Models and Methods of Finite-Dimensional Variational Inequalities

Algorithms for solving finite-dimensional inequalities are studied. The emphasis is on numerical methods based on the optimization approach. Examples of economic equilibrium models that assist in solving variational inequalities are presented.     

Bicriteria Decision Making and Financial Equilibrium: A Variational Inequality Perspective

In this paper we develop a framework for the study of financial equilibrium in the case of sectors in the economy, each of which is faced with two objectives/criteria in his portfolio selection decision making. In particular, we first present the bicriteria decision model of an individual financial sector, who seeks an optimal portfolio composition, given that the wishes to minimize his risk and to maximize his return. We utilize a value function approach to reformulate a sector's bicriteria optimization problem as a single optimization problem and argue that constant weight value functions may not adequately reveal a sector's preference over the return and the risk. Hence, we introduce state-dependent weights for the modeling of a sector's decision-making problem. We, subsequently, provide qualitative properties of the value function. We state the economic system conditions governing the instrument prices, define the financial equilibrium conditions, and show that they can be formulated as a variational inequality problem. Finally, qualitative properties of existence and uniqueness of the equilibrium are obtained. This work is the first to establish the connections among bicriteria problems (in the general setting of value functions), financial equilibrium problems, and variational inequality problems.     

A Posteriori Error Estimators for a Class of Variational Inequalities

In this paper, we present an a posteriori error analysis for the finite element approximation of a variational inequality. We derive a posteriori error estimators of residual type, which are shown to provide upper bounds on the discretization error for a class of variational inequalities provided the solutions are sufficiently regular. Furthermore we derive sharp a posteriori error estimators with both lower and upper error bounds for a subclass of the obstacle problem which are frequently met in many physical models. For sufficiently regular solutions, these estimates are shown to be equivalent to the discretization error in an energy type norm. Our numerical tests show that these sharp error estimators are both reliable and efficient in guiding mesh adaptivity for computing the free boundaries.     

The Nonlocal Newton Method for Convex Optimization Problems and Monotone Variational Inequalities

The nonlocal Newton method is developed for nonlinear problems of conditional convex optimization and monotone variational inequalities in a finite-dimensional space. The Newton direction vector is calculated from a solution of a linear-approximating variational inequality. A new penalty function is proposed to define a step length.     

Penalty Method for Variational Inequalities with Multivalued Mappings. III

By means of penalty operators, a method of regularization of variational inequalities is generalized to a class of variational inequalities with multivalued mappings.     

Penalty Method for Variational Inequalities with Multivalued Mappings. Part II

By means of penalty operators, a method of regularization of variational inequalities is generalized to a class of variational inequalities with multivalued mappings.     

The Nonlocal Newton's Method for Monotone Variational Inequalities on a Polyhedron

A nonlocally converging Newton method is developed in the paper. In this method, a solution of one approximating linear variational inequality is used on an iteration. The method is more efficient as compared with the existing Newton-type algorithms.     

Dantzig—Wolfe Decomposition of Variational Inequalities

The creation and ongoing management of a large economic model can be greatly simplified if the model is managed in separate smaller pieces defined, e.g. by region or commodity. For this purpose, we define an extension of Dantzig–Wolfe decomposition for the variational inequality (VI) problem, a modeling framework that is widely used for models of competitive or oligopolistic markets. The subproblem, a collection of independent smaller models, is a relaxed VI missing some “difficult” constraints. The subproblem is modified at each iteration by information passed from the last solution of the master problem in a manner analogous to Dantzig–Wolfe decomposition for optimization models. The master problem is a VI which forms convex combinations of proposals from the subproblem, and enforces the difficult constraints. A valid stopping condition is derived in which a scalar quantity, called the “convergence gap,” is monitored. The convergence gap is a generalization of the primal-dual gap that is commonly monitored in implementations of Dantzig–Wolfe decomposition for optimization models. Convergence is proved under conditions general enough to be applicable to many models. An illustration is provided for a two-region competitive model of Canadian energy markets.     

Numerical Schemes for Variational Inequalities Arising in International Asset Pricing

This paper introduces a valuation model of international pricing in the presence of political risk. Shipments between countries are charged with shipping costs and the country specific production processes are modelled as diffusion processes. The political risk is modelled as a continous time jump process that affects the drift of the returns in the politically unstable countries. The valuation model gives rise to a singular stochastic control problem that is analyzed numerically. The fundamental tools come from the theory of viscosity solutions of the associated Hamilton–Jacobi–Bellman equation which turns out to be a system of integral-differential Variational Inequalities with gradient constraints.     

Existence Theorems for Nonconvex Problems of Variational Calculus

A solution to a variational calculus problem is studied under the conditions of integrant convexity. The existence theorem is proved. As an example, a function is given, which satisfies all the conditions of the theorem but is not convex.     

Twelve monotonicity conditions arising from algorithms for equilibrium problems

In the last years many solution methods for equilibrium problems (EPs) have been developed. Several different monotonicity conditions have been exploited to prove convergence. The paper investigates all the relationships between them in the framework of the so-called abstract EP. The analysis is further detailed for variational inequalities and linear EPs, which include also Nash EPs with quadratic payoffs.     

A Posteriori Error Estimates for Parabolic Variational Inequalities

We study a posteriori error estimates in the energy norm for some parabolic obstacle problems discretized with a Euler implicit time scheme combined with a finite element spatial approximation. We discuss the reliability and efficiency of the error indicators, as well as their localization properties. Apart from the obstacle resolution, the error indicators vanish in the so-called full contact set. The case when the obstacle is piecewise affine is studied before the general case. Numerical examples are given.     

An Approximate Solution for a Class of Second-Order Elliptic Variational Inequalities in Arbitrary-Form Domains

Penalty and dummy-domain methods are used to approximate second-order elliptic variational inequalities with a restriction inside a domain by nonlinear boundary-value problems in a rectangle. Difference schemes, with the order of accuracy O(h ^1/2) in the grid norm W ₂ ¹(), are constructed for these problems.     

同时配准-分割脑MR图像的耦合变分模型

提出一种新颖的变分耦合模型,同时实现配准与分割.模型中使用耦合函数将非刚性配准信息与基于区域信息的曲线演化理论结合在一起,构造总能量函数,通过求解该能量函数的极值达到配准-分割的目的.该方法可以分割多模态医学图像,即使在两图间的强度信息区别较大时,也可以得到较好的分割结果.实验结果表明该方法具有较好的鲁棒性.     

A Comparison of the Performance of Flexible Functional Forms for Use in Applied General Equilibrium Modelling

This paper describes a procedure for testing the global properties of functional forms which recognizes their specific role in economic equilibrium modelling. This procedure is employed to investigate the global regularity and third-order curvature properties of three widely used flexible functional forms, the Translog, the Generalized Leontieff and the Normalized Quadratic functional forms. We contrast the properties of these functions with a globally regular flexible form, the Non-separable Nested Constant-Elasticity-of-Substitution functional form. Our results indicate that inherently regular     

供应链网络双渠道均衡

研究了制造商通过分销商实体链和电子商务直销渠道,将其产品经由零售商销售给具有随机需求的消费市场的供应链网络双渠道均衡问题.利用有限维变分不等式理论,分别刻画了存在生产能力限制的供应市场、分销市场、零售市场以及存在限定性价格上限的消费市场的均衡,建立了整个供应链网络双渠道均衡模型,并且设计了供应链网络双渠道均衡的投影收缩算法.数值算例结果表明:当政府对竞争市场实行限制性价格上限时,将造成消费市场中的商品短缺,并导致制造商、分销商以及零售商的总利润减少,当存在产能约束时情况更为严重.     

A new approach to supply chain network equilibrium models

We establish nonlinear complementarity formulations for the supply chain network equilibrium models. The formulations have simple structures and facilitate us to study qualitative properties of the models. In this setting, we obtain weaker conditions to guarantee the existence and uniqueness of the equilibrium pattern for a supply chain. A smoothing Newton algorithm that exploits the network structure is proposed for solving these models. Not only is the smoothing Newton algorithm proved to be globally convergent without requiring the assumptions of monotonicity and Lipschitz continuity, but also it can overcome the flaw that the performance of the modified projection method heavily depends on the choice of the predetermined step size. Numerical results indicate the advantages of the nonlinear complementarity formulation and the smoothing Newton algorithm.     

A Variational Framework for Retinex

Retinex theory addresses the problem of separating the illumination from the reflectance in a given image and thereby compensating for non-uniform lighting. This is in general an ill-posed problem. In this paper we propose a variational model for the Retinex problem that unifies previous methods. Similar to previous algorithms, it assumes spatial smoothness of the illumination field. In addition, knowledge of the limited dynamic range of the reflectance is used as a constraint in the recovery process. A penalty term is also included, exploiting a-priori knowledge of the nature of the reflectance image. The proposed formulation adopts a Bayesian view point of the estimation problem, which leads to an algebraic regularization term, that contributes to better conditioning of the reconstruction problem.Based on the proposed variational model, we show that the illumination estimation problem can be formulated as a Quadratic Programming optimization problem. An efficient multi-resolution algorithm is proposed. It exploits the spatial correlation in the reflectance and illumination images. Applications of the algorithm to various color images yield promising results.     

A variational inequality approach for marketable pollution permits

In this paper, we present a variational inequality framework for the modeling, qualitative analysis, and computation of equilibria in markets in pollution permits. The models developed herein allow for different modes of competitive behavior, including oligopolistic behavior, while taking into account the cost of polluting plus the prices of the licenses to pollute at different receptor points. The models deal explicitly with spatial differentiation and also ensure that the imposed environmental quality standards are met through the initial allocation of licenses. An algorithm is proposed, along with convergence results, to compute the profit-maximized quantities of the firms' products and quantities of emissions, as well as the equilibrium allocation of licenses and their prices. Numerical examples are included to illustrate the approach. This is the first time that this methodology is utilized in environmental economics.     

A Study on Network Design Problems for Multi-modal Networks by Probit-based Stochastic User Equilibrium

This paper develops a multi-modal transport network model considering various travel modes including railway, bus, auto, and walking. Travellers are assumed to choose their multi-modal routes so as to minimise their perceived disutilities of travel following the Probit Stochastic User Equilibrium (SUE) condition. Factors influencing the disutility of a multi-modal route include actual travel times, discomfort on transit systems, expected waiting times, fares, and constants specific to transport modes. The paper then deals with the multi-modal network design problem (NDP). The paper employs the method of sensitivity analysis to define linear approximation functions between the Probit SUE link flows and the design parameters, which are then used as constraints in the sub-problem of the NDP instead of the original SUE condition. Based on this reformulated NDP, an efficient algorithm for solving the problem is proposed in the paper. Two instances of this general NDP formulation are then presented in the paper: the optimal frequency design problem for public transport services (FDP), and the anti-freezing admixture dispersion problem (AADP).