Systems Optimization Methodology, Part 1

This monograph deals with theoretical fundamentals and numerical methods of optimizing nondetermined models of systems. The main body of this work is devoted to investigation and optimization of system models under incomplete information. Much consideration is given to one-, two- and multistage problems of stochastic programming, solution methods and problems of solution stability. Optimization problems with fuzzy variables and optimization problems in function spaces are investigated. Examples are given for implementation of specific models of optimization under incomplete information. The book is based on lectures given by the author since 1965 for undergraduates and postgraduates at St Petersburg State University Ch. 1. Philosophical problems of the methodology for systems modeling. 1. Philosophical problems of the methodology of scientific cognition. 2. Philosophical problems of modeling. 3. Theoretical aspects of modeling systems in the process of scientific cognition. 4. Systems modeling. 5. Mathematical logic as a means of cognition -- ch. 2. Problems of quantitative analysis in natural and social sciences. 6. Problems of systems quantitative analysis. 7. Problems of systems description. 8. Definitions of systems -- ch. 3. Dantzig-Wulf decomposition. 9. Dantzig-Wulf decomposition method. 10. Dual approach in block programming. 11. Transportation problem solution by the decomposition method. 12. Decomposition for problems with a block-staircase structure. 13. Solution of the interval programming problem. 14. Extension of the Dantzig-Wulf decomposition principle to nonlinear programming problems -- ch. 4. Parametric decomposition. 15. Kornai-Liptack method. 16. Solution technique for block-separable nonlinear problems. 17. On parametric decomposition of the resources allocation problem. 18. One method of parametric decomposition for linear and separable programming problems -- ch. 5. Decomposition based on separation of variables. 19. Constraint relaxation method for the convex programming problem. 20. The Ritter method for the block problem with the tying variables and constraints. 21. The Rosen division method for linear programming problems. 22. The Rosen division method for nonlinear programming. 23. Benders method for a special mathematical programming problem -- ch. 6. Decomposition based on optimization technique. 24. Application of the componentwise descent method for solving the problems of mathematical programming and optimal management. 25. Conditional gradient method and decomposition of problems of mathematical programming and optimal control. 26. Utilization of a penalty constant in decomposition of the mathematical programming problem. 27. Decomposition based on simplex method modifications -- ch. 7. Decomposition and aggregation. 28. Aggregation method for solving a system of linear equations. 29. Aggregation method for the block problem of linear programming. 30. Decomposition and aggregation based on perturbations method. 31. Decomposition method based on aggregation of variables from different blocks -- ch. 8. Application of solution techniques for large dimension problems to grain farming optimization. 32. Grain chain. 33. Grain farming optimization problem. 34. Solution technique. 35. Application of the algorithm to the grain farming optimization problem -- ch. 9. Major problems of multiobjective optimization. 36. Formulation of the multiobjective optimization problem. 37. The study of the maximal efficiency principle. 38. The study of the parametric principle of maximal efficiency in multiobjective optimization -- ch. 10. The study of improvability and priority issues in multiobjective optimization problems. 39. The study of the problem of feasible solution improvability. 40. The study of the priority problem in multiobjective optimization -- ch. 11. Problems of multiobjective optimization under information deficiency. 41. Problems of decision multiobjective optimization under uncertainty. 42. Multiobjective optimization problems for dynamic control systems -- ch. 12. Methodology of vector optimization. 43. Optimization methodology for hierarchical sequence of quality criteria. 44. Optimization of hierarchical sequence of quality criteria. 45. Finding the set of unimprovable points. 46. Determination of the solution based on a particular tradeoff -- Conclusion "This monograph defines the notion of a "system" by reference to those systems which exhibit goal-oriented behavior and utilize the notion of decision making and controls. Such systems allow for phenomenological description and fix the nature of causal transformations of input effects into output quantities. The study of consequences of the fact that the systems possess some properties constitutes the content of systems optimization methodology which goes beyond the scope of descriptive classification of systems. Chapter 1 deals with philosophical problems of systems methodology. An attempt is made to systematize and analyze the problems of scientific methodology as applied to systems modeling methodology which is viewed as the most general concept utilized in modern science. Chapter 2 focuses on problems of qualitative analysis in natural and social sciences. Attention is drawn to problems of measurement theory and quantitative analysis of systems. Approaches and methods of systems analysis and synthesis form the central portion of the book. Much study is given to the methods of systems decomposition, an integration using both discrete and continuous descriptions of objects, processes, and phenomena. Examples of complex goal-oriented systems are also provided. The remaining part of the book is largely centered around the methodology of multiobjective systems optimization Covering the subject of systems optimization methodology, this work discusses areas including: philosophical problems of the methodology for systems modelling; decomposition and aggregation; and vector optimization.