Variational Inequality Problems

Between 1970 and 1974 he held staff positions successively at Logicon and Hughes xGround Systems Group, where he did applied research in program verification, higher level language computer architecture, and digital hardware design languages. From 1974 to 1980 he was an Assistant Professor in the Department of Information Engineering at the University of Illinois, Chicago, where he taught computer science and conducted basic research in inductive learning and production systems. Since 1980 he has been with the Information Systems Research Section at the Jet Propulsion Laboratory, Pasadena, CA. His present research interests are intelligent planning, reasoning, and learning systems in the context of space exploration and national defense.

Dynamic Traffic Assignment (DTA) has evolved substantially since the pioneering work of Merchant and Nemhauser. Numerous formulations and solutions approaches have been introduced ranging from mathematical programming, to variational inequality, optimal control, and simulation-based. The aim of this special issue is to document the main existing DTA approaches for future reference. This opening paper will summarize the current understanding of

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.

In this paper, we suggest and consider a class of new three-step approximation schemes for general variational inequalities. Our results include Ishikawa and Mann iterations as special cases. We also study the convergence criteria of these schemes.

We propose a prox-type method with efficiency estimate O(−1) for approximating saddle points of convex-concave C 1,1 functions and solutions of variational inequalities with monotone Lipschitz continuous operators. Application examples include matrix games, eigenvalue minimization and computing Lovasz capacity number of a graph and are illustrated by numerical experiments with large-scale matrix games and Lovasz capacity problems.

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.

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.

... Optimal control of variational inequalities. Post a Comment. CONTRIBUTORS: Author: Barbu, V. PUBLISHER: Pitman Advanced Pub. Program (Boston). SERIES TITLE: YEAR: 1984. PUB TYPE: Book (ISBN 0273086294 [pbk]). VOLUME/EDITION: PAGES (INTRO/BODY): 298 p. ...

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,

In this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization, convex-concave games and variaüonal inequaliües. We outline the ideas underlying these methods and present rate-of-convergence estimates.

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 variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disdphnes, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modehng, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equihbrium problems.

In this paper, we introduce and study a new class of variational inequalities: Projection technique is used to suggest an iterative algorithm for finding the approximate solution of this class. We also discuss the convergence criteria of the iterative algorithm. Several special cases are discussed, which can be obtained from the general result.

The existence and uniqueness of the solution of a backward SDE, on a random (possibly inÿnite) time interval, involving a subdi erential operator is proved. We then obtain a probabilistic interpretation for the viscosity solution of some parabolic and elliptic variational inequalities.

We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Fréchet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued differential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton-Jacobi equations. By the infimum-convolution description of the Hamilton-Jacobi solutions, this approach provides a clear view of the connection between logarithmic Sobolev inequalities and transportation cost inequalities investigated recently by F. Otto and C. Villani. In particular, we recover in this way transportation from Brunn-Minkowski inequalities and for the exponential measure.  2001 Éditions scientifiques et médicales Elsevier SAS

In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic Mirror-Prox (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the

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.

We make use of the auxiliary problem principle to develop iterative algorithms for solving equilibrium problems. The first one is an extension of the extragradient algorithm to equilibrium problems. In this algorithm the equilibrium bifunction is not required to satisfy any monotonicity property, but it must satisfy a certain Lipschitztype condition. To avoid this requirement we propose linesearch procedures commonly used in variational inequalities to obtain projection-type algorithms for solving equilibrium problems. Applications to mixed variational inequalities are discussed. A special class of equilibrium problems is investigated and some preliminary computational results are reported.

In this paper we study the existence of bounded weak solutionsfor some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the plaplacian operator, and the lower order terms, which depend on the solution u and its gradient ∇u, have a power growth of order p − 1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.

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 dynamic framework for the modeling and analysis of supply chain networks with corporate social responsibility through integrated environmental decision-making. Through a multilevel supply chain network, we model the multicriteria decision-making behavior of the various decision-makers (manufacturers, retailers, and consumers), which includes the maximization of profit, the minimization of emission (waste), and the minimization of risk. We explore the dynamic evolution of the product flows, the associated product prices, as well as the levels of social responsibility activities on the network until an equilibrium pattern is achieved. We provide some qualitative properties of the dynamic trajectories, under suitable assumptions, and propose a discrete-time algorithm which is then applied to track the evolution of the levels of social responsibility activities, product flows and prices over time. We illustrate the model and computational procedure with several numerical examples.

In this paper we suggest new dual methods for solving variational inequalities with monotone operators. We show that with an appropriate stepsize strategy, our method is optimal both for Lipschitz continuous operators (O(1) iterations), and for the operators with bounded variations (O(1 2) iterations). Our technique can be applied for solving non-smooth convex minimization problems with known structure. In this case the worst-case complexity bound is O(1) iterations.

A very general optimization problem with a variational inequality constraint, inequality constraints and an abstract constraint is studied. Fritz John type and Kuhn-Tucker type necessary optimality conditions involving Mordukhovich coderivatives are derived. Several constraint quali cations for the Kuhn-Tucker type necessary optimality conditions involving Mordukhovich coderivatives are introduced and their relationships are studied. Applications to bilevel programming problems are also given.

Utility maximization problems of mixed optimal stopping/control type are considered, which can be solved by reduction to a family of related pure optimal stopping problems. Sufficient conditions for the existence of optimal strategies are provided in the context of continuoustime, Itô process models for complete markets. The mathematical tools used are those of optimal stopping theory, continuous-time martingales, convex analysis, and duality theory. Several examples are solved explicitly, including one which demonstrates that optimal strategies need not always exist.

Methods of maximal monotone operators are used in order to study, from a general point of view, duality numerical algorithms for solving variational inequalities. With classical algorithms, such as Uxawa's method for the standard and augmented Lagrangian, this paper presents some new algorithms, which appear to have very good numerical performances.

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.

Given a point-to-set operator T , we introduce the operator T ε defined as T ε (x) = {u : u -v, x -y ≥ -ε for all y ∈ R n , v ∈ T (y)}. When T is maximal monotone T ε inherits most properties of the ε-subdifferential, e.g. it is bounded on bounded sets, T ε (x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T ε , and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consists of solving problems of the form 0 ∈ H ε (x), while the subproblems of the exact method are of the form 0 ∈ H(x). If ε k is the coefficient used in the k-th iteration and the ε k 's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is enough well behaved, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.

The concept of cognitive radio (CR) has recently received great attention from the research community as a promising paradigm to achieve efficient use of the frequency resource by allowing the coexistence of licensed (primary) and unlicensed (secondary) users in the same bandwidth. In this paper, we propose a novel Nash equilibrium (NE) problem to model concurrent communications of cognitive secondary users who compete against each other to maximize their information rate. The formulation contains constraints on the transmit power (and possibly spectral masks) as well as aggregate interference tolerable at the primary users' receivers. The coupling among the strategies of the players due to the interference constraints presents a new challenge for the analysis of this class of Nash games that cannot be addressed using the game theoretical models proposed in the literature. For this purpose, we need the framework given by the more advanced theory of finite-dimensional variational inequalities (VI). This provides us with all the mathematical tools necessary to analyze the proposed NE problem (e.g., existence and uniqueness of the solution) and to devise alternative distributed algorithms along with their convergence properties.

Sensitivity analysis results for variational inequalities are presented which give conditions for existence and equations for calculating the derivatives of solution variables with respect to perturbation parameters. The perturbations are of both the variational inequality function and the feasible region. Results for the special case of nonlinear complementarity are also presented. A numerical example demonstrates the results for variational inequalities.

Projection Technique is used to suggest a unified and general iterative algorithm for computing the approximate solution of a new class of quasi variational inequalities. The convergence properties of this algorithm are also considered. Several special cases which can be obtained from the general results are also discussed.

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algorithm for solving several classes of convex programs.

We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior, without taking explicitly into account the constraints which define the convex set. We establish full convergence to a solution with minimal conditions upon the monotone operator F, weaker than strong monotonicity or Lipschitz continuity, for instance, and including cases where the solution needs not be unique. We apply our algorithm to several relevant classes of convex sets, including orthants, boxes, polyhedra and balls, for which Bregman functions are presented which give rise to explicit iteration formulae, up to the determination of two scalar stepsizes, which can be found through finite search procedures.

We consider equilibrium problems in the framework of the formulation proposed by Blum and Oettli, which includes variational inequalities, Nash equilibria in noncooperative games, and vector optimization problems, for instance, as particular cases. We establish new su cient and=or necessary conditions for existence of solutions of such problems. Our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems, together with extensions to equilibrium problems of gap functions for variational inequalities. Then we apply our results to some particular instances of equilibrium problems, obtaining results which include, among others, a new lemma of the alternative for convex optimization problems.

This paper proposes a time-dependent network equilibrium model that simultaneously considers a trav-elerÕs choice of departure time, route, parking location and parking duration in road networks with multiple user classes and multiple parking facilities. In the proposed model, travelers are differentiated by their trip purpose and parking duration, parking locations are characterized by facility type and parking charge, and the decision-making process of travelers on travel and parking choices is assumed to follow a hierarchical choice structure. The model is formulated as a variational inequality problem, and is solved by a heuristic solution algorithm. Numerical results for two example networks are presented to show the solution quality and investigate the solution sensitivities to some input data. It is found that parking behavior is significantly affected by travel demand, walking distance, parking capacity, and parking charge. The proposed model provides a useful tool for studying the complex temporal and spatial interaction between road traffic and parking congestion, and can be used to assess the effects of various parking policies and infrastructure improvements at a strategic level.

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.

We consider the asymmetric continuous traffic equilibrium network model with fLxed demands where the travel cost on each link of the transportation network may depend on the flow on this as well as other links of the network and we perform stability and sensitivity analysis, Assuming that the travel cost functions are monotone we first show that the traffic equlibrium pattern depends continuously upon the assigned travel demands and travel cost functions. We then focus on the delicate question of predicting the direction of the change in the traffic pattern and the incurred travel costs resulting from changes in the travel cost functions and travel demands and attempt to elucidate certain counter intuitive phenomena such as 'Braess' paradox'. Our analysis depends crucially on the fact that the governing equilibrium conditions can be formulated as a variational inequality.

This paper provides a means for comparing various computer codes for solving large scale mixed complementarity problems. We discuss inadequacies in how solvers are currently compared, and present a testing environment that addresses these inadequacies. This testing environment consists of a library of test problems, along with GAMS and MATLAB interfaces that allow these problems to be easily accessed. The environment is intended for use as a tool by other researchers to better understand both their algorithms and their implementations, and to direct research toward problem classes that are currently the most challenging. As an initial benchmark, eight different algorithm implementations for large scale mixed complementarity problems are briefly described and tested with default parameter settings using the new testing environment.

In this paper, we develop a dynamic framework for the modeling and analysis of supply chain networks with corporate social responsibility through integrated environmental decision-making. Through a multilevel supply chain network, we model the multicriteria decision-making behavior of the various decision-makers (manufacturers, retailers, and consumers), which includes the maximization of profit, the minimization of emission (waste), and the minimization of risk. We explore the dynamic evolution of the product flows, the associated product prices, as well as the levels of social responsibility activities on the network until an equilibrium pattern is achieved. We provide some qualitative properties of the dynamic trajectories, under suitable assumptions, and propose a discrete-time algorithm which is then applied to track the evolution of the levels of social responsibility activities, product flows and prices over time. We illustrate the model and computational procedure with several numerical examples.

In this paper, we introduce a system of vector equilibrium problems and prove the existence of a solution. As an application, we derive some existence results for the system of vector variational inequalities. We also establish some existence results for the system of vector optimization problems, which includes the Nash equilibrium problem as a special case.

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.

The dynamic user-optimal (DUO) departure time and route choice problem is to determine travelers' best departure times and route choices at each instant of time. In a previous paper, we presented a route-based two-level optimal control model for the DUO departure time/route choice problem. However, this model is not appropriate for large scale transportation networks because some degree of route enumeration is necessary to solve the model. In this paper, we present a link-based variational in equality (VI) formulation for the DUO departure time/route choice problem so that route enumeration can be avoided in both the formulation and the solution procedure. The model extends our previous VI model for the DUO route choice problem to the case where both departure time and route over a general road network must be chosen simultaneously. By proving the necessity and sufficiency of this VI, we establish the equivalence of the VI formulation and the link-based DUO departure time/route choice conditions.