized kinematics, the method is not applicable as is to tensegrity structures with internal mechanisms or where the geometric stiffness is significant compared to the material stiffness. The method however plays a role in a more general nonlinear kinematics formulation.

We consider a setting of a two settlement power market where firms compete in the forward market and an uncertain real-time market. A recourse-based framework is proposed where firms make simultaneous bids in the forward market and take recourse in the real-time market contingent on the realization of uncertainty. The market participants include both generation firms as well as the independent system operator (ISO), the latter of which is assumed to maximize wheeling revenue. The resulting stochastic game-theoretic problem is seen to be a Nash game with coupled strategy sets, often referred to as a generalized Nash game. In general, the primal variational conditions of such problems are given by quasi-variational inequality. Yet, the associated complementarity problem in a primal-dual space admits a monotonicity property that allows us to derive an appropriate existence statement. Computation of equilibria is complicated by the challenge arising from the size of the sample-space. We present a distributed iterative regularization technique that is shown to scale well with the size of the sample-space. Finally, the paper concludes with the application of this model on an electrical network and provides insights on market design and operation.

During the last decade Virtual Reality has benefited of numerous works dedicated to the modeling and the simulation of multibody systems. Even if a large set of methods are already used by the community of computer graphics and virtual reality, some progresses need still to be made to bridge the gap between accurate mechanical simulation and efficient simulations in real-time of realistic systems. Particularly, the problem of the three dimensional frictional contact with impacts is a key point for the applications with haptic feedback. The work aims at adapting the outstanding methods in computational mechanics to the real-time constraints induced by Virtual Reality. For efficiency reasons, our work is based on the Non Smooth Contact Dynamics (NSCD) framework introduced by Moreau (1988). Two major advantages of the method can be exhibited for the real-time context: the method uses a time-stepping numerical scheme without explicit event-handling procedure and an unilateral contact impact formulation associated with the 3D Coulomb's friction law. In the present paper, we use and compare different iterative algorithms (Non Smooth Gauss-Seidel, projected conjugate gradient, the PATH solver) for complementarity problems with a direct one (Lemke). The efficiency of the methods is compared in terms of complexity, of quality ratio between visual feed-back of solution and convergence criterion and of CPU time keeping in mind the real-time constraint.

In this paper, we suggest and analyze some three-step iterative schemes for finding the common elements of the set of the solutions of the Noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of the suggested iterative schemes under some mild conditions. Since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.

The generalized order linear complementarity problem (in the setting of a finite dimensional vector lattice) is the problem of finding a solution to the piecewise-linear system x A (Mix T ql) A (M2x T q2) A... A (Mkx + at:) O, where Mi's are linear transformations and qi's are vectors. This problem is equivalent to the generalized linear complementarity problem considered by Cottle and Dantzig [J. Combin. Theory, 8 (1970), pp. 79-90.]. Using degree theory, a comprehensive analysis of existence, uniqueness, and stability aspects of this problem is presented.

This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of nth-order linear time-invariant (LTI) systems with nu (respectively, n y ) independent inputs (respectively, outputs). The algorithm is based on a "cone complementarity" formulation of the problem and is guaranteed to produce a stabilizing controller of order m n 0 max(n u ; n y ), matching a generic stabilizability result of Davison and Chatterjee [7]. Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura's generic stabilizability result (m n0nu 0ny+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis.

The problem of the numerical solution of a (linear or nonlinear) complementary ty problem is approached here by exploiting the fact that, as shown by Mangasarian, a complementarity problem can be reformulated as a problem of solving a system of nonlinear equations; and the idea is considered of solving such a system of equations by means of a method developed by the present authors, which is based on following a numerical solution trajectory of a second-order differential equation inspired by clas sical mechanics. The method has been suitably modified in order to handle problems with many unknowns, by means of an "inexact conjugate-gradients" method which is used to solve the linear system arising in the numerical integration of the differential equation. The method has been tested on some complementarity problems obtained from the discretization of variational inequalities of mathematical physics, and some numerical results are reported.

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 signi cant advance in robustness at no cost in e ciency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously di erentiable, pseudo-monotone mixed nonlinear complementarity problem. QP-COMP 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 e ectiveness of the algorithm.

The origins and some motivational details of a collection of nonlinear mixed complementarity problems are given. This collection serves two purposes. Firstly, it gives a uniform basis for testing currently available and new algorithms for mixed complementarity problems. Function and Jacobian evaluations for the resulting problems are provided via a GAMS interface, making thorough testing of algorithms on practical complementarity problems possible. Secondly, it gives examples of how to formulate many popular problem formats as mixed complementarity problems and how to describe the resulting problems in GAMS format. We demonstrate the ease and power of formulating practical models in the MCP format. Given these examples, it is hoped that this collection will grow to include many problems that test complementarity algorithms more fully.

Dominance-based Rough Set Approach (DRSA) has been introduced to deal with multiple criteria classification (also called multiple criteria sorting, or ordinal classification with monotonicity constraints), where assignments of objects may be inconsistent with respect to dominance principle. In this paper, we consider an extension of DRSA to the context of imprecise evaluations of objects on condition criteria and imprecise assignments of objects to decision classes. The imprecisions are given in the form of intervals of possible values. In order to solve the problem, we reformulate the dominance principle and introduce second-order rough approximations. The presented methodology preserves well-known properties of rough approximations, such as rough inclusion, complementarity, identity of boundaries and precisiation. Moreover, the meaning of the precisiation property is extended to the considered case. The paper presents also a way to reduce decision tables and to induce decision rules from rough approximations.

Two basic market designs are used for the trading of electricity in meshed electricity networks with transmission constraints. Analytical results show that, in two-node networks, a market design integrating transmission and energy markets reduces the ability of electricity generators to exercise market power, relative to a design with separated markets for transmission and energy designs. In multi-node networks, countervailing effects make an analytic analysis difficult. We present a formulation of both market designs as an equilibrium problem with equilibrium constraints, and apply our model to a realistic node network for Belgium, Germany, France and the Netherlands. We find that the integrated market design results in lower prices.

A nodal electric power network with Cournot-Nash interaction among power generators is formulated as a mixed complementarity problem. The model incorporates a direct current (DC) power flow approximation with thermal line losses to model real-time flows. We include constant wheeling rate and variable congestion charges for transmission of electricity. Market power and welfare effects are measured in an aggregated Indiana electric grid model. We find that imposing DC power flow constraints in a model results in significant changes in social welfare estimates. Line losses are also an important factor affecting market power and welfare of market participants in the case study. D

We make use the Banach contraction mapping principle to prove linear convergence of a regularization algorithm for strongly monotone Ky Fan inequalities that satisfy a Lipschitz-type condition recently introduced by Mastroeni. We then modify the proposed algorithm to obtain a linesearch-free algorithm which does not require the Lipschitz-type condition. We apply the proposed algorithms to implement inexact proximal methods for solving monotone (not necessarily strongly) Ky Fan inequalities. Applications to variational inequality and complementarity problems are discussed. As a consequence a linearly convergent derivative-free algorithm without linesearch for strongly monotone nonlinear complementarity problem is obtained. Application to a Nash-Cournot equilibria model is discussed and some preliminary computation results are reported.

This article describes an effective and simple primal heuristic to greedily encourage a reduction in the number of binary or 0 Á/1 logic variables before an implicit enumerative-type search heuristic is deployed to find integer-feasible solutions to 'hard' production scheduling problems. The basis of the technique is to employ well-known smoothing functions used to solve complementarity problems to the local optimization problem of minimizing the weighted sum over all binary variables the product of themselves multiplied by their complement. The basic algorithm of the 'smooth-and-dive accelerator' (SDA) is to solve successive linear programming (LP) relaxations with the smoothing functions added to the existing problem's objective function and to use, if required, a sequence of binary variable fixings known as 'diving'. If the smoothing function term is not driven to zero as part of the recursion then a branch-and-bound or branch-and-cut search heuristic is called to close the procedure finding at least integerfeasible primal infeasible solutions. The heuristic's effectiveness is illustrated by its application to an oil-refinery's crude-oil blendshop scheduling problem, which has commonality to many other production scheduling problems in the continuous and semicontinuous (CSC) process domains. #

In this paper, we consider some classes of merit functions for general variational inequalities. Using these functions, we obtain error bounds for the solution of general variational inequalities under some mild conditions. Since the general variational inequalities include variational inequalities, quasivariational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. In this respect, results obtained in this paper represent a refinement of previously known results for classical variational inequalities.  2005 Elsevier Inc. All rights reserved.

In this paper, we consider and study the implicit complementarity problem in the setting of a Hilbert lattice. It has been shown that this problem can be formulated as a fixed-point problem by using a suitable change of variables. Moreover, this formulation allows us to prove the existence and uniqueness of solutions of the implicit complementarity problem.

In this article, we extend two classes of merit functions for the second-order complementarity problem (SOCP) to infinite-dimensional SOCP. These two classes of merit functions include several popular merit functions, which are used in nonlinear complementarity problem, (NCP)/(SDCP) semidefinite complementarity problem, and SOCP, as special cases. We give conditions under which the infinite-dimensional SOCP has a unique solution and show that all these merit functions provide an error bound for infinite-dimensional SOCP and have bounded level sets. These results are very useful for designing solution methods for infinite-dimensional SOCP.

In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [14] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [18] to the SCCP. By exploiting the Cartesian P -properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R 02 -property. All of these results generalize some recent important works in ] under a unified framework.

Although one of the central questions in the global strategy field is how multinational firms successfully navigate multiple and often conflicting institutional environments, we know relatively little about the effect of conflicting labor market institutions on multinational firms' strategic choice and operating performance. With its decision to invest in manufacturing operations in nearly every one of the world's largest welding markets, Lincoln Electric offers us a quasiexperiment. We leverage a unique data set covering 1996-2005 that combines data on each host country's labor market institutions with data on each subsidiary's strategic choices and historical operating performance. We find that Lincoln Electric performed significantly better in countries with labor laws and regulations supporting manufacturers' interests and that Lincoln Electric performed significantly better in countries that allowed unconstrained use of incentive payfor-performance. Furthermore, we find that in countries with labor market institutions unfriendly to manufacturers, the company was still able to enhance its performance significantly by what we term flexible intermediate adaptation.

A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.

In this paper, we introduce some new notions of (α, γ)-exceptional family of elements (in short, (α, γ)-(EFE)) and (α, β, γ)-exceptional family of elements (in short, (α, β, γ)-(EFE)) for a pair of continuous functions involved in the implicit complementarity problem (in short, ICP). Based upon these notions and the topological degree theory, we studied the feasibility and strictly feasibility of (ICP) in R n and an infinite-dimensional Hilbert space H , respectively. As special cases, we obtain the feasibility and strictly feasibility of complementarity problems and partly answered the second open problem (P2) proposed by Isac [G. Isac, Exceptional families of elements, feasibility and complementarity,

This article describes an effective and simple primal heuristic to greedily encourage a reduction in the number of binary or 0-1 logic variables before an implicit enumerative-type search heuristic is deployed to find integer-feasible solutions to "hard" production scheduling problems. The basis of the technique is to employ well-known smoothing functions used to solve complementarity problems to the local optimization problem of minimizing the weighted sum over all binary variables the product of themselves multiplied by their complement. The basic algorithm of the "smooth-and-dive accelerator" (SDA) is to solve successive linear programming (LP) relaxations with the smoothing functions added to the existing problem's objective function and to use, if required, a sequence of binary variable fixings known as "diving". If the smoothing function term is not driven to zero as part of the recursion then a branch-and-bound or branch-and-cut search heuristic is called to close the procedure finding at least integer-feasible primal infeasible solutions. The heuristic's effectiveness is illustrated by its application to an oil-refinery's crude-oil blendshop scheduling problem, which has commonality to many other production scheduling problems in the continuous and semi-continuous (CSC) process domains.