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Full Description
In this book, the theory, methods and applications of separable optimization are considered. Some general results are presented, techniques of approximating the separable problem by linear programming problem, and dynamic programming are also studied. Convex separable programs subject to inequality/ equality constraint(s) and bounds on variables are also studied and convergent iterative algorithms of polynomial complexity are proposed. As an application, these algorithms are used in the implementation of stochastic quasigradient methods to some separable stochastic programs. The problems of numerical approximation of tabulated functions and numerical solution of overdetermined systems of linear algebraic equations and some systems of nonlinear equations are solved by separable convex unconstrained minimization problems. Some properties of the Knapsack polytope are also studied. This second edition includes a substantial amount of new and revised content. Three new chapters, 15-17, are included. Chapters 15-16 are devoted to the further analysis of the Knapsack problem. Chapter 17 is focused on the analysis of a nonlinear transportation problem. Three new Appendices (E-G) are also added to this edition and present technical details that help round out the coverage.
Optimization problems and methods for solving the problems considered are interesting not only from the viewpoint of optimization theory, optimization methods and their applications, but also from the viewpoint of other fields of science, especially the artificial intelligence and machine learning fields within computer science. This book is intended for the researcher, practitioner, or engineer who is interested in the detailed treatment of separable programming and wants to take advantage of the latest theoretical and algorithmic results. It may also be used as a textbook for a special topics course or as a supplementary textbook for graduate courses on nonlinear and convex optimization.
Contents
Preface to the New Edition.- Preface.-1 Preliminaries: Convex Analysis and Convex Programming.- Part I. Separable Programming.- 2 Introduction: Approximating the Separable Problem.- 3. Convex Separable Programming.- 4. Separable Programming: A Dynamic Programming Approach.- Part II. Convex Separable Programming With Bounds on the Variables.- 5. Statement of the Main Problem. Basic Result.- 6. Version One: Linear Equality Constraints.- 7. The Algorithms.- 8. Version Two: Linear Constraint of the Form \geq.- 9. Well-Posedness of Optimization Problems. On the Stability of the Set of Saddle Points of the Lagrangian.- 10. Extensions.- 11. Applications and Computational Experiments.- Part III. Selected Supplementary Topics and Applications.- 12. Applications of Convex Separable Unconstrained Nondifferentiable Optimization to Approximation Theory.- 13. About Projections in the Implementation of Stochastic Quasigradient Methods to Some Probabilistic Inventory Control Problems.- 14. Valid Inequalities, Cutting Planes and Integrality ofthe Knapsack Polytope.- 15. Relaxation of the Equality Constrained Convex Continuous Knapsack Problem.- 16. On the Solution of Multidimensional Convex Separable Continuous Knapsack Problem with Bounded Variables.- 17. Characterization of the Optimal Solution of the Convex Generalized Nonlinear Transportation Problem.- Appendices.- A. Some Definitions and Theorems from Calculus.- B. Metric, Banach and Hilbert Spaces.- C. Existence of Solutions to Optimization Problems — A General Approach.- D. Best Approximation: Existence and Uniqueness.- E. On the Solvability of a Quadratic Optimization Problem with a Feasible Region Defined as a Minkowski Sum of a Compact Set and Finitely Generated Convex Closed Cone- F. On the Cauchy-Schwarz Inequality Approach for Solving a Quadratic Optimization Problem.- G. Theorems of the Alternative.- Bibliography.- List of Notation.- List of Statements.- Index.
