Patrick T. Harker

Bio: Patrick T. Harker is an academic researcher from University of Pennsylvania. The author has contributed to research in topics: Variational inequality & Complementarity theory. The author has an hindex of 48, co-authored 104 publications receiving 10009 citations. Previous affiliations of Patrick T. Harker include University of California, Santa Barbara & University of Chicago.

Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications

Abstract: Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

The theory of ratio scale estimation: Saaty's analytic hierarchy process

Abstract: The Analytic Hierarchy Process developed by Saaty (Saaty, T. L. 1980. The Analytic Hierarchy Process. McGraw-Hill, New York.) has proven to be an extremely useful method for decision making and planning. However, some researchers in these areas have raised concerns over the theoretical basis underlying this process. This paper addresses currently debated issues concerning the theoretical foundations of the Analytic Hierarchy Process. We also illustrate through proof and through examples the validity or fallaciousness of these criticisms.

Competition and Outsourcing with Scale Economies

Abstract: Scale economies are commonplace in operations, yet because of analytical challenges, relatively little is known about how firms should compete in their presence. This paper presents a model of competition between two firms that face scale economies; (i.e., each firm's cost per unit of demand is decreasing in demand). A general framework is used, which incorporates competition between two service providers with price- and time-sensitive demand (a queuing game), and competition between two retailers with fixed-ordering costs and pricesensitive consumers (an Economic Order Quantity game). Reasonably general conditions are provided under which there exists at most one equilibrium, with both firms participating in the market. We demonstrate, in the context of the queuing game, that the lower cost firm in equilibrium may have a higher market share and a higher price, an enviable situation. We also allow each firm to outsource their production process to a supplier. Even if the supplier's technology is no better than the firms' technology and the supplier is required to establish dedicated capacity (so the supplier's scale can be no greater than either firm's scale), we show that the firms strictly prefer to outsource. We conclude that scale economies provide a strong motivation for outsourcing that has not previously been identified in the literature.

The Analytic hierarchy process : applications and studies

Abstract: Management science is a di scipl ine dedicated to the development of techniques that enable decision makers to cope with the increasing complexity of our world. The early burst of excitement which was spawned by the development and successful applications of linear programming to problems in both the public and private sectors has challenged researchers to develop even more sophisticated methods to deal with the complex nature of decision making. Sophistication, however, does not always trans 1 ate into more complex mathematics. Professor Thomas L. Saaty was working for the U. S. Defense Department and for the U. S. Department of State in the late 1960s and early 1970s. In these positions, Professor Saaty was exposed to some of the most complex decisions facing the world: arms control, the Middle East problem, and the development of a transport system for a Third World country. While having made major contributions to numerous areas of mathematics and the theory of operations research, he soon realized that one did not need complex mathematics to come to grips with these decision problems, just the right mathematics! Thus, Professor Saaty set out to develop a mathematically-based technique for analyzing complex situations which was sophisticated in its simplicity. This technique became known as the Analytic Hierarchy Process (AHP) and has become very successful in helping decision makers to structure and analyze a wide range of problems."

Tabu Search

Abstract: From the Publisher: This book explores the meta-heuristics approach called tabu search, which is dramatically changing our ability to solve a hostof problems that stretch over the realms of resource planning,telecommunications, VLSI design, financial analysis, scheduling, spaceplanning, energy distribution, molecular engineering, logistics,pattern classification, flexible manufacturing, waste management,mineral exploration, biomedical analysis, environmental conservationand scores of other problems. The major ideas of tabu search arepresented with examples that show their relevance to multipleapplications. Numerous illustrations and diagrams are used to clarifyprinciples that deserve emphasis, and that have not always been wellunderstood or applied. The book's goal is to provide ''hands-on' knowledge and insight alike, rather than to focus exclusively eitheron computational recipes or on abstract themes. This book is designedto be useful and accessible to researchers and practitioners inmanagement science, industrial engineering, economics, and computerscience. It can appropriately be used as a textbook in a masterscourse or in a doctoral seminar. Because of its emphasis on presentingideas through illustrations and diagrams, and on identifyingassociated practical applications, it can also be used as asupplementary text in upper division undergraduate courses. Finally, there are many more applications of tabu search than canpossibly be covered in a single book, and new ones are emerging everyday. The book's goal is to provide a grounding in the essential ideasof tabu search that will allow readers to create successfulapplications of their own. Along with the essentialideas,understanding of advanced issues is provided, enabling researchers togo beyond today's developments and create the methods of tomorrow.

Proximal Algorithms

Abstract: This monograph is about a class of optimization algorithms called proximal algorithms. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Here, we discuss the many different interpretations of proximal operators and algorithms, describe their connections to many other topics in optimization and applied mathematics, survey some popular algorithms, and provide a large number of examples of proximal operators that commonly arise in practice.