Nondifferentiable Optimization: Motivations and Applications : Proceedings of an IIASA (International Institute for Applied Systems Analysis) Workshop on Nondifferentiable Optimization Held at Sopron, Hungary, September 17–22, 1984

Understanding the stochastic enviornment is as much important to the manager as to the economist. From production and marketing to financial management, a manager has to assess various costs imposed by uncertainty. The economist analyzes the role of incomplete and too often imperfect information structures on the optimal decisions made by a firm. The need for understanding the role of uncertainty in quantitative decision models, both in economics and management science provide the basic motivation of this monograph. The stochastic environment is analyzed here in terms of the following specific models of optimization: linear and quadratic models, linear programming, control theory and dynamic programming. Uncertainty is introduced here through the para­ meters, the constraints, and the objective function and its impact evaluated. Specifically recent developments in applied research are emphasized, so that they can help the decision-maker arrive at a solution which has some desirable charac­ teristics like robustness, stability and cautiousness. Mathematical treatment is kept at a fairly elementary level and applied as­ pects are emphasized much more than theory. Moreover, an attempt is made to in­ corporate the economic theory of uncertainty into the stochastic theory of opera­ tions research. Methods of optimal decision rules illustrated he re are applicable in three broad areas: (a) applied economic models in resource allocation and economic planning, (b) operations research models involving portfolio analysis and stochastic linear programming and (c) systems science models in stochastic control and adaptive behavior. The International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria, has been involved in research on nondifferentiable optimization since 1976. IIASA-based East-West cooperation in this field has been very productive, leading to many important theoretical, algorithmic and applied results. Nondifferentiable optimi­ zation has now become a recognized and rapidly developing branch of mathematical programming. To continue this tradition, and to review recent developments in this field, IIASA held a Workshop on Nondifferentiable Optimization in Sopron (Hungary) in September 1964. The aims of the Workshop were: 1. To discuss the state-of-the-art of nondifferentiable optimization (NDO), its origins and motivation; 2. To compare-various algorithms; 3. To evaluate existing mathematical approaches, their applications and potential; 4. To extend and deepen industrial and other applications of NDO. The following topics were considered in separate sessions: General motivation for research in NDO: nondifferentiability in applied problems, nondifferentiable mathematical models. Numerical methods for solving nondifferentiable optimization problems, numerical experiments, comparisons and software. Nondifferentiable analysis: various generalizations of the concept of subdifferen­ tials. Industrial and other applications. This volume contains selected papers presented at the Workshop. It is divided into four sections, based on the above topics: I. Concepts in Nonsmooth Analysis II. Multicriteria Optimization and Control Theory III. Algorithms and Optimization Methods IV. Stochastic Programming and Applications We would like to thank the International Institute for Applied Systems Analysis, particularly Prof. V. Kaftanov and Prof. A.B. Kurzhanski, for their support inorganiz­ ing this meeting. The International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria, has been involved in research on nondifferentiable optimization since 1976. IIASA-based East-West cooperation in this field has been very productive, leading to many important theoretical, algorithmic and applied results. Nondifferentiable optimi­ zation has now become a recognized and rapidly developing branch of mathematical programming. To continue this tradition, and to review recent developments in this field, IIASA held a Workshop on Nondifferentiable Optimization in Sopron (Hungary) in September 1964. The aims of the Workshop were: 1. To discuss the state-of-the-art of nondifferentiable optimization (NDO), its origins and motivation; 2. To compare-various algorithms; 3. To evaluate existing mathematical approaches, their applications and potential; 4. To extend and deepen industrial and other applications of NDO. The following topics were considered in separate sessions: General motivation for research in NDO: nondifferentiability in applied problems, nondifferentiable mathematical models. Numerical methods for solving nondifferentiable optimization problems, numerical experiments, comparisons and software. Nondifferentiable analysis: various generalizations of the concept of subdifferen­ tials. Industrial and other applications. This volume contains selected papers presented at the Workshop. It is divided into four sections, based on the above topics: I. Concepts in Nonsmooth Analysis II. Multicriteria Optimization and Control Theory III. Algorithms and Optimization Methods IV. Stochastic Programming and Applications We would like to thank the International Institute for Applied Systems Analysis, particularly Prof. V. Kaftanov and Prof. A.B. Kurzhanski, for their support in organiz­ ing this meeting Optimal Control theory has been increasingly used in Economi- and Management Science in the last fifteen years or so. It is now commonplace, even at textbook level. It has been applied to a great many areas of Economics and Management Science, such as Optimal Growth, Optimal Population, Pollution control, Natural Resources, Bioeconomics, Education, International Trade, Monopoly, Oligopoly and Duopoly, Urban and Regional Economics, Arms Race control, Business Finance, Inventory Planning, Marketing, Maintenance and Replacement policy and many others. It is a powerful tool of dynamic optimization. There is no doubt social sciences students should be familiar with this tool, if not for their own research, at least for reading the literature. These Lecture Notes attempt to provide a plain exposition of Optimal Control Theory, with a number of economic examples and applications designed mainly to illustrate the various techniques and point out the wide range of possible applications rather than to treat exhaustively any area of economic theory or policy. Chapters 2,3 and 4 are devoted to the Calculus of Variations, Chapter 5 develops Optimal Control theory from the Variational approach, Chapter 6 deals with the problems of constrained state and control variables, Chapter 7, with Linear Control models and Chapter 8, with stabilization models. Discrete systems are discussed in Chapter 9 and Sensitivity analysis in Chapter 10. Chapter 11 presents a wide range of Economics and Management Science applications. In this book Professor Beckmann, with considerable ingenuity, offers a mathematical analysis of productive organizations in the widest sense. Starting with descriptive features he builds up, step by step, production functions, profiting from the rigor of a set of axioms or assumptions and their logical implications. Among the organizations studied hierarchies play a predominant role and are compared with such forms of cooperation as partnerships and'ladders'. A number of well-known basic concepts such as span of control, rank, line vs. staff and others serve as starting points. His analysis leads to such refinements as balanced, regular or degenerated organization patterns and interesting comparisons of the efficiency of various structures. Empirical verification of the axioms or assumptions is not the objective chosen by the author--except a few concrete illustra­ tions--but the book constitutes an excellent basis for such research. Several of the results obtained take simpler forms for very large hierarchies. The renewed interest, shown in political discussions, in the bureaucratization of both large enterprises and government machinery makes Dr. Beckmann's work highly topical. Discussions (by Bahro) of the GDR and by many other authors of Japanese management as compared with American or western European are cases in point. Some additional variables may then have to be added, of a psychological nature: for instance satisfaction from work or irritation evoked by excessive supervision. In this book Professor Beckmann, with considerable ingenuity, offers a mathematical analysis of productive organizations in the widest sense. Starting with descriptive features he builds up, step by step, production functions, profiting from the rigor of a set of axioms or assumptions and their logical implications. Among the organizations studied hierarchies play a predominant role and are compared with such forms of cooperation as partnerships and "ladders". A number of well-known basic concepts such as span of control, rank, line vs. staff and others serve as starting points. His analysis leads to such refinements as balanced, regular or degenerated organization patterns and interesting comparisons of the efficiency of various structures. Empirical verification of the axioms or assumptions is not the objective chosen by the author--except a few concrete illustra­ tions--but the book constitutes an excellent basis for such research. Several of the results obtained take simpler forms for very large hierarchies. The renewed interest, shown in political discussions, in the bureaucratization of both large enterprises and government machinery makes Dr. Beckmann's work highly topical. Discussions (by Bahro) of the GDR and by many other authors of Japanese management as compared with American or western European are cases in point. Some additional variables may then have to be added, of a psychological nature: for instance satisfaction from work or irritation evoked by excessive supervision