H. W. Corley

Most vital links and nodes in weighted networks

The n most vital links (or nodes) in a weighted network are those n links (nodes) whose removal from the network results in the greatest increase in shortest distance between two specified nodes. Preliminary results are presented for obtaining these entities.

Optimality conditions for maximizations of set-valued functions

The maximization with respect to a cone of a set-valued function into possibly infinite dimensions is defined, and necessary and sufficient optimality conditions are established. In particular, an analogue of the Fritz John necessary optimality conditions is proved using a notion of derivative defined in terms of tangent cones.

Existence and Lagrangian duality for maximizations of set-valued functions

The maximization with respect to a cone of a set-valued function into possibly infinite dimensions is defined; some existence results are established; and a Lagrangian duality theory is developed.

Shortest paths in networks with vector weights

For a directed network in which vector weights are assigned to arcs, the Pareto analog to the shortest path problem is analyzed. An algorithm is presented for obtaining all Pareto shortest paths from a specified node to every other node.

Optimization Theory for n-Set Functions

Abstract An optimization theory is developed for functions of n sets. Optimality conditions are established, and a Lagrangian duality is obtained.

An existence result for maximizations with respect to cones

A sufficient condition is given for the existence of a solution to a generalized Pareto maximization problem in which maximization is defined in terms of cones. This result generalizes the fact that an upper semicontinuous real-valued function achieves its maximum on a compact set.

Duality theory for maximizations with respect to cones

Generalizations of Pareto optimality have been studied by a number of authors. In finite dimensions such work is exemplified by Corley [S 1, DaCunha and Polak [lo], Goeffrion [ 131, Hartley [ 151. Lin [ 171, Tanino and Sawaragi 1211, Wendell and Lee [22], and Yu [23]. The optimization of functions into possibly infinite dimensions has been considered by Borwein 121, Cesari and Suryanarayana [3], Christopeit [4], Corley [6,7], Craven [S, 91, Hurwicz [ 16 1, Neustadt [ 19 1. and Ritter [20]. An extensive bibliography on Pareto optimality, its extensions, and applications is given in 111. In this paper a duality theory is developed using the concept of saddlepoints for a problem in which the maximization of a function into possibly infinite dimensions is defined in terms of a cone. The results here extend the work of Tanino and Sawaragi [21] to infinite dimensions. A distinction is also made here between the notions of weak and strong optimality. Distinguishing between the two concepts allows the removal of the assumption of properness in [ 211 in establishing a relationship between the primal and dual problems, as well as permits additional duality relationships to be proved.

Games with vector payoffs

Two-person games are defined in which the payoffs are vectors. Necessary and sufficient conditions for optimal mixed strategies are developed, and examples are presented.

A Partitioning Problem with Applications in Regional Design

Many problems in two-dimensional location analysis can be formulated as one of optimally dividing a given region into n subregions with specified areas. Examples are problems involving districting, facility design, warehouse layout, and urban planning. This paper contains a study of such a partitioning problem. Theoretical results are presented for a problem of optimally partitioning a given set of points in k-dimensional Euclidean space into n subsets, where each subset has a specified Lebesgue measure. The existence of an optimal solution is established, and necessary and sufficient optimality conditions are proved. Models are then formulated in terms of this partitioning problem for specific districting and warehouse-layout problems.

A new scalar equivalence for Pareto optimization

A new scalar equivalence is presented for Pareto optimization. This equivalence involves the maximization of a real-valued function subject to parametric constraints. The method extends to maximizations defined in terms of polyhedral cones.