A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εk-global minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an ε-global minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.

We present a modeling and analysis approach that offers a way to generate preference optimal flight paths in one-on-one air combat. The pilot's sequential maneuvering decisions are modeled by a multistage influence diagram. The influence diagram graphically describes the elements of the decision process, contains a point-mass model for the dynamics of an aircraft and takes into account the decision maker's preferences under conditions of uncertainty. Optimal trajectories with respect to the given preference model are obtained by converting the multistage influence diagram into a discrete time dynamic optimization problem that is solved by nonlinear programming. The presented approach is illustrated by analyzing two one-on-one air combat scenarios.

We provide a new mathematical model for strategic traffic management, formulated and analyzed as a mathematical program with equilibrium constraints (MPEC). The model includes two types of control (upper-level) variables, which may be used to describe such traffic management actions as traffic signal setting, network design, and congestion pricing. The lower-level problem of the MPEC describes a traffic equilibrium model in the sense of Wardrop, in which the control variables enter as parameters in the travel costs. We consider a (small) variety of model settings, including fixed or elastic demands, the possible presence of side constraints in the traffic equilibrium system, and representations of traffic flows and management actions in both link-route and link-node space. For this model, we also propose and analyze a descent algorithm. The algorithm utilizes a new reformulation of the MPEC into a constrained, locally Lipschitz minimization problem in the product space of controls and traffic flows. The reformulation is based on the Minty (1967) parameterization of the graph of the normal cone operator for the traffic flow polyhedron. Two immediate advantages of making use of this reformulation are that the resulting descent algorithm can be operated and established to be convergent without requiring that the travel cost mapping is monotone, and without having to ever solve the lower-level equilibrium problem. We provide example realizations of the algorithm, establish their convergence, and interpret their workings in terms of the traffic network.

In this paper we study the relationship between bilevel optimization and multicriteria optimization. Given a bilevel optimization problem, we introduce an order relation such that the optimal solutions of the bilevel problem are the nondominated points with respect to the order relation. In the case where the lower level problem of the bilevel optimization problem is convex and continuously dierentiable in the lower level variables, this order relation is equivalent to a second, more tractable order relation. Then, we show how to construct a (nonconvex) cone for which we can prove that the nondominated points with respect to the order relation induced by the cone are also nondominated points with respect to any of the two order relations mentioned before. We also comment on the practical and computational implications of our approach.

Network equilibrium models for that have traditionally been used for transportation planning have penetrated in recent years to other scientific fields. These models have recently been introduced in telecommunications networks literature, as well as in the in the field of game theory. Researchers in the latter fields are not always aware of the very rich literature on equilibrium models outside of their application area. On the other hand, researchers that have used network equilibrium models in transportation may not be aware of new application areas of their tools. The aim of this paper is to present some central research issues and tools in network equilibria and pricing that could bring closer the three mentioned research communities.

Economists model economic activity based on fairly standard assumptions about human behavior and decision making. The rationality assumption, in which all economic agents optimize well-defined objectives under complete information, has been the prevalent model in economics since its inception. Optimization formulations also arise in modeling market structures other than perfect competition and in games of strategic interaction. This sets up a close association between models in microeconomics, game theory, and industrial organization on the one hand, and mathematical programming, optimization, and complementarity theory on the other. Many mathematical programming solution techniques are applicable to the formulation and solution of economics models. This talk provides a brief overview of the connections between economics and mathematical programming / equilibrium modeling.

Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner. Example applications are taken from chemical engineering.

Abstract: We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer's network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.

The bilevel programming problem (BLPP) is equivalent to a two-person Stackelberg game in which the leader and follower pursue individual objectives. Play is sequential and the choices of one affect the choices and attainable payoffs of the other. The purpose of this paper is to investigate an extension of the linear BLPP where both players' objective functions are bilinear. To overcome certain discontinuities in the master problem, a regularized term is added to the follower's objective function. Using ideas from parametric programming, the directional derivatives of the regularized follower's solution function are computed along with its generalized Jacobian. This allows us to develop a bundle trust region algorithm. Theoretical results related to the existence of solutions are presented as well as a convergence analysis of the proposed methodology. Key words: bilevel programming, bundle algorithm, Lipschitz continuity, generalized gradients, nondifferentiable optimization. 1 Freiberg University of Mining and Technology, Germany, dempe@math.tu-freiberg.de 2 Graduate Program in Operations Research, University of Texas, Austin, U.S.A., jbard@mail.utexas.edu 1

In this paper we consider the bilevel programming problem (BLPP) which is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level prob-lem. We extend well-known constraint qualifications for nonlinear programming problems such as the Abadie constraint qualification, the Kuhn-Tucker constraint qualification, the Zangwill constraint qualification, the Arrow-Hurwicz-Uzawa con-straint qualification and the weak reverse convex constraint qualification to BLPPs and derive a Karash-Kuhn-Tucker (KKT) type necessary optimality condition un-der these constraint qualifications without assuming the lower level problem sat-isfying the Mangasarian Fromovitz constraint qualification. Relationships among various constraint qualifications are also given. Key words: necessary optimality conditions, constraint qualifications, nonsmooth analysis, value function, bilevel programming problems.