A variational approach for large deflection of ends supported nanorod under a uniformly distributed load,using intrinsic coordinate finite elements

This paper presents a variational formulation that can be used for large deflection analysis of ends supported nanorod including the coupled effects of nonlocal elasticity and surface stress under a uniformly distributed load. The variational formulation involving the strain energy due to bending of nonlocal elasticity including the surface stress effect and virtual work done by a uniformly distributed load, is expressed in terms of the intrinsic coordinates. The Lagrange multiplier technique is applied to impose the boundary conditions which accomplished in the formulation. The validity of the variational approach is ensured by Euler's equation, which identical to the one derived by the force equilibrium consideration of an infinitesimal nanorod segment. The finite element method and Newton–Raphson iterative procedure based on the variational formulation are used to solve a system of nonlinear equations. Moreover, the very large deflection configurations of ends supported nanorod are highlighted in this study.     

优化和均衡的等价性

通过向量优化问题, 向量变分不等式问题以及向量变分原理来分析优化问题及均衡问题的一致性.从而显然, 可以用统一的观点来处理数值优化、向量优化以及博弈论等问题.进而为非线性分析提供了一个新的发展空间.     

Super Efficiency of Multicriterion Network Equilibrium Model and Vector Variational Inequality

The super efficiency of a vector variational inequality is considered in this paper. We show that for both the single and multiple criteria cases, a network equilibrium model can be recast as super efficient solutions to a kind of variational inequality. For the network equilibrium model with a vector-valued cost function, we derive the necessary and sufficient condition in terms of the super efficiency of a vector variational inequality by using the Gerstewitz’s function without any convex assumptions.     

Dynamic Walrasian price equilibrium problem: evolutionary variational approach with sensitivity analysis

Many authors have been devoted to the study of the static general economic equilibrium problem regulated to Walras’ law (see e.g. Arrow and Debreu in Econometrica 22:265–290, 1954; Arrow and Hahn in General competitive analysis, 1991; Arrow et al. in Econometrica 27:82–109, 1959; Border in Fixed point theorems with application to economics and game theory, Cambridge University Press, Cambridge, 1985; Dafermos in Math Programm 46:391–402, 1990; Dafermos and Zhao in Oper Res Lett 10:396–376, 1991; Donato et al. in J Glob Optim, 2007; Hahn in Stability, North Holland, Amsterdam, 1982; Jofré et al. in Math Oper Res, 2007; Nagurney in Network economics—a variational inequality approach, Kluwer, Dordrecht, 1999; Nagurney and Zhao in Network formalism for pure exchange economic equilibria, World Scientific Press, Singapore, 1993; Walker in J Polit Econ 94(4), 1987; Walras in Elements d’Economique Politique Pure, Corbaz, Lausanne, Switzerland, 1874; Zhao in Variational inequalities in general equilibrium: analysis and computation, PhD thesis, Brown University, 1988; and their bibliography). The aim of this paper is to provide a first approach to a particular dynamic general economic equilibrium problem: a Walrasian price equilibrium problem when the data are time-dependent. The equilibrium conditions that describe this pure exchange economic model are expressed in terms of an evolutionary variational inequality, for which existence and sensitivity results are given. Moreover, our problem can be expressed in a common way to many other equilibrium problems.     

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, 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 modeling, 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 equilibrium problems.     

Variational inequalities and discrete and continuum models of network equilibrium problems

In this paper, we present different equilibrium models (the traffic equilibrium model in the static, in the dynamic and in the continuum case, the market equilibrium problem in the static and in the dynamic case) and we find the variational inequality related to each problem. We propose also a computational procedure for the calculation of the equilibrium solution and present the dual formulation of the variational inequality related to the traffic equilibrium problem.     

Noncompliant oligopolistic firms and marketable pollution permits: Statics and dynamics

In this paper, we consider the modeling, analysis, and computation of solutions to both static and dynamic models of multiproduct, multipollutant noncompliant oligopolistic firms who engage in a market for pollution permits. In the case of the static model, we utilize variational inequality theory for the formulation of the governing equilibrium conditions as well as the qualitative analysis of the equilibrium pattern, including sensitivity analysis. We then propose a dynamic model, using the theory of projected dynamical systems, whose set of stationary points coincides with the set of solutions to the variational inequality problem. We propose an algorithm, which is a discretization in time of the dynamic adjustment process, and provide convergence results using the stability analysis results that are also provided herein. Finally, we apply the algorithm to several numerical examples to compute the profit-maximized quantities of the oligopolistic firms' products and the quantities of emissions, along with the equilibrium allocation of licenses and their prices, as well as the possible noncompliant overflows and underflows. This is the first time that these methodologies have been utilized in conjunction to study a problem drawn from environmental policy modeling and analysis.     

An existence result of a quasi-variational inequality associated to an equilibrium problem

In this paper we consider a Walrasian pure exchange economy with utility function which is a particular case of a general economic equilibrium problem, without production. We assume that each agent is endowed with at least of a commodity, his preferences are expressed by an utility function and it prevails a competitive behaviour: each agent regards the prices payed and received as independent of his own choices. The Walrasian equilibrium can be characterized as a solution to a quasi-variational inequality. By using this variational approach, our goal is to prove an existence result of equilibrium solutions.     

A variational inequality formulation of equilibrium models for end-of-life products with nonlinear constraints

Variational inequality theory facilitates the formulation of equilibrium problems in economic networks. Examples of successful applications include models of supply chains, financial networks, transportation networks, and electricity networks. Previous economic network equilibrium models that were formulated as variational inequalities only included linear constraints; in this case the equivalence between equilibrium problems and variational inequality problems is achieved with a standard procedure because of the linearity of the constraints. However, in reality, often nonlinear constraints can be observed in the context of economic networks. In this paper, we first highlight with an application from the context of reverse logistics why the introduction of nonlinear constraints is beneficial. We then show mathematical conditions, including a constraint qualification and convexity of the feasible set, which allow us to characterize the economic problem by using a variational inequality formulation. Then, we provide numerical examples that highlight the applicability of the model to real-world problems. The numerical examples provide specific insights related to the role of collection targets in achieving sustainability goals.     

Sufficient conditions to compute any solution of a quasivariational inequality via a variational inequality

We define the concept of reproducible map and show that, whenever the constraint map defining the quasivariational inequality (QVI) is reproducible then one can characterize the whole solution set of the QVI as a union of solution sets of some variational inequalities (VI). By exploiting this property, we give sufficient conditions to compute any solution of a generalized Nash equilibrium problem (GNEP) by solving a suitable VI. Finally, we define the class of pseudo-Nash equilibrium problems, which are (not necessarily convex) GNEPs whose solutions can be computed by solving suitable Nash equilibrium problems.     

Bicriterion Weight Varying Spatial Price Networks

We consider a spatial price equilibrium problem in which the consumers take their decisions according to the transportation cost and transportation time necessary for obtaining a given commodity. In particular, each consumer market can give a different weight to each component of a generalized cost, and we suppose that this weight can depend on time. Thus, we are faced with a time-dependent equilibrium problem which we cast within the framework of variational inequalities. We give existence results and, by using the example of a linear operator, we propose also a discretization procedure for equilibrium problems which can be modeled by the same type of variational inequality.     

弹性理论中各种变分原理的分类

本文按弹性理论中各种变分原理的约束条件的不同,对所有变分原理进行分类.我们在前文中业已指出,应力应变关系这样的约束条件是不能用拉氏乘子法解除的.剩下的可能约束条件共有四种:(1)平衡方程,(2)应变位移关系,(3)边界外力已知的边界条件,和(4)边界位移已知的边界条件.弹性理论的各种变分原理中,有的只有一种约束条件,有的有两种或三种,最多只能有四种约束条件.这样一共可能有15种变分原理,但是每种变分原理既可以用应变能A表示,又可以用余能B表示.这样,我们一共应有30种形式完全不同的变分原理,我们全部列出了这三十种形式的变分原理.     

On Equivalent Equilibrium Problems

We give sufficient conditions for the equivalence between two equilibrium problems. In particular we deduce that, under suitable assumptions, an equilibrium problem has an equivalent reformulation as a generalized variational inequality. Such conditions are satisfied when the equilibrium bifunction is lower semicontinuous, coercive and quasiconvex with respect to the second variable. We also show that the equivalent generalized variational inequality inherits the same generalized monotonicity properties of the original nonconvex equilibrium problem.     

A network formulation of market equilibrium problems and variational inequalities

This paper shows that market equilibrium problems of production may generally be modelled as equilibrium flow problems in networks and that their equilibrium conditions can be visualized as a variational inequality. This connection would allow us to transplant directly elements of the well-developed theory of equilibrium flow in networks to the theory of market equilibrium.     

基于非局部弹性应力场理论的纳米尺度效应研究:纳米梁的平衡条件、控制方程以及静态挠度

该文成功地解答了3个关于非局部应力理论用于纳米梁的问题:(ⅰ) 在绝大多数研究中,非局部效应增加导致纳米结构体刚度下降,其现象表现为弯曲挠度增加,固有频率减少,屈曲载荷下降,但为什么Eringen 的非局部弹性理论给出了完全相反的结论;(ⅱ) 为什么在某些研究结果中,非局部效应消失或是对研究结果无影响,比如纳米悬臂梁在集中载荷作用下的弯曲挠度; (ⅲ) 在高阶控制方程中,为什么高阶边界条件不存在．通过应用非局部弹性理论和精确变分原理分析纳米梁的弯曲问题,推导出全新的平衡条件、控制方程、边界条件和静态响应．这些方程和条件包含了与之前的相关研究结果符号相反的高阶微分项,这一差别导致了纳米效应对结构体的影响结果完全相反． 还证明之前为大家所公认的纳米梁静态或动态平衡条件实际上没有达到平衡,只有用等效弯矩代替非局部弯矩时,才可达到平衡．这些结论通常是可以被其它方法,比如应变梯度理论、耦合应力模型以及相关实验所证明．     

A supply chain network equilibrium model with random demands

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.     

Pareto Optimality in Nonconvex Economies with Infinite-dimensional Commodity Spaces

The paper contains applications of variational analysis to the study of Pareto optimality in nonconvex economies with infinite-dimensional commodity spaces satisfying the Asplund property. Our basic tool is a certain extremal principle that provides necessary conditions for set extremality and can be treated as a variational extension of the classical convex separation principle to systems of nonconvex sets. In this way we obtain new versions of the generalized second welfare theorem for nonconvex economies in terms of appropriate normal cones of variational analysis.     

Inverse Variational Inequality Approach and Applications

It is well known that a dynamic oligopolistic market equilibrium problem can be studied as an evolutionary variational inequality and this problem is approached as a problem of profit optimization for the firms. On the contrary, in this article, with the help of an inverse variational formulation, the behavior of control policies for an oligopolistic market equilibrium problem, whose aim is to regulate the exportation through the adjustment of taxes on the firms, is studied. This is considered as a policymaker optimization problem. More precisely, a definition of equilibrium for the firms by using the Lagrange multipliers is provided together to the optimal regulatory tax definition. Moreover an existence result is given and, at last, a numerical example is analyzed.     

Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.     

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan's equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.