Variational Inequalities

We develop elements of calculus of variational sets for set-valued mappings, which were recently introduced in Khanh and Tuan (2008) [1,2] to replace generalized derivatives in establishing optimality conditions in nonsmooth optimization. Most of the usual calculus rules, from chain and sum rules to rules for unions, intersections, products and other operations on mappings, are established. Direct applications in stability and optimality conditions for various vector optimization problems are provided.

In this paper, an equilibrium model of a competitive supply chain network is developed. Such a model is sufficiently general to handle many decision-makers and their independent behaviors. The network structure of the supply chain is identified and equilibrium conditions are derived. A finite-dimensional variational inequality formulation is established. Qualitative properties of the equilibrium model and numerical examples are given. Ó

The purpose of this paper is to prove the existence of solutions of the Stampacchia variational inequality for a quasimonotone multivalued operator without any assumption on the existence of inner points. Moreover, the operator is not supposed to be bounded valued. The result strengthens a variety of other results in the literature.

We explore convergence notions for bivariate functions that yield convergence and stability results for their maxinf (or minsup) points. This lays the foundations for the study of the stability of solutions to variational inequalities, the solutions of inclusions, of Nash equilibrium points of non-cooperative games and Walras economic equilibrium points, of fixed points, the primal and dual solutions of convex optimization problems and of zero-sum games. These applications will be dealt with in a couple of accompanying papers.

We explore convergence notions for bivariate functions that yield convergence and stability results for their maxinf (or minsup) points. This lays the foundations for the study of the stability of solutions to variational inequalities, the solutions of inclusions, of Nash equilibrium points of non-cooperative games and Walras economic equilibrium points, of fixed points, the primal and dual solutions of convex optimization problems and of zero-sum games. These applications will be dealt with in a couple of accompanying papers.

In this paper, the equivalence between variational inclusions and a generalized type of Weiner-Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brezis-Crandall-Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.

This paper concerns developing two hybrid proximal point methods (PPMs) for finding a common solution of some optimization-related problems. First we construct an algorithm to solve simultaneously an equilibrium problem and a variational inequality problem, combing the extragradient method for variational inequalities with an approximate PPM for equilibrium problems. Next we develop another algorithm based on an alternate approximate PPM for finding a common solution of two different equilibrium problems. We prove the global convergence of both algorithms under pseudomonotonicity assumptions.

In this paper, we develop an integrated framework for the modeling of reverse supply chain management of electronic waste, which includes recycling. We describe the behavior of the various decision-makers, consisting of the sources of electronic waste, the recyclers, the processors, as well as the consumers associated with the demand markets for the distinct products. We construct the multitiered e-cycling network equilibrium model, establish the variational inequality formulation, whose solution yields the material flows as well as the prices, and provide both qualitative properties of the equilibrium pattern as well as numerical examples that are solved using the proposed algorithm.

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inversestrongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.

It is well known that in the standard traffic network equilibrium model with a single value of time (VOT) for all users, a so-called marginal-cost toll can drive a user equilibrium flow pattern to a system optimum. This result holds when either cost (money) or time units are used in expressing the objective function of the system optimum and the criterion for user equilibrium. This paper examines the multi-criteria or the costversus-time network equilibrium and system optimum problem in a network with a discrete set of VOTs for several user classes. Specifically, the following questions are investigated: Are the user-optimal flows dependent upon the unit (time or money) used in measuring the travel disutility in the presence of road pricing? Are there any uniform link tolls across all individuals (link tolls that are identical for all user classes) that can support a multi-class user equilibrium flow pattern as a system optimum when the system objective function is measured by either money or time units? What are the general properties of the valid toll set? Ó equilibrium model and network performance evaluation because it describes how users make tradeoffs between cost and time in response to toll charges. Conventionally, VOTs are assumed to be identical for all users (homogeneous users). In the case of homogeneous users, the first-best congestion pricing theory, namely, the theory of marginal cost pricing is well established in general traffic networks. In line with this theory, a toll that is equal to the difference between the marginal social cost and the marginal private cost is charged on each link, so as to achieve a system optimum flow pattern in the network . Investigations have been conducted on how this classical economic principle would work in a general congested road network with queuing (Yang and Huang, 1998) and in a congested network in a stochastic equilibrium . In spite of its perfect theoretical basis, the principle of marginal cost pricing can be difficult to apply in real situations. Apart from public and political resistance, a primary reason is due to the high extra cost spent on the equipment for toll collection over the entire network. This has motivated a number of researchers to consider various forms of second-best pricing schemes where only a subset of links is subjected to toll charges. A typical simple example of the second-best pricing involves two parallel routes where an untolled alternative exists. This problem has been investigated in both static and dynamic situations by, for example,

This paper presents two neural networks to find the optimal point in convex optimization problems and variational inequality problems, respectively. The domain of the functions that define the problems is a convex set, which is determined by convex inequality constraints and affine equality constraints. The neural networks are based on gradient descent and exact penalization and the convergence analysis is based on a control Liapunov function analysis, since the dynamical system corresponding to each neural network may be viewed as a so-called variable structure closed loop control system.

We generalize some results due to Pappalardo and Passacantando [10]. We prove necessary and sufficient conditions for the monotonicity of a trajectory of an autonomous dynamical system with locally Lipschitz data, by means of Clarke’s generalized Jacobian. Some of the results are developed in the framework of variational inequalities.

In this paper we propose a new model for the equilibrium multi-modal assignment problem with combined modes (MAPCM) for the case of asymmetric costs. MAPCM is stated on a generic passenger assignment equilibrium model, on a generalized traffic assignment model, and on a nested logit distribution as demand model which explicitly takes into account the choice of mode of transport and transfer node among modal networks. This model is formulated as a variational inequality problem in the space of the hyperpath flows and then solved by the disaggregate simplicial decomposition (DSD) algorithm.

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.

This is a selection of facts, old and recent, about quadrature domains. The text, written in the form of a survey, is addressed to non-experts and covers a vari- ety of phenomena related to quadrature domains. Such as: the dierence between quadrature domains for subharmonic, harmonic and respectively complex analytic functions, geometric properties of the boundary, instability in the reverse Hele- Shaw flow, dependence and non-uniqueness on the quadrature data, interpretation in terms of function theory on Riemann surfaces, a matrix model and a reconstruc- tion algorithm. Plus some low degree/order examples where computations can be carried out in detail.

Currently the modeling of check valves and flow control valves in water distribution systems is based on heuristics intermixed with solving the set of nonlinear equations governing flow in the network. At the beginning of a simulation, the operating status of these valves is not known and must be assumed. The system is then solved. The status of the check valves and flow control valves are then changed to try to determine their correct operating status, at times leading to incorrect solutions even for simple systems. This paper proposes an entirely different approach. Content and co-content theory is used to define conditions that guarantee the existence and uniqueness of the solution. The work here focuses solely on flow control devices with a defined head discharge versus head loss relationship. A new modeling approach for water distribution systems based on subdifferential analysis that deals with the nondifferentiable flow versus head relationships is proposed in this paper. The water distribution equations are solved as a constrained nonlinear programming problem based on the content model where the Lagrangian multipliers have important physical meanings. This new method gives correct solutions by dealing appropriately with inequality and equality constraints imposed by the presence of the flow regulating devices ͑check valves, flow control valves, and temporarily closed isolating valves͒. An example network is used to illustrate the concepts.

A number of phenomenological models for the adhesive contact problem are presented in this paper. The nonmonotone nature of the adhesive contact laws and the inequalities that are introduced by unilateral contact effects lead to nonsmooth and nonconvex potential energy optimization problems. For discretised problems the potential energy function is in general assumed to be qua&differentiable and/or the sets that describe the inequality subsidiary conditions are also assumed to be described by quasidifferentiable functions.

AbstractÐIn this paper, we develop a multimodal trac network equilibrium model with vehicular emission pollution permits using the theory of variational inequalities. We consider both the case of compliance in which travelers emit pollutants no more than that mandated by their license holdings and that of noncompliance. In the former case, we prove that environmental standards imposed by the authorities are met provided that the initial license allocation meets the environmental target. For the latter model, we establish a penalty scheme that guarantees that there will be no nonconcompliant behavior. Qualitative analysis of the model is conducted and existence and uniqueness results of the solution obtained. An algorithm is proposed, with convergence results, to compute the multimodal equilibrium link load, travel demand, marginal cost of emission abatement, emission under¯ow, over¯ow, license, and license price pattern. The algorithm is then applied to several numerical examples. # ). It is clear that transportation policies to reduce pollution must be coupled with congestion reduction. Indeed, this point has been reiterated by many studies (cf. .

In this paper, we use uniform quartic polynomial splines to develop a new method, which is used for computing approximations to the solution and its ÿrst, second as well as third derivatives for a system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. It is shown that the present method is of order two and gives approximations which are better than those produced by other collocation and ÿnite di erence methods. Numerical examples are presented to illustrate the applicability of the new method. (E.A. Al-Said), [email protected] (M.A. Noor).

We study a new class of electromagnetostatic problems in the variational framework of the subspace of W1,p(Ω) of vector functions with zero divergence and zero normal trace, for [Formula: see text], in smooth, bounded and simply connected domains Ω of ℝ3. We prove a Poincaré–Friedrichs type inequality and we obtain the existence of steady-state solutions for an electromagnetic induction heating problem and for a quasi-variational inequality modelling a critical state generalized problem for type-II superconductors.

In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primaldual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. The proposed schemes differ from the classical approaches (divergent series methods, mirror descent methods) by presence of two control sequences. The first sequence is responsible for aggregating the support functions in the dual space, and the second one establishes a dynamically updated scale between the primal and dual spaces. This additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set (however, we always assume the uniform boundedness of subgradients). We present the variants of subgradient schemes for nonsmooth convex minimization, minimax problems, saddle point problems, variational inequalities, and stochastic optimization. In all situations our methods are proved to be optimal from the view point of worst-case black-box lower complexity bounds.

A modified Gilbert equation for micromagnetics is considered, obtained by augmenting the standard viscous-like dissipation with a rate-independent term. We prove existence of a weak solution both with and without viscous dissipation. By scaling time we show that, if the applied field varies very slowly, then gyromagnetic effects and viscous dissipation become negligible. In the infinitesimally-slow-loading limit, the system thus becomes fully rate-independent.

The Australian River Assessment System (AusRivAS) is a nation-wide programme designed to assess the health of Australian rivers and streams. In order to produce river health assessments, the AusRivAS method uses the outcomes of cluster analysis applied to macroinvertebrate data from a number of different locations. At present, the clustering step is conducted using the statistical Unweighted Pair Group Arithmetic Averaging (UPGMA) method. A potential shortcoming of this approach is that it uses a linear performance measure for grouping similar data points. A recently developed approach for clustering ecological data (MIR-max) overcomes this limitation by using mutual information as the performance measure. However, MIR-max uses a hillclimbing approach for optimising mutual information, which could become trapped in local optima of the search space. In this paper, the potential of using evolutionary algorithms (EAs), such as genetic algorithms and ant colony optimisation algorithms, for maximising the mutual information of ecological data clusters is investigated. The MIR-max and EA-based approaches are applied to the South Australian combined season riffle AusRivAS data, and the results obtained are compared with those obtained using the UPGMA method. The results indicate that the overall mutual information values of the clusters obtained using MIR-max and the EA-based approaches are significantly higher than those obtained using the UPGMA method, and that the use of genetic and ant colony optimisation algorithms is successful in determining clusters with higher overall mutual information values compared with those obtained using MIR-max for the case study considered.

Moving from the seminal papers of Han and Reddy, we propose a ÿxed-point algorithm for the solution of hardening plasticity two-dimensional problems. The continuous problem may be classiÿed as a mixed non-linear non-di erentiable variational inequality of the second type and is discretized by means of a truly mixed ÿnite-element scheme. One of the main peculiarities of our approach is the use of the composite triangular element of Johnson and Mercier for the approximation of the stress ÿeld. The nondi erentiability is coped with via regularization whereas the non-linearity is approached with a ÿxedpoint iterative strategy. Numerical results are proposed that investigate the sensitivity of the approach with respect to the mesh size and the regularization parameter . The simplicity of the proposed ÿxedpoint scheme with respect to more classical return mapping approaches seems to represent one of the key features of our algorithm. 84 P. VENINI AND R. NASCIMBENE mixed formulations in elastoplasticity include the following noteworthy ones:

We propose a variational model for one of the most important problems in traffic networks, namely, the network equilibrium flow that is, traditionally in the context of operations research, characterized by minimum cost flow. This model has the peculiarity of being formulated by means of a suitable variational inequality (VI) and its solution is called "equilibrium". This model becomes a minimum cost model when the cost function is separable or, more general, when the Jacobian of the cost operator is symmetric; in such cases a functional representing the total network utility exists. In fact in these cases we can write the first order optimality conditions which turn out to be a VI. In the other situations (i.e., when global utility functional does not exist), which occur much more often in the real problems, we can study the network by looking for equilibrium solutions instead of minimum points. The Lagrangean approach to the study of the VI allows us to introduce dual variables, associated to the constraints of the feasible set, which may receive interesting interpretations in terms of potentials associated to the arcs and the nodes of the network. This interpretation is an extension and generalization of the classic Bellman conditions. Finally, we deepen the analysis of the networks having capacity constraints.

Nous nous intéressons à la valeur asymptotique dans les jeux répétés à somme nulle avec une évaluation générale de la suite des paiements d'étapes. Nous montrons l'existence de la valeur asymptotique dans un sens robuste dans les jeux répétés à information incomplète, les jeux de splitting et les jeux absorbants. La technique de preuve consiste (1) à plonger le jeu répété en temps discret dans un jeu en temps continu et (2) à utiliser les solutions de viscosités.

In this paper, we suggest and analyze some new iterative methods for solving general monotone mixed variational inequalities, which are being used to study odd-order and nonsymmetric boundary value problems arising in pure and applied sciences. These new methods can be viewed as generalizations and extensions of Ž . the methods of He, Solodov and Tseng, and Noor for solving monotone mixed variational inequalities.

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.

Moving from the seminal papers of Han and Reddy, we propose a ÿxed-point algorithm for the solution of hardening plasticity two-dimensional problems. The continuous problem may be classiÿed as a mixed non-linear non-di erentiable variational inequality of the second type and is discretized by means of a truly mixed ÿnite-element scheme. One of the main peculiarities of our approach is the use of the composite triangular element of Johnson and Mercier for the approximation of the stress ÿeld. The nondi erentiability is coped with via regularization whereas the non-linearity is approached with a ÿxedpoint iterative strategy. Numerical results are proposed that investigate the sensitivity of the approach with respect to the mesh size and the regularization parameter . The simplicity of the proposed ÿxedpoint scheme with respect to more classical return mapping approaches seems to represent one of the key features of our algorithm. 84 P. VENINI AND R. NASCIMBENE mixed formulations in elastoplasticity include the following noteworthy ones:

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.

We consider the development of single-timescale schemes for the distributed computation of equilibria associated with Nash games in which each player solves a convex program. Equilibria associated with such games are wholly captured by the solution set of a variational inequality. Our focus is on a class of games, termed as monotone Nash games, that lead to monotone variational inequalities. Distributed extensions of standard approaches for solving such variational problems are characterized by two challenges: (1) Unless suitable assumptions (such as strong monotonicity) are imposed on the mapping arising in the specification of the variational inequality, iterative methods often require the solution of a sequence of regularized problems, a naturally two-timescale process that is harder to implement in practice; (2) Additionally, algorithm parameters for all players (such as steplengths, regularization parameters, etc.) have to be chosen centrally and communicated to all players; importantly, these parameters cannot be independently chosen by a player. Motivated by these shortcomings, we present two practically implementable distributed regularization schemes that work on a single-timescale; specifically, each scheme requires precisely one gradient or projection step at every iteration. Of these, the first is an iterative Tikhonov regularization (ITR) scheme while the second is an analogously constructed iterative proximal-point (IPP) method. Both schemes are characterized by the property that the regularization/centering parameter are updated after every iteration, rather than when one has approximately solved the regularized problem. To aid in distributed settings requiring limited coordination across players, the schemes allow players to select their parameters independently and do not insist on central prescription of such parameters. We conclude with an application of these schemes on a networked Cournot game with nonlinear prices.

In this paper, we use uniform quartic polynomial splines to develop a new method, which is used for computing approximations to the solution and its ÿrst, second as well as third derivatives for a system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. It is shown that the present method is of order two and gives approximations which are better than those produced by other collocation and ÿnite di erence methods. Numerical examples are presented to illustrate the applicability of the new method. (E.A. Al-Said), [email protected] (M.A. Noor).

This work pleads for the use of the concept of strategies, and their network-theoretic representation as hyperpaths, for modeling network assignment problems. While this concept describes adequately the behavior of users in transit systems, we show that it can apply as well to networks where arc capacities are rigid. This opens up a whole new field of research and raises several questions, from both the theoretical and computational points of view. These are investigated in the paper.

In this paper, we develop a supply chain network model in which both physical and electronic transactions are allowed and in which supply side risk as well as demand side risk are included in the formulation. The model consists of three tiers of decision-makers: the manufacturers, the distributors, and the retailers, with the demands associated with the retail outlets being random. We model the optimizing behavior of the various decisionmakers, with the manufacturers and the distributors being multicriteria decision-makers and concerned with both profit maximization and risk minimization. We derive the equilibrium conditions and establish the finite-dimensional variational inequality formulation. We provide qualitative properties of the equilibrium pattern in terms of existence and uniqueness results and also establish conditions under which the proposed computational procedure is guaranteed to converge. We illustrate the supply chain network model through several numerical examples for which the equilibrium prices and product shipments are computed. This is the first multitiered supply chain network equilibrium model with electronic commerce and with supply side and demand side risk for which modeling, qualitative analysis, and computational results have been obtained.

In this paper, we develop a supply chain network model consisting of manufacturers and retailers in which the demands associated with the retail outlets are random. We model the optimizing behavior of the various decision-makers, derive the equilibrium conditions, and establish the finite-dimensional variational inequality formulation. We provide qualitative properties of the equilibrium pattern in terms of existence and uniqueness results and also establish conditions under which the proposed computational procedure is guaranteed to converge. Finally, we illustrate the model through several numerical examples for which the equilibrium prices and product shipments are computed. This is the first supply chain network equilibrium model with random demands for which modeling, qualitative analysis, and computational results have been obtained.

Based on the logarithmic-quadratic proximal method, a descent alternating direction method is introduced for structured variational inequalities in the paper. The proposal method generates the new iteration by searching the optimal step size along the integrated descent direction from two descent directions. Global convergence of the proposal method is proved under certain assumptions. To illustrate the proposal method and demonstrate its efficiency, some applications and their numerical results are also provided.

In this paper, we develop a supply chain network model in which both physical and electronic transactions are allowed and in which supply side risk as well as demand side risk are included in the formulation. The model consists of three tiers of decision-makers: the manufacturers, the distributors, and the retailers, with the demands associated with the retail outlets being random. We model the optimizing behavior of the various decisionmakers, with the manufacturers and the distributors being multicriteria decision-makers and concerned with both profit maximization and risk minimization. We derive the equilibrium conditions and establish the finite-dimensional variational inequality formulation. We provide qualitative properties of the equilibrium pattern in terms of existence and uniqueness results and also establish conditions under which the proposed computational procedure is guaranteed to converge. We illustrate the supply chain network model through several numerical examples for which the equilibrium prices and product shipments are computed. This is the first multitiered supply chain network equilibrium model with electronic commerce and with supply side and demand side risk for which modeling, qualitative analysis, and computational results have been obtained.

The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first perform the analysis of this system. We then propose a discretization, where the displacements are approximated by standard finite elements and the action by a local postprocessing which admits an equivalent mixed reformulation. We prove the well-posedness of the discrete problem and establish optimal a priori error estimates.

The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the upper level decision variable, then using certain uniform parametric error bounds as penalty functions gives single level problems equivalent to the GBLP. Several local and global uniform parametric error bounds are presented, and assumptions guaranteeing that they apply are discussed. We then derive Kuhn-Tucker-type necessary optimality conditions by using exact penalty formulations and nonsmooth analysis.

A variational inequality problem with a mapping g : < n ! < n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in < n. Recently, several homotopy methods, such as interior-point and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods Chen-Mangasarian and Gabriel-Mor e proposed a class of smooth functions to approximate F. In this paper, we study several properties of the trajectory de ned by solutions of these smooth systems. We propose a homotopy-smoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. Furthermore, if the involved function g is an a ne function, the method nds a solution of the problem in nite steps. Preliminary numerical results indicate that the method is promising.

In this paper, we introduce a modified Mann iterative process for approximating a common fixed point of a finite family of strict pseudo-contractions in Hilbert spaces. We establish the strong convergence theorem of the general iteration scheme under some mild conditions. Our results extend and improve the recent ones announced by many others.