Electrical and electronic equipment waste (e-waste) is growing fast. Due to its potential economic value as well as its possible negative impacts on the environment, tracing e-waste flow is a major concern for stakeholders of e-waste management. Especially, whether or not adequate amounts of electrical and electronic equipment waste (WEEE) flow into the designed recycling systems is a fundamental issue for sustainable operations. In this paper, we analyze how technical, market, and legislative factors influence the total amount of e-waste that is collected, recycled, exported and (legally and illegally) disposed off. We formulate the e-waste network flow model as a variational inequality problem. The results of the numerical examples highlight the importance of considering the interaction between the supply and the demand side for precious materials in policy-decisions. Low collection rates of e-waste lead to low profits for stakeholders and make it difficult to establish sustainable recycling operations. Increasing WEEE collection rates increases recyclers' profits; however, it only increases smelters' profits up to a certain limit, after which smelters cannot benefit further due to limited demand for precious materials. Furthermore, the results emphasize the importance of establishing international control regimes for WEEE flows and reveal possible negative consequences of the recent trend of dematerialization. More precisely, product dematerialization tends to decrease recyles' and smelters' profits as well as to increase the outflow of e-waste from the designated recycling system. (authors' abstract)

This work aims to investigate some applications of the conjugate duality for scalar and vector optimization problems to the construction of gap functions for variational inequalities and equilibrium problems. The basic idea of the approach is to reformulate variational inequalities and equilibrium problems into optimization problems depending on a fixed variable, which allows us to apply duality results from optimization problems. Based on some perturbations, first we consider the conjugate duality for scalar optimization. As applications, duality investigations for the convex partially separable optimization problem are discussed. Afterwards, we concentrate our attention on some applications of conjugate duality for convex optimization problems in finite and infinite-dimensional spaces to the construction of a gap function for variational inequalities and equilibrium problems. To verify the properties in the definition of a gap function weak and strong duality are used. The remainder of this thesis deals with the extension of this approach to vector variational inequalities and vector equilibrium problems. By using the perturbation functions in analogy to the scalar case, different dual problems for vector optimization and duality assertions for these problems are derived. This study allows us to propose some set-valued gap functions for the vector variational inequality. Finally, by applying the Fenchel duality on the basis of weak orderings, some variational principles for vector equilibrium problems are investigated.

by Yan Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 78-80). / Abstract also in Chinese. / Chapter 1 --- Admissible Points of Convex Sets --- p.7 / Chapter 1.1 --- Introduction and Notations --- p.7 / Chapter 1.2 --- The Main Result --- p.7 / Chapter 1.2.1 --- The Proof of Theoreml.2.1 --- p.8 / Chapter 1.3 --- An Application --- p.10 / Chapter 2 --- A Generalization on The Theorems of Admissible Points --- p.12 / Chapter 2.1 --- Introduction and Notations --- p.12 / Chapter 2.2 --- Fundamental Lemmas --- p.14 / Chapter 2.3 --- The Main Result --- p.16 / Chapter 3 --- Introduction to Variational Inequalities --- p.21 / Chapter 3.1 --- Variational Inequalities in Finite Dimensional Space --- p.21 / Chapter 3.2 --- Problems Which Relate to Variational Inequalities --- p.25 / Chapter 3.3 --- Some Variations on Variational Inequality --- p.28 / Chapter 3.4 --- The Vector Variational Inequality Problem and Its Relation with The Vector Optimization Problem --- p.29 / Chapter 3.5 --- Variational Inequalities in Hilbert Space --- p.31 / Chapter 4 --- Vector Variational Inequalities --- p.36 / Chapter 4.1 --- Preliminaries --- p.36 / Chapter 4.2 --- Notations --- p.37 / Chapter 4.3 --- Existence Results of Vector Variational Inequality --- p.38 / Chapter 5 --- The Generalized Quasi-Variational Inequalities --- p.44 / Chapter 5.1 --- Introduction --- p.44 / Chapter 5.2 --- Properties of The Class F0 --- p.46 / Chapter 5.3 --- Main Theorem --- p.53 / Chapter 5.4 --- Remarks --- p.58 / Chapter 6 --- A set-valued open mapping theorem and related re- sults --- p.61 / Chapter 6.1 --- Introduction and Notations --- p.61 / Chapter 6.2 --- An Open Mapping Theorem --- p.62 / Chapter 6.3 --- Main Result --- p.63 / Chapter 6.4 --- An Application on Ordered Normed Spaces --- p.66 / Chapter 6.5 --- An Application on Open Decomposition --- p.70 / Chapter 6.6 --- An Application on Continuous Mappings from Order- infrabarreled Spaces --- p.72 / Bibliography

He Yiran. / Includes index. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (p. 74-80) and index.. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

L'objectif de cette thèse est d'étudier des propriétés des points d'équilibre dans des réseaux de transport multi-critères et de développer des méthodes numériques permettant de trouver l'ensemble des points d'équilibre ou une partie représentative de cet ensemble. Le travail est structure comme suit. Dans le premier chapitre nous donnons une introduction de la thèse. Le chapitre 2 est un rappel de certaines notions que nous utilisons dans les autres. Nous y rappelons le concept de point optimal de Pareto, les fonctions multivoques et les problèmes d'inégalité variationnelle. Nous introduisons certaines fonctions de scalarisation et puis établissons quelques propriétés importantes. Dans le chapitre 3, nous décrivons les réseaux de transport qui sont étudiés dans cette thèse. Dans chaque modèle, nous rappelons les définitions des points d'équilibre et donnons une relation entre ces définitions. Dans le chapitre 4 nous traitons les réseaux de transport multi-critères mono-produit sans contraintes de capacité. Tout d'abord, nous construisons deux problèmes d'optimisation dont les solutions sont exactement l'ensemble des points d'équilibre du modèle initiale et établissons certaines propriétés importantes de continuité et de dérivabilité génériques des fonctions objectifs. Puis nous donnons une formule permettant de calculer le gradient des fonctions objectifs. Nous proposons également un algorithme et prouvons sa convergence pour générer une représentation de l'ensemble des points d'équilibre. Puisque les fonctions objectifs de nos problèmes d'optimisation ne sont pas continues, une méthode de lissage est également considérée afin d'utiliser quelques techniques d'optimisation globale. En fin, nous introduisons le concept de point d'équilibre robuste, puis nous établissons des critères de robustesse et une formule permettant de calculer le rayon de robustesse. Dans le chapitre 5 nous étudions des points d'équilibre vectoriel dans le réseau de transport multi-critères mono-produit sous contraintes de capacité.Tout d'abord, nous proposons un problème d'optimisation équivalent. En utilisant des techniques analogues à celles du chapitre 4 nous obtenons également un sous-ensemble des points d'équilibre du modèle proposé. Dans le dernier chapitre nous considérons des points d'équilibre fort dans le réseau de transport multi-critères multi-produit sous contraintes de capacité. Nous établissons des conditions d'existence des points d'équilibre fort, des relations entre des points d'équilibre fort et des points d'équilibre par rapport à une famille de fonctions ainsi qu'une relation entre des points d'équilibre fort et les points efficaces de l'ensemble des valeurs de la fonction de coût. En plus nous construisons des problèmes d'inéqualité variationnellle, dont les solutions sont les points d'équilibre fort. La dernière partie de ce chapitre est consacrée à un algorithme permettant de trouver des points d'équilibre d'un réseau multi-critères sous contraintes de capacité. Certains exemples numériques sont donnés pour illustrer notre méthode. Nous fermons la thèse avec une liste de références et appendice contenant le code matlab de nos algorithmes. / The purpose of this thesis is to study equilibria in multi-criteria trafficnetworks and develop numerical methods to find the set of all equilibria oronly one representative part of this set. The thesis is structured as follows.In the first chapter we present an introduction of the thesis. Chapter 2is of preliminary character. We recall the concept of Pareto minimal pointsand some notions related to set-valued maps and variational inequality pro-blem. We introduce some scalarizing functions, in particular the so-calledaugmented biggest/smallest monotone functions and augmented signed distance functions, and establish some properties we shall use later.Chapter 3 describes the traffic network models to be studied in this thesis.We define equilibrium for each model and determine a relationship betweenthem. We also give some counter examples for some existing results in therecent literature on this topic.In Chapter 4 we develop a new solution method for multi-criteria net-work equilibrium problems without capacity constraints. To this end we shallconstruct two optimization problems the solutions of which are exactly theset of equilibria of the model, and establish some important generic conti-nuity and differentiability properties of the objective functions. Then we givethe formula to calculate the gradient of the objective functions which enablesus to modify Frank-Wolfe's reduced gradient method to get descent directiontoward an optimal solution. We prove the convergence of the method whichgenerates a nice representative set of equilibria. Since the objective functionsof our optimization problems are not continuous, a method of smoothingthem is also considered in order to see how global optimization algorithmsmay help.We shall also introduce the concept of robust equilibrium, establishcriteria for robustness and a formula to compute the radius of robustness.In Chapter 5 we consider vector equilibrium in the multi-criteria single-product traffic network with capacity constraints.We propose an equivalent optimization problem and establish some im-portant generic continuity and differentiability properties of the objectivefunction. Then we give a formula which allows us to calculate the gradientof the objective function. After that we apply the approach of Chapter 4 toobtain an algorithm for generating equilibria of this network. We also givesome numerical examples to illustrate our approach.In the last chapter we consider strong vector equilibrium in the multi-criteria multi-product traffic network with capacity constraints.We establish conditions for existence of strong vector equilibrium.We alsoestablish relations between equilibrium and efficient points of the value set ofthe cost function and with equilibrium with respect to a family of functions.Moreover we exploit particular increasing functions discussed in Chapter 2 toconstruct variational inequality problems, solutions of which are equilibriumflows. The final part of this chapter is devoted to an algorithm for findingequilibrium flows of a multi-criteria network with capacity constraints. Somenumerical examples are given to illustrate our method and its applicability.A list of references and appendices containing the code Matlab of ouralgorithms follow.

This paper studies the problem of multi-period pricing for perishable products in a competitive (oligopolistic) market. We study non cooperative Nash equilibrium policies for sellers. At the beginning of the time horizon, the total inventories are given and additional production is not an available option. The analysis for periodic production-review models, where production decisions can be made at the end of each period at some production cost after incurring holding or backorder costs, does not extend to this model. Using results from game theory and variational inequalities we study the existence and uniqueness of equilibrium policies. We also study convergence results for an algorithm that computes the equilibrium policies. The model in this paper can be used in a number of application areas including the airline, service and retail industries. We illustrate our results through some numerical examples. / Singapore-MIT Alliance (SMA)

In this paper, we establish several strong convergence results of general approximate proximal algorithm and general Bregman-function-based approximate proximal algorithm for solving the generalized variational inequality problem with pseudomonotone multifunction.

M. Tech. Mathematical Technology. / Discusses the main problems in terms of how to derive mathematical models for a free, a forced and a damped spring pendulum and determining numerical solutions using a computer algebra system (CAS), because exact analytical solutions are not obvious. Hence this mini-dissertation mainly deals with how to derive mathematical models for the spring pendulum using the Euler-Lagrange equations both in the Cartesian and polar coordinate systems and finding solutions numerically. Derivation of the equations of motion are done for the free, forced and damped cases of the spring pendulum. The main objectives of this mini-dissertation are: firstly, to derive the equations of motion governing the oscillatory and rotational components of the spring pendulum for the free, the forced and damped cases of the spring pendulum ; secondly, to solve these equations numerically by writing the equations as initial value problems (IVP); and finally, to introduce a novel way of incorporating nonlinear damping into the Euler-Lagrange equations of motion as introduced by Joubert, Shatalov and Manzhirov (2013, [20]) for the spring pendulum and interpreting the numerical solutions using CAS-generated graphics.

Many mathematical models generally consist of either a continuous system like that of a system of differential equations, or a discrete system such as a discrete game theoretic model; however, there exist phenomena in which neither modeling approach alone is sufficient for capturing the behaviour of the intended real world system. This leads to the need to explore the use of combinations of such discrete and continuous processes, namely the use of mathematical modeling with what are known as hybrid dynamical systems. In what follows, we provide a blueprint for one approach to study several classes of equilibrium problems in non-equilibrium states through the direct use of hybrid dynamical systems. The motivation of our work stems from the fact that the real world is rarely, if ever, in a state of perfect equilibrium and that the behaviour of equilibrium problems in non-equilibrium states is just as complex and interesting (if not more so) than standard equilibrium solutions. Our approach consists of an association of classes of traffic equilibrium problems, noncooperative games, minimization problems, and complementarity problems to a class of hybrid dynamical system called projected dynamical systems. The purposed connection between equilibrium problems and projected dynamical system is made possible through mutual connections to the robust framework of variational inequalities. The results of our work include theoretical contributions such as showing how evolution solutions (non-equilibrium solutions) can be analyzed from a theoretical point of view and how they relate to equilibrium solutions; computational methods for tracking and visualizing evolution solutions and the development of numerical algorithms for simulation; and applications such as the effect of population vaccination decisions in the spread of infectious disease, dynamic traffic networks, dynamic vaccination games, and nonsmooth electrical circuits.

This thesis consists of five scientific papers dealing with equations related to the optimal switching problem, mainly backward stochastic differential equations and variational inequalities. Besides the scientific papers, the thesis contains an introduction to the optimal switching problem and a brief outline of possible topics for future research. Paper I concerns systems of variational inequalities with operators of Kolmogorov type. We prove a comparison principle for sub- and supersolutions and prove the existence of a solution as the limit of solutions to iteratively defined interconnected obstacle problems. Furthermore, we use regularity results for a related obstacle problem to prove Hölder continuity of this solution. Paper II deals with systems of variational inequalities in which the operator is of non-local type. By using a maximum principle adapted to this non-local setting we prove a comparison principle for sub- and supersolutions. Existence of a solution is proved using this comparison principle and Perron's method. In Paper III we study backward stochastic differential equations in which the solutions are reflected to stay inside a time-dependent domain. The driving process is of Wiener-Poisson type, allowing for jumps. By a penalization technique we prove existence of a solution when the bounding domain has convex and non-increasing time slices. Uniqueness is proved by an argument based on Ito's formula. Paper IV and Paper V concern optimal switching problems under incomplete information. In Paper IV, we construct an entirely simulation based numerical scheme to calculate the value function of such problems. We prove the convergence of this scheme when the underlying processes fit into the framework of Kalman-Bucy filtering. Paper V contains a deterministic approach to incomplete information optimal switching problems. We study a simplistic setting and show that the problem can be reduced to a full information optimal switching problem. Furthermore, we prove that the value of information is positive and that the value function under incomplete information converges to that under full information when the noise in the observation vanishes.