Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method

On the Efficient Solution of Variational Inequalities; Complexity and Computational Efficiency

In this paper we combine ideas from cutting plane and interior point methods in order to solve variational inequality problems efficiently. In particular, we introduce a general framework that incorporates nonlinear as well as linear "smarter" cuts. These cuts utilize second order information on the problem through the use of a gap function. We establish convergence as well as complexity results for this framework. Moreover, in order to devise more practical methods, we consider an affine scaling method as it applies to symmetric, monotone variationalinequality problems and demonstrate its convergence. Finally, in order to further improve the computational efficiency of the methods in this paper, we combine the cutting plane approach with the affine scaling approach

Auxiliary problem principles for equilibria

The auxiliary problem principle allows solving a given equilibrium problem (EP) through an equivalent auxiliary problem with better properties. The paper investigates two families of auxiliary EPs: the classical auxiliary problems, in which a regularizing term is added to the equilibrium bifunction, and the regularized Minty EPs. The conditions that ensure the equivalence of a given EP with each of these auxiliary problems are investigated exploiting parametric definitions of different kinds of convexity and monotonicity. This analysis leads to extending some known results for variational inequalities and linear EPs to the general case together with new equivalences. Stationarity and convexity properties of gap functions are investigated as well in this framework. Moreover, both new results on the existence of a unique solution and new error bounds based on gap functions with good convexity properties are obtained under weak quasimonotonicity or weak concavity assumptions

Solving Variational Inequalities Defined on A Domain with Infinitely Many Linear Constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples for the proposed methods

Existence and solution methods for equilibria

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algorithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them

An Augmented Lagrangian Approach to Conically Constrained Non-monotone Variational Inequality Problems

In this paper we consider a non-monotone (mixed) variational inequality model with (nonlinear) convex conic constraints. Through developing an equivalent Lagrangian function-like primal-dual saddle-point system for the VI model in question, we introduce an augmented Lagrangian primal-dual method, to be called ALAVI in the current paper, for solving a general constrained VI model. Under an assumption, to be called the primal-dual variational coherence condition in the paper, we prove the convergence of ALAVI. Next, we show that many existing generalized monotonicity properties are sufficient -- though by no means necessary -- to imply the above mentioned coherence condition, thus are sufficient to ensure convergence of ALAVI. Under that assumption, we further show that ALAVI has in fact an $o(1/\sqrt{k})$ global rate of convergence where $k$ is the iteration count. By introducing a new gap function, this rate further improves to be $O(1/k)$ if the mapping is monotone. Finally, we show that under a metric subregularity condition, even if the VI model may be non-monotone the local convergence rate of ALAVI improves to be linear. Numerical experiments on some randomly generated highly nonlinear and non-monotone VI problems show practical efficacy of the newly proposed method

Essays on variational inequalities and competitive supply chain models

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2004.Includes bibliographical references (p. 103-107).In the first part of the thesis we combine ideas from cutting plane and interior point methods to solve variational inequality problems efficiently. In particular, we introduce "smarter" cuts into two general methods for solving these problems. These cuts utilize second order information on the problem through the use of a gap function. We establish convergence results for both methods, as well as complexity results for one of the methods. Finally, we compare the performance of an approach that combines affine scaling and cutting plane methods with other methods for solving variational inequalities. The second part of the thesis considers a supply chain setting where several capacitated suppliers compete for orders from a single retailer in a multi-period environment. At each period the retailer places orders to the suppliers in response to the prices and capacities they announce. Our model allows the retailer to carry inventory. Furthermore, suppliers can expand their capacity at an additional cost; the retailer faces exogenous, price-dependent, stochastic demand. We analyze discrete as well as continuous time versions of the model: (i) we illustrate the existence of equilibrium policies; (ii) we characterize the structure of these policies; (iii) we consider coordination mechanisms; and (iv) we present some computational results. We also consider a modified model that uses option contracts and finally present some extensions.by Marina Zaretsky.Ph.D

Adaptive Coarse Spaces for FETI-DP and BDDC Methods

Iterative substructuring methods are well suited for the parallel iterative solution of elliptic partial differential equations. These methods are based on subdividing the computational domain into smaller nonoverlapping subdomains and solving smaller problems on these subdomains. The solutions are then joined to a global solution in an iterative process. In case of a scalar diffusion equation or the equations of linear elasticity with a diffusion coefficient or Young modulus, respectively, constant on each subdomain, the numerical scalability of iterative substructuring methods can be proven. However, the convergence rate deteriorates significantly if the coefficient in the underlying partial differential equation (PDE) has a high contrast across and along the interface of the substructures. Even sophisticated scalings often do not lead to a good convergence rate. One possibility to enhance the convergence rate is to choose appropriate primal constraints. In the present work three different adaptive approaches to compute suitable primal constraints are discussed. First, we discuss an adaptive approach introduced by Dohrmann and Pechstein that draws on the operator P_D which is an important ingredient in the analysis of iterative substructuring methods like the dual-primal Finite Element Tearing and Interconnecting (FETI-DP) method and the closely related Balancing Domain Decomposition by Constraints (BDDC) method. We will also discuss variations of the method by Dohrmann and Pechstein introduced by Klawonn, Radtke, and Rheinbach. Secondly, we describe an adaptive method introduced by Mandel and Sousedík which is also based on the P_D-operator. Recently, a proof for the condition number bound in this method was provided by Klawonn, Radtke, and Rheinbach. Thirdly, we discuss an adaptive approach introduced by Klawonn, Radtke, and Rheinbach that enforces a Poincaré- or Korn-like inequality and an extension theorem. In all approaches generalized eigenvalue problems are used to compute a coarse space that leads to an upper bound of the condition number which is independent of the jumps in the coefficient and depend on an a priori prescribed tolerance. Proofs and numerical tests for all approaches are given in two dimensions. Finally, all approaches are compared

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal