Dean A. Carlson

Large-scale convex optimization methods for air quality policy assessment

This paper presents the implementation of a large-scale systems analytic method in a model permitting the assessment of air quality policies in urban regions. This method is based on the use of a convex optimization technique called the analytic center cutting plane method. One realizes the coupling of a photochemical model (TAPOM-Lite), used to simulate ozone creation and dispersion in the region under study, and of a technoeconomic model (MARKAL-Lite) that represents the technology and energy uses by different economic sectors in the same region. Although the models correspond to different time and space scales, one realizes the coupling through a series of approximating convex reduced order optimization problems with constraints that are implicitly defined by the photochemical and the technoeconomic model, respectively. The implementation of the method in a case study corresponding to the region of Geneva, Switzerland is described.

An Observation on Two Methods of Obtaining Solutions to Variational Problems

In this paper, we compare two methods for obtaining solutions for free problems in the calculus of variations. The first is due to Carathéodory (Ref. 1) and the second due to Leitmann (Ref. 2). Both methods introduce the notion of equivalent variational problems. Using either approach, an auxiliary problem is obtained for which the solution is more easily obtained. We compare both approaches by using each to solve the same class of examples. We conclude our discussion by unifying the two approaches into one and illustrating the potential of this new method through the use of an elementary example.

Infinite Horizon Dynamic Games with Coupled State Constraints

In this paper we investigate the existence of equilibrium solutions to a class of infinite-horizon differential games with a coupled state constraint. These differential games typically represent dynamic oligopoly models where each firm controls its own dynamics (decoupled controls). In the absence of a coupled state constraint, the interactions among players take place essentially in the objective function. The introduction of a coupled state constraint in such models can be motivated, for example, by the consideration of a global environmental constraint imposed on competing firms. Our approach for the characterization of asymptotic equilibria under such a constraint is an adaptation of Rosen’s normalized equilibrium concept [10] to this infinite-horizon dynamic game framework. We utilize the Turnpike property and the notion of an attracting normalized steady-state equilibrium for the definition of a time invariant multiplier to provide conditions for the existence of dynamic equilibria. We consider discounted and undiscounted dynamic game models in our investigation.

Overtaking Equilibria for Switching Regulator and Tracking Games

We consider a nonzero-sum stochastic differential game with linear dynamics and quadratic costs. The random disturbances are defined as a Markov chain which switches the dynamics’ modes and the targets which are tracked by the different players. Due to these random switches the overtaking equilibrium concept has to be used when the time horizon is infinite. We characterize an affine feedback overtaking equilibrium for the p-player case, prove existence for the completely symmetric game case and establish a comparison theorem for the solution of the coupled Riccati equations in the two-player deterministic case.

Carathéodory's method for a class of dynamic games ✩

The method of equivalent variational methods, originally due to Caratheodory for free problems in the calculus of variations is extended to investigate feedback Nash equilibria for a class of n-person differential games. Both the finite-horizon and infinite-horizon cases are considered. Examples are given to illustrate the presented results.  2002 Elsevier Science (USA). All rights reserved.

Uniformly overtaking and weakly overtaking optimal solutions in infinite-horizon optimal control: when optimal solutions are agreeable

In this paper, we investigate the relationship between two classes of optimality which have arisen in the study of dynamic optimization problems defined on an infinite-time domain. We utilize an optimal control framework to discuss our results. In particular, we establish relationships between limiting objective functional type optimality concepts, commonly known as overtaking optimality and weakly overtaking optimality, and the finite-horizon solution concepts of decision-horizon optimality and agreeable plans. Our results show that both classes of optimality are implied by corresponding uniform limiting objective functional type optimality concepts, referred to here as uniformly overtaking optimality and uniformly weakly overtaking optimality. This observation permits us to extract sufficient conditions for optimality from known sufficient conditions for overtaking and weakly overtaking optimality by strengthening their hypotheses. These results take the form of a strengthened maximum principle. Examples are given to show that the hypotheses of these results can be realized.

Nonconvex and relaxed infinite-horizon optimal control problems

In this paper, we consider the Lagrange problem of optimal control defined on an unbounded time interval in which the traditional convexity hypotheses are not met. Models of this form have been introduced into the economics literature to investigate the exploitation of a renewable resource and to treat various aspects of continuous-time investment. An additional distinguishing feature in the models considered is that we do not assume a priori that the objective functional (described by an improper integral) is finite, and so we are led to consider the weaker notions of overtaking and weakly overtaking optimality. To treat these models, we introduce a relaxed optimal control problem through the introduction of chattering controls. This leads us naturally to consider the relationship between the original problem and the convexified relaxed problem. In particular, we show that the relaxed problem may be viewed as a limiting case for the original problem. We also present several examples demonstrating the applicability of our results.

The Existence and Uniqueness of Equilibria in Convex Games with Strategies in Hilbert Spaces

In this paper we extend the approach of Rosen [7] for existence and uniqueness of Nash equilibria for finite-dimensional convex games to an abstract setting in which the strategies of each player are in separable Hilbert spaces. Through the use of an extension of the Kakutani fixed-point theorem, we are able to extend Rosen’s existence result to this setting. Our uniqueness results are obtained by extending Rosen’s notion of strict diagonal convexity to this setting. Several examples, in the context of open-loop dynamic games, to which our results may be applied are presented.

Infinite-horizon optimal controls for problems governed by a Volterra integral equation

In this work, we concern ourselves with the existence of optimal solutions to optimal control problems defined on an unbounded time interval with states governed by a nonlinear Volterra integral equation. These results extend both the work of Baum and others in infinite-horizon control of ordinary differential equations as well as the work of Angell concerning integral equations. In addition, we incorporate into the objective functional (described by an improper integral) a discount factor which reflects a hereditary dependence on both state and control. In this manner, we are able to generalize the recent results of Becker, Boyd, and Sung in which they establish an existence theorem in the calculus of variations with objective functionals of the so-called recursive type. Our results are obtained through the use of appropriate lower-closure theorems and compactness conditions. Examples are presented in which the applicability of our results is demonstrated.

The Existence of Optimal Controls for Problems Defined on Time Scales

In this paper we investigate the existence of optimal solutions for dynamic optimization problems defined on time scales. We use the classical convexity and seminormality conditions originating in the works of L. Tonelli and E. J. McShane for problems in the calculus of variations and in the works of L. Cesari, C. Olech, R. T. Rockafellar and others for problems in optimal control theory, thus extending these classical results to optimal control problems whose states satisfy a dynamic equation on an arbitrary time scale. As applications of our results we focus on three examples of time scales—the real line, the integers and a monotone sequence of points converging to one.