The application of variational inequality theory to the study of spatial equilibrium and disequilibrium

Includes bibliographical references (p. 26-29).Supported by the National Science Foundation VPW Program. RII-880361by A. Nagurney

Parallel Computation of Large-Scale Nonlinear Network Problems in the Social and Economic Sciences

In this paper we focus on the parallel computation of large - scale equilibrium and optimization problems arising in the social and economic sciences. In particular, we consider problems which can be visualized and conceptualized as nonlinear network flow problems. The underlying network structure is then exploited in the development of parallel decomposition algorithms. We first consider market equilibrium problems, both dynamic and static, which are formulated as variational inequality problems, and for which we propose parallel decomposition algorithms by time period and by commodity, respectively. We then turn to the parallel computation of large-scale constrained matrix problems which are formulated as optimization problems and discuss the results of parallel decomposition by row/column

An exact solution method for binary equilibrium problems with compensation and the power market uplift problem

We propose a novel method to find Nash equilibria in games with binary decision variables by including compensation payments and incentive-compatibility constraints from non-cooperative game theory directly into an optimization framework in lieu of using first order conditions of a linearization, or relaxation of integrality conditions. The reformulation offers a new approach to obtain and interpret dual variables to binary constraints using the benefit or loss from deviation rather than marginal relaxations. The method endogenizes the trade-off between overall (societal) efficiency and compensation payments necessary to align incentives of individual players. We provide existence results and conditions under which this problem can be solved as a mixed-binary linear program. We apply the solution approach to a stylized nodal power-market equilibrium problem with binary on-off decisions. This illustrative example shows that our approach yields an exact solution to the binary Nash game with compensation. We compare different implementations of actual market rules within our model, in particular constraints ensuring non-negative profits (no-loss rule) and restrictions on the compensation payments to non-dispatched generators. We discuss the resulting equilibria in terms of overall welfare, efficiency, and allocational equity

Dynamic pricing with demand learning under competition

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 199-204).In this thesis, we focus on oligopolistic markets for a single perishable product, where firms compete by setting prices (Bertrand competition) or by allocating quantities (Cournot competition) dynamically over a finite selling horizon. The price-demand relationship is modeled as a parametric function, whose parameters are unknown, but learned through a data driven approach. The market can be either in disequilibrium or in equilibrium. In disequilibrium, we consider simultaneously two forms of learning for the firm: (i) learning of its optimal pricing (resp. allocation) strategy, given its belief regarding its competitors' strategy; (ii) learning the parameters in the price-demand relationship. In equilibrium, each firm seeks to learn the parameters in the price-demand relationship for itself and its competitors, given that prices (resp. quantities) are in equilibrium. In this thesis, we first study the dynamic pricing (resp. allocation) problem when the parameters in the price-demand relationship are known. We then address the dynamic pricing (resp. allocation) problem with learning of the parameters in the price-demand relationship. We show that the problem can be formulated as a bilevel program in disequilibrium and as a Mathematical Program with Equilibrium Constraints (MPECs) in equilibrium. Using results from variational inequalities, bilevel programming and MPECs, we prove that learning the optimal strategies as well as the parameters, is achieved. Furthermore, we design a solution method for efficiently solving the problem. We prove convergence of this method analytically and discuss various insights through a computational study.(cont.) Finally, we consider closed-loop strategies in a duopoly market when demand is stochastic. Unlike open-loop policies (such policies are computed once and for all at the beginning of the time horizon), closed loop policies are computed at each time period, so that the firm can take advantage of having observed the past random disturbances in the market. In a closed-loop setting, subgame perfect equilibrium is the relevant notion of equilibrium. We investigate the existence and uniqueness of a subgame perfect equilibrium strategy, as well as approximations of the problem in order to be able to compute such policies more efficiently.by Carine Simon.Ph.D

Parallel computation of large-scale network equilibria and variational inequalities.

Equilibrium of a network is obtained when each user who competes to optimize his utility can not improve his utility any further. Equilibrium problems governed by distinct equilibrium concepts can be formulated in one general framework--that of variational inequalities. The synthesis of variational inequalities and networks induces the creation of highly efficient algorithms which are especially suited for the large-scale equilibrium problems. Motivated by the recent technological advances in parallel computing architectures, parallel algorithms of large-scale equilibrium problems were developed using the theory of variational inequalities. In the case where the feasible constraint set of a network equilibrium problem can be expressed as a Cartesian product of subsets, the application of variational inequality decomposition algorithms for the parallel computation becomes possible. A new spatial price equilibrium model, which is not based on the path flows, but, rather, on the link flows to allow the decomposition by time periods, was developed and used as a prototype of large-scale network equilibrium problems. The variational inequality formulations were decomposed first by commodities, then by time periods, and, subsequently, by markets. The coarse grain parallel architectures used were the IBM 3090-600E and the IBM 3090-600J at the Cornell Theory Center with six processors each. The maximum speed-ups obtained were 1.93 for two processors, 3.74 for four processors, and 5.15 for six processors. The market subproblems were further decomposed by links, resulting in a fine grain parallel implementation. The Thinking Machine\u27s Connection Machine, CM-2, with 32,768 processors was used for the numerical experimentation. The fine grain parallel algorithm solved input/output matrix problems more than 20 times faster, when compared to the results on the IBM 3090-600J. It is expected that further enhancements to parallel languages and parallel architectures will make even more efficient implementations realizable, and that parallel computing and the theory of variational inequalities can be successfully applied to solve more efficiently other large-scale problems with an underlying network structure, such as traffic equilibrium problems, general economic equilibrium problems, and financial equilibrium problems

An Exact Solution Method for Binary Equilibrium Problems with Compensation and the Power Market Uplift Problem

We propose a novel method to find Nash equilibria in games with binary decision variables by including compensation payments and incentive-compatibility constraints from non-cooperative game theory directly into an optimization framework in lieu of using first order conditions of a linearization, or relaxation of integrality conditions. The reformulation offers a new approach to obtain and interpret dual variables to binary constraints using the benefit or loss from deviation rather than marginal relaxations. The method endogenizes the trade-off between overall (societal) efficiency and compensation payments necessary to align incentives of individual players. We provide existence results and conditions under which this problem can be solved as a mixed-binary linear program. We apply the solution approach to a stylized nodal power-market equilibrium problem with binary on-off decisions. This illustrative example shows that our approach yields an exact solution to the binary Nash game with compensation. We compare different implementations of actual market rules within our model, in particular constraints ensuring non-negative profits (no-loss rule) and restrictions on the compensation payments to non-dispatched generators. We discuss the resulting equilibria in terms of overall welfare, efficiency, and allocational equity

Abstracts of National PhD Students Days 2018 (NPSD’2018) 1st Edition, May 4-5, 2018, Université Ibn Zohr Agadir Maroc, Ouarzazate, Morocco

Abstracts of National PhD Students Days 2018 (NPSD’2018) 1st Edition, May 4-5, 2018, Université Ibn Zohr Agadir Maroc, Ouarzazate, Morocc

Road network maintenance and repair considering day-to-day traffic dynamics and transient congestion

Road maintenance and repair (M&R) are essential for keeping the performance of traffic infrastructure at a satisfactory level, and extending their lifetime to the fullest extent possible. For road networks, effective M&R plans should not be constructed in a myopic or ad-hoc fashion regardless of the subsequent benefits and costs associated with those projects considered. A hallmark of road M&R studies is the use of user equilibrium (UE) models to predict network traffic for a given set of road conditions with or without M&R. However, UE approaches ignore the traffic disequilibrium states and transient congestion as a result of M&R derived disruptions to network traffic on a day-to-day (DTD) time scale, which could produce additional substantial travel costs. As shown in the numerical studies on a M&R plan of the Sioux Falls network, the additional maintenance derived travel cost is about 4 billion, which is far exceed the actual M&R construction cost of 0.2 billion. Therefore, it is necessary to recognise the substantial social costs induced by maintenance-derived disruptions in the form of transient congestion when planning M&R. This realistic and pressing issue is not properly addressed by the road M&R planning problems with traffic equilibrium constraints. This thesis proposes a dual-time-scale road network M&R model aiming to simultaneously capture the long-term effects of M&R activities under traffic equilibria, and the maintenance-derived transient congestion using day-to-day (DTD) traffic evolutionary dynamics. The notion of ‘day’ is arbitrarily defined (e.g. weeks or months). The proposed M&R model consists of three sub-models: (1) a within-day dynamic network loading (DNL) model; (2) a day-to-day dynamic traffic assignment (DTD DTA) model; and (3) a day-to-day road quality model. The within-day traffic dynamics is captured by the Lighthill-Whitham-Richards (LWR) fluid dynamic network loading model. The day-to-day phase of the traffic dynamics specify travellers’ route and departure time choices in a stochastic manner based on a sequential mixed multinomial or nested Logit model. Travel information sharing behaviour is further integrated into this macroscopic doubly dynamic (both within-day and day-to-day dynamic) traffic assignment (DDTA) model to account for the impact of incomplete information on travel experiences. A deterministic day-to-day road quality model based on an exponential form of traffic flow is employed to govern the road deterioration process, where a quarter-car index (QI) is applied. All these dynamics are incorporated in a holistic dual-time-scale M&R model, which captures realistic phenomena associated with short-term and long-term effects of M&R, including physical queuing and spillback, road capacity reduction, temporal-spatial shift of congestion due to on-going M&R activities, and the tendency to converge to an equilibrium after M&R actions. Following the dual-time-scale road network M&R model, a bi-level road M&R optimisation model is proposed, where the aforementioned three sub-models are incorporated into the lower-level problem, while the upper-level is to minimise M&R expenditure and network travel costs while maintaining a satisfactory level of road quality. The M&R planning horizon is long yet finite (e.g. years or decades). A ‘quality-usage’ feedback mechanism is investigated in the proposed bi-level M&R model, namely, (1) the DTD road quality evolution as a result of DTD traffic loads and the M&R effectiveness; and (2) the evolution of DTD traffic in response to both DTD road deterioration and the improved road quality after M&R activities. The effectiveness of developed M&R optimisation model is demonstrated through case studies on the Sioux Falls network. A metaheuristic Genetic Algorithm (GA) approach is employed to solve the M&R problems given its highly nonlinear, nonconvex and non-differentiable nature. Explicit travellers’ choice behaviour dynamics and complex traffic phenomena such as network paradoxes arising from M&R activities are illustrated. Through a comparison with the results under the dynamic user equilibrium (DUE) method, the proposed DTD method achieves significant reduction in network travel cost of $ 25 million, approximately 20% of the total cost. This points to the benefit of using the DTD dynamics for capturing network’s responses to M&R in a more realistic way. The M&R model proposed in this thesis could provide valuable managerial insights for road M&R planning agencies.Open Acces