International series in management science/operations research, 2007
ABSTRACT In this paper we are concerned with a class of mathematical problems that arise when mod... more ABSTRACT In this paper we are concerned with a class of mathematical problems that arise when modelling dynamic competition among conflicting agents that may or may not reach an equilibrium. We call this class of problems differential variational inequalities. We are further interested in the application of models belonging to this class of problems to the study of (1) policy constrained, limited warfareand (2) Internet vulnerability.
In this paper we give a mathematical statement of a di¤erential spatial monopoly wherein transact... more In this paper we give a mathematical statement of a di¤erential spatial monopoly wherein transactions are facilitated by a freight transportation network that is based on an underlying graph. The monopolist has a presence at every node, and freight services are available for paths connecting stipulated origin-destination (OD) pairs. The monopolist produces a single homogeneous product that may be manufactured and inventoried, as well as sold, at every node. The dynamics take the form of ordinary di¤erential equations that describe ‡ow conservation. We present the necessary conditions of the monopolist's optimal control problem and observe their interpretation is that marginal revenue equals marginal cost at each node for each instant of continuous time. Existence of an optimal solution to the monopolist's problem is proven and a numerical example solved using two algorithms: one implemented in discrete time and the other in continuous time.
International series in management science/operations research, 2010
... Variational Inequalities and Complementarity Problems Volume II Springer ... Finite-Dimension... more ... Variational Inequalities and Complementarity Problems Volume II Springer ... Finite-Dimensional Variational Inequalities and Complementarity Problems Volume ...
Lecture Notes in Economics and Mathematical Systems, 1981
In this paper we offer a statement of the continuous multiobjective equilibrium network design pr... more In this paper we offer a statement of the continuous multiobjective equilibrium network design problem and discuss how it may be solved by an adaptation of Geoffrion’s interactive method. The formulation is fully multimodal and employs no symmetry restrictions.
International series in management science/operations research, 2010
The theory of optimal economic growth is a branch of economic theory that makes direct and sophis... more The theory of optimal economic growth is a branch of economic theory that makes direct and sophisticated use of the theory of optimal control. As such, the models of optimal economic growth that have been devised and reported in the economics literature are relatively easy for a person who has mastered the material of Chapters 3 and 4 of this book to comprehend. Among other things, this chapter shows how aspatial optimal economic growth theory may be extended to study optimal growth of interdependent regions in a national economy. Moreover, working through the analyses presented in this chapter provides a means of assessing and improving one’s mastery of the key mathematical concepts from the theory of optimal control that were introduced in previous chapters, especially the analysis and interpretation of optimality conditions and singular controls.
International series in management science/operations research, 2010
The primary intent of this chapter is to introduce the reader to the theoretical foundations of n... more The primary intent of this chapter is to introduce the reader to the theoretical foundations of nonlinear programming as well as the theoretical foundations of deterministic discrete-time optimal control. In fact, deterministic discrete-time optimal control problems, as we shall see, are actually nonlinear mathematical programs with a very particular type of structure. In a later chapter, we will also discover that deterministic continuous-time optimal control problems are specific instances of mathematical programs in topological vector spaces. Consequently, it is imperative for the student of optimal control to have a command of the foundations of nonlinear programming. Particularly important are the notions of local and global optimality in mathematical programming, the Kuhn-Tucker necessary conditions for optimality in nonlinear programming, and the role played by convexity in making necessary conditions sufficient. Readers already comfortable with finite-dimensional nonlinear programming may wish to go immediately to Section 2.9. We do caution, however, that subsequent chapters of this book assume substantial familiarity with finitedimensional nonlinear programming, so that an overestimate of one's nonlinear programming knowledge can be very detrimental to ultimately obtaining a deep understanding of optimal control theory and differential games.
Vibration generated by highway traffic is a significant form of environmental pollution that has ... more Vibration generated by highway traffic is a significant form of environmental pollution that has not been studied as comprehensively as noise pollution. Vibration sources and the mechanisms of vibration propagation and attenuation must be better understood before uniform criteria for vibration pollution and corrective measures can be established. The theory of ground-borne vibration provides considerable insight into the nature of traffic-induced vibration. The relevant theory and associated experimental data have not heretofore been put into a form useful to highway engineers and transportation scientists not trained in the problem area. The highway irregularity spectrum is the most important cause of vibration. The Rayleigh wave is of primary importance to traffic-induced vibration assessment. Significant structural damage from traffic- induced vibration is rare. However, human response to vibration is both physiological and psychological, and human often classify vibration as unacceptable at levels lower than those that cause structural damage. The nature of the long wave lengths of traffic-induced vibrations makes it unlikely that attenuation measures such as trenches will be successful. Indications are that careful maintenance of the highway will be the most effective vibration preventive and that special abatement measures will be used only whete complaints have occurred.
ABSTRACT We develop a generalized Harrod‐Domar type of model for optimal regional growth which al... more ABSTRACT We develop a generalized Harrod‐Domar type of model for optimal regional growth which allows determination of the optimal allocation of regional public investments and which considers multiple growth objectives, as well as both normal and singular fiscal instruments. This general model is shown to include most of the previous optimal regional growth models as special cases. Necessary conditions for some special cases of the general model are analyzed in detail, and decision rules associated with derived optimal regional growth policies are articulated. These special cases verify that singular controls do exist in certain instances, and that they must be considered for the complete specification of optimal regional growth policies; this is significant because singular controls have not been previously analyzed in the literature. We conclude the paper with a discussion of sufficiency conditions for optimal regional growth models which is more general than that given by previous authors.
International series in management science/operations research, 2010
In this chapter we focus on extending the notion of a noncooperative Nash equilibrium to a dynami... more In this chapter we focus on extending the notion of a noncooperative Nash equilibrium to a dynamic, continuous-time setting. The dominant mathematical perspective we will employ is that of a differential variational inequality. In fact we shall see that many of the results obtained in the previous chapter for finite-dimensional variational inequalities and static games carry over with some slight modifications to the dynamic, continuous-time setting we now address.
... This trade-off is forced on the urban freight model builder because of the network detail nee... more ... This trade-off is forced on the urban freight model builder because of the network detail needed to accurately describe routing choices in an urban setting. We argue that the best trade-off possible with present computational tools is to determine the demand for finished and semi ...
On the use of network-equilibrium models in urban and regional analyses: the emergence of dynamic... more On the use of network-equilibrium models in urban and regional analyses: the emergence of dynamic concepts of equilibrium A great deal of intellectual energy has been expended on the development of network-equilibrium models for urban and regional analyses. In some recent papers, , Nagurney (1987a) have reviewed the vast literature on this subject. What is of concern in this commentary is that the networkequilibrium literature shows a growing tendency for scholars to question the relevance of the fundamental paradigm of equilibrium as a steady-state balance of supply and demand, this balance being appropriately defined to reflect the spatial separation of economic activities and the consequent need for transportation services. This questioning stems from two general observations: first, the traditional concept of equilibrium appears to have a relatively low explanatory power in terms of being able to predict the behavior of real systems; and second, the static perspective inherent in models of steady-state equilibrium excludes evident temporal fluctuations in traffic costs, traffic flows, commodity consumption levels, commodity production levels, and commodity prices. Indeed, the very time scales assumed in the traditional models of network equilibrium seem to be inconsistent with the time needed for the systems studied to approach a steady state. An example is provided by so-called Wardropian user equilibrium, the preferred model for the modelling of urban traffic networks, an equilibrium under which, when reached, no user may unilaterally alter his or her route choice without incurring greater cost. This equilibrium is assumed to be reached within the time span of a typical 'rush hour'. The minor assumptions implicit in this grand assumption are numerous, and include such simplifications as there being no temporal fluctuations in costs experienced by users, as would occur for example when unannounced maintenance is conducted or a traffic accident is encountered. Equally unaccounted for in the user-equilibrium paradigm is the 'new' user who learns his or her best route through trial and error. Indeed, such 'learning by doing' must have gone on during previous rush hours if a user equilibrium is attained in the current rush hour. Furthermore, this 'pre-equilibrium' behavior is of interest in and of itself for determining the impacts of any action which alters the network. Similar criticisms may be offered concerning the Samuelson -Takayama -Judge model of static spatial price equilibrium and its variants (see Nagurney, 1987a), which are used to model interregional commodity flows. In particular, models of this type require that delivered prices be equated to local prices for nontrivial commodity flows. Again, temporal fluctuations may occur owing to effects as widely varied as: weather patterns which alter transportation costs, technological innovations which alter commodity supply functions, and income changes which alter commodity demand functions. Dynamic generalizations of the Samuelson-Takayama-Judge model have been proposed by Nagurney (1987b) and Friesz et al (1988a), although these efforts have still not resulted in a commonly agreed upon approach for treating the dynamic interactions of spatially separated markets. The above remarks concerning the failure of the most commonly used traffic and spatial price equilibrium models to treat dynamic considerations in a
International series in management science/operations research, 2007
ABSTRACT In this paper we are concerned with a class of mathematical problems that arise when mod... more ABSTRACT In this paper we are concerned with a class of mathematical problems that arise when modelling dynamic competition among conflicting agents that may or may not reach an equilibrium. We call this class of problems differential variational inequalities. We are further interested in the application of models belonging to this class of problems to the study of (1) policy constrained, limited warfareand (2) Internet vulnerability.
In this paper we give a mathematical statement of a di¤erential spatial monopoly wherein transact... more In this paper we give a mathematical statement of a di¤erential spatial monopoly wherein transactions are facilitated by a freight transportation network that is based on an underlying graph. The monopolist has a presence at every node, and freight services are available for paths connecting stipulated origin-destination (OD) pairs. The monopolist produces a single homogeneous product that may be manufactured and inventoried, as well as sold, at every node. The dynamics take the form of ordinary di¤erential equations that describe ‡ow conservation. We present the necessary conditions of the monopolist's optimal control problem and observe their interpretation is that marginal revenue equals marginal cost at each node for each instant of continuous time. Existence of an optimal solution to the monopolist's problem is proven and a numerical example solved using two algorithms: one implemented in discrete time and the other in continuous time.
International series in management science/operations research, 2010
... Variational Inequalities and Complementarity Problems Volume II Springer ... Finite-Dimension... more ... Variational Inequalities and Complementarity Problems Volume II Springer ... Finite-Dimensional Variational Inequalities and Complementarity Problems Volume ...
Lecture Notes in Economics and Mathematical Systems, 1981
In this paper we offer a statement of the continuous multiobjective equilibrium network design pr... more In this paper we offer a statement of the continuous multiobjective equilibrium network design problem and discuss how it may be solved by an adaptation of Geoffrion’s interactive method. The formulation is fully multimodal and employs no symmetry restrictions.
International series in management science/operations research, 2010
The theory of optimal economic growth is a branch of economic theory that makes direct and sophis... more The theory of optimal economic growth is a branch of economic theory that makes direct and sophisticated use of the theory of optimal control. As such, the models of optimal economic growth that have been devised and reported in the economics literature are relatively easy for a person who has mastered the material of Chapters 3 and 4 of this book to comprehend. Among other things, this chapter shows how aspatial optimal economic growth theory may be extended to study optimal growth of interdependent regions in a national economy. Moreover, working through the analyses presented in this chapter provides a means of assessing and improving one’s mastery of the key mathematical concepts from the theory of optimal control that were introduced in previous chapters, especially the analysis and interpretation of optimality conditions and singular controls.
International series in management science/operations research, 2010
The primary intent of this chapter is to introduce the reader to the theoretical foundations of n... more The primary intent of this chapter is to introduce the reader to the theoretical foundations of nonlinear programming as well as the theoretical foundations of deterministic discrete-time optimal control. In fact, deterministic discrete-time optimal control problems, as we shall see, are actually nonlinear mathematical programs with a very particular type of structure. In a later chapter, we will also discover that deterministic continuous-time optimal control problems are specific instances of mathematical programs in topological vector spaces. Consequently, it is imperative for the student of optimal control to have a command of the foundations of nonlinear programming. Particularly important are the notions of local and global optimality in mathematical programming, the Kuhn-Tucker necessary conditions for optimality in nonlinear programming, and the role played by convexity in making necessary conditions sufficient. Readers already comfortable with finite-dimensional nonlinear programming may wish to go immediately to Section 2.9. We do caution, however, that subsequent chapters of this book assume substantial familiarity with finitedimensional nonlinear programming, so that an overestimate of one's nonlinear programming knowledge can be very detrimental to ultimately obtaining a deep understanding of optimal control theory and differential games.
Vibration generated by highway traffic is a significant form of environmental pollution that has ... more Vibration generated by highway traffic is a significant form of environmental pollution that has not been studied as comprehensively as noise pollution. Vibration sources and the mechanisms of vibration propagation and attenuation must be better understood before uniform criteria for vibration pollution and corrective measures can be established. The theory of ground-borne vibration provides considerable insight into the nature of traffic-induced vibration. The relevant theory and associated experimental data have not heretofore been put into a form useful to highway engineers and transportation scientists not trained in the problem area. The highway irregularity spectrum is the most important cause of vibration. The Rayleigh wave is of primary importance to traffic-induced vibration assessment. Significant structural damage from traffic- induced vibration is rare. However, human response to vibration is both physiological and psychological, and human often classify vibration as unacceptable at levels lower than those that cause structural damage. The nature of the long wave lengths of traffic-induced vibrations makes it unlikely that attenuation measures such as trenches will be successful. Indications are that careful maintenance of the highway will be the most effective vibration preventive and that special abatement measures will be used only whete complaints have occurred.
ABSTRACT We develop a generalized Harrod‐Domar type of model for optimal regional growth which al... more ABSTRACT We develop a generalized Harrod‐Domar type of model for optimal regional growth which allows determination of the optimal allocation of regional public investments and which considers multiple growth objectives, as well as both normal and singular fiscal instruments. This general model is shown to include most of the previous optimal regional growth models as special cases. Necessary conditions for some special cases of the general model are analyzed in detail, and decision rules associated with derived optimal regional growth policies are articulated. These special cases verify that singular controls do exist in certain instances, and that they must be considered for the complete specification of optimal regional growth policies; this is significant because singular controls have not been previously analyzed in the literature. We conclude the paper with a discussion of sufficiency conditions for optimal regional growth models which is more general than that given by previous authors.
International series in management science/operations research, 2010
In this chapter we focus on extending the notion of a noncooperative Nash equilibrium to a dynami... more In this chapter we focus on extending the notion of a noncooperative Nash equilibrium to a dynamic, continuous-time setting. The dominant mathematical perspective we will employ is that of a differential variational inequality. In fact we shall see that many of the results obtained in the previous chapter for finite-dimensional variational inequalities and static games carry over with some slight modifications to the dynamic, continuous-time setting we now address.
... This trade-off is forced on the urban freight model builder because of the network detail nee... more ... This trade-off is forced on the urban freight model builder because of the network detail needed to accurately describe routing choices in an urban setting. We argue that the best trade-off possible with present computational tools is to determine the demand for finished and semi ...
On the use of network-equilibrium models in urban and regional analyses: the emergence of dynamic... more On the use of network-equilibrium models in urban and regional analyses: the emergence of dynamic concepts of equilibrium A great deal of intellectual energy has been expended on the development of network-equilibrium models for urban and regional analyses. In some recent papers, , Nagurney (1987a) have reviewed the vast literature on this subject. What is of concern in this commentary is that the networkequilibrium literature shows a growing tendency for scholars to question the relevance of the fundamental paradigm of equilibrium as a steady-state balance of supply and demand, this balance being appropriately defined to reflect the spatial separation of economic activities and the consequent need for transportation services. This questioning stems from two general observations: first, the traditional concept of equilibrium appears to have a relatively low explanatory power in terms of being able to predict the behavior of real systems; and second, the static perspective inherent in models of steady-state equilibrium excludes evident temporal fluctuations in traffic costs, traffic flows, commodity consumption levels, commodity production levels, and commodity prices. Indeed, the very time scales assumed in the traditional models of network equilibrium seem to be inconsistent with the time needed for the systems studied to approach a steady state. An example is provided by so-called Wardropian user equilibrium, the preferred model for the modelling of urban traffic networks, an equilibrium under which, when reached, no user may unilaterally alter his or her route choice without incurring greater cost. This equilibrium is assumed to be reached within the time span of a typical 'rush hour'. The minor assumptions implicit in this grand assumption are numerous, and include such simplifications as there being no temporal fluctuations in costs experienced by users, as would occur for example when unannounced maintenance is conducted or a traffic accident is encountered. Equally unaccounted for in the user-equilibrium paradigm is the 'new' user who learns his or her best route through trial and error. Indeed, such 'learning by doing' must have gone on during previous rush hours if a user equilibrium is attained in the current rush hour. Furthermore, this 'pre-equilibrium' behavior is of interest in and of itself for determining the impacts of any action which alters the network. Similar criticisms may be offered concerning the Samuelson -Takayama -Judge model of static spatial price equilibrium and its variants (see Nagurney, 1987a), which are used to model interregional commodity flows. In particular, models of this type require that delivered prices be equated to local prices for nontrivial commodity flows. Again, temporal fluctuations may occur owing to effects as widely varied as: weather patterns which alter transportation costs, technological innovations which alter commodity supply functions, and income changes which alter commodity demand functions. Dynamic generalizations of the Samuelson-Takayama-Judge model have been proposed by Nagurney (1987b) and Friesz et al (1988a), although these efforts have still not resulted in a commonly agreed upon approach for treating the dynamic interactions of spatially separated markets. The above remarks concerning the failure of the most commonly used traffic and spatial price equilibrium models to treat dynamic considerations in a
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