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.     

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.     

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.     

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.     

一种非精确求解结构型变分不等式的渐近点算法

近来,交替方向法成为了学者们研究的热点。对于一类子问题能够精确求解的变分不等式,该算法是有效的。然而,在实际问题中,变分不等式的子问题是非常困难甚至是不可能精确求解的。在渐近点算法的基础上得到一种非精确的渐近点算法,使得变分不等式子问题具有显式解,通过简单的预测校正步得到子问题的解。在合理的假设下,算法的收敛性得到了证明,一些数值实验表明了所提算法的有效性。     

基于模糊函数估计和变分泛函数优化的含噪声模糊图像复原算法

针对摄影图像中常见的噪声和模糊同时退化图像的问题,提出一种Radon变换估计模糊函数结合变分泛函优化的图像复原算法。算法首先利用方向滤波器对图像去噪,然后利用Radon变换以去除方向滤波在模糊函数估计中引起的偏差,再利用模糊函数通过去卷积操作去除模糊,最后在去模糊的基础上利用 L0梯度范数的变换泛函的优化算法,去除剩余噪声从而对图像进行复原。实验表明,该算法可以有效地恢复摄影图像中的场景。     

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.     

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     

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.     

利用点扩展函数信息的变分图像去噪及增晰方法

基于偏微分方程(PDE)及变分法的图像去噪方法利用其数学特性得到了优于传统方法的结果,但很多模型只考虑了去噪的问题。通过对最小化凸能量函数模型引入点扩展函数信息,构造了具有去模糊效果的变分去噪模型,采用了Kacˇanov线性化方法进行求解,得到了更好的结果,实验结果及数据证明了模型的有效性。     

求解最优控制问题的混合变量变分方法及其航天控制应用

针对非线性最优控制导出的Hamiltonian系统两点边值问题,提出一种以离散区段右端状态和左端协态为混合独立变量的数值求解方法, 将非线性Hamiltonian系统两点边值问题的求解通过混合独立变量变分原理转化为非线性方程组求解.所提出的算法综合了求解最优控制 的"直接法"和"间接法"的特征,既满足最优控制理论的一阶必要条件,又不需要对协态初值的准确猜测,避免了求解大规模非线性规划问题. 通过两个航天控制算例讨论了本文算法的精度和效率等问题.与近年来在航空航天控制中备受关注的高斯伪谱方法相比较,本文算法无论是在 精度还是效率上都具有明显的优势.     

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.