Mathematical Programming Theory and Methods

Mathematical Programming, a branch of Operations Research, is perhaps the most efficient technique in making optimal decisions. It has a very wide application in the analysis of management problems, in business and industry, in economic studies, in military problems and in many other fields of our p...

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Hlavní autor: Sinha, S. M
Médium: E-kniha
Jazyk:angličtina
Vydáno: Chantilly Elsevier Science & Technology 2006
Elsevier Science
Vydání:1
Témata:
Mathematics
Programming (Mathematics)
ISBN:9788131203767, 813120376X
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  • Front Cover -- Mathematical Programming: Theory and Methods -- Copyright Page -- Contents -- Chapter 1. Introduction -- 1.1 Background and Historical Sketch -- 1.2. Linear Programming -- 1.3. Illustrative Examples -- 1.4. Graphical Solutions -- 1.5. Nonlinear Programming -- PART 1: MATHEMATICAL FOUNDATIONS -- Chapter 2. Basic Theory of Sets and Functions -- 2.1. Sets -- 2.2. Vectors -- 2.3. Topological Properties of Rn -- 2.4. Sequences and Subsequences -- 2.5. Mappings and Functions -- 2.6. Continuous Functions -- 2.7. Infimum and Supremum of Functions -- 2.8. Minima and Maxima of Functions -- 2.9. Differentiable Functions -- Chapter 3. Vector Spaces -- 3.1. Fields -- 3.2. Vector Spaces -- 3.3. Subspaces -- 3.4. Linear Dependence -- 3.5. Basis and Dimension -- 3.6. Inner Product Spaces -- Chapter 4. Matrices and Determinants -- 4.1. Matrices -- 4.2. Relations and Operations -- 4.3. Partitioning of Matrices -- 4.4. Rank of a Matrix -- 4.5. Determinants -- 4.6. Properties of Determinants -- 4.7. Minors and Cofactors -- 4.8. Determinants and Rank -- 4.9. The Inverse Matrix -- Chapter 5. Linear Transformations and Rank -- 5.1. Linear Transformations and Rank -- 5.2. Product of Linear Transformations -- 5.3. Elementary Transformations -- 5.4. Echelon Matrices and Rank -- Chapter 6. Quadratic Forms and Eigenvalue Problems -- 6.1. Quadratic Forms -- 6.2. Definite Quadratic Forms -- 6.3. Characteristic Vectors and Characteristic Values -- Chapter 7. Systems of Linear Equations and Linear Inequalities -- 7.1. Linear Equations -- 7.2. Existence Theorems for Systems of Linear Equations -- 7.3. Basic Solutions and Degeneracy -- 7.4. Theorems of the Alternative -- Chapter 8. Convex Sets and Convex Cones -- 8.1. Introduction and Preliminary Definitions -- 8.2. Convex Sets and their Properties -- 8.3. Convex Hulls -- 8.4. Separation and Support of Convex Sets
  • 8.5. Convex Polytopes and Polyhedra -- 8.6. Convex Cones -- Chapter 9. Convex and Concave Functions -- 9.1. Definitions and Basic Properties -- 9.2. Differentiable Convex Functions -- 9.3. Generalization of Convex Functions -- 9.4. Exercises -- PART 2: LINEAR PROGRAMMING -- Chapter 10. Linear Programming Problems -- 10.1. The General Problem -- 10.2. Equivalent Formulations -- 10.3. Definitions and Terminologies -- 10.4. Basic Solutions of Linear Programs -- 10.5. Fundamental Properties of Linear Programs -- 10.6. Exercises -- Chapter 11. Simplex Method: Theory and Computation -- 11.1. Introduction -- 11.2. Theory of the Simplex Method -- 11.3. Method of Computation: The Simplex Algorithm -- 11.4. The Simplex Tableau -- 11.5. Replacement Operation -- 11.6. Example -- 11.7. Exercises -- Chapter 12. Simplex Method: Initial Basic Feasible Solution -- 12.1. Introduction: Artificial Variable Techniques -- 12.2. The Two-Phase Method [ 117] -- 12.3. Examples -- 12.4. The Method of Penalties [71 ] -- 12.5. Examples: Penalty Method -- 12.6. Inconsistency and Redundancy -- 12.7. Exercises -- Chapter 13. Degeneracy in Linear Programming -- 13.1. Introduction -- 13.2. Charnes' Perturbation Method -- 13.3. Example -- 13.4. Exercises -- Chapter 14. The Revised Simplex Method -- 14.1. Introduction -- 14.2. Outline of the Procedure -- 14.3. Example -- 14.4. Exercises -- Chapter 15. Duality in Linear Programming -- 15.1. -- 15.2. Cannonical Dual Programs and Duality Theorems -- 15.3. Equivalent Dual Forms -- 15.4. Other Important Results -- 15.5. Lagrange Multipliers and Duality -- 15.6. Duality in the Simplex Method -- 15.7. Example -- 15.8. Applications -- 15.9. Economic Interpretation of Duality -- 15.9. Exercises -- Chapter 16. Variants of the Simplex Method -- 16.1. Introduction -- 16.2. The Dual Simplex Method -- 16:3. The Dual Simplex Algorithm
  • 16.4. Initial Dual - Feasible Basic Solution -- 16.5. Example -- 16.6. The Primal - Dual Algorithm -- 16.7. Summary of the Primal-Dual Algorithm -- 16.8. Example -- 16.9. The Initial Solution to the Dual Problem: The Artificial Constraint Technique -- 16.9. Exercises -- Chapter 17. Post-Optimization Problems: Sensitivity Analysis and Parametric Programming -- 17.1. Introduction -- 17.2. Sensitivity Analysis -- 17.3. Changes in the Cost Vector -- 17.4. Changes in the Requirement Vector -- 17.5. Changes in the Elements of the Technology Matrix -- 10.6. Addition of a Constraint -- 17.7. Addition of a Variable -- 17.8. Parametric Programming -- 17.9. Parametric Changes in the Cost Vector -- 17.10. Parametric Changes in the Requirement Vector -- 17.11. Exercises -- Chapter 18. Bounded Variable Problems -- 18.2. Bounded from Below -- 18.3. Bounded from Above -- 18.4. The Optimality Criterion -- 18.5. Improving a Basic Feasible Solution -- 18.6. Example -- 18.7. Exercises -- Chapter 19. Transportation Problems -- 19.1. Introduction -- 19.2. The Mathematical Formulation -- 19.3. Fundamental Properties of Transportation Problems -- 19.4. Initial Basic Feasible Solution -- 19.5. Duality and Optimality Criterion -- 19.6. Improvement of a Basic Feasible Solution -- 19.7. The Transportation Algorithm -- 19.8. Degeneracy -- 19.9. Examples -- 19.10. Unbalanced Transportation Problem -- 19.11. The Transhipment Problem -- 19.12. Exercises -- Chapter 20. Assignment Problems -- 20.1. Introduction and Mathematical Formulation -- 20.2. The Hungarian Method -- 20.3. The Assignment Algorithm -- 20.4. Variations of the Assignment Model -- 20.5. Some Applications of the Assignment Model -- 20.6. Exercises -- Chapter 21. The Decomposition Principle for Linear Programs -- 21.1 Introduction -- 21.2. The Original Problem and its Equivalent -- 21.3. The Decomposition Algorithm
  • Chapter 28. Some Special Topics in Mathematical Programming -- 28.1. Goal Programming -- 28.2. Multiple Objective Linear Programming -- 28.3. Fractional Programming -- 28.4. Exercises -- Chapter 29. Dynamic Programming -- 29.1. Introduction -- 29.2. Basic Features of Dynamic Programming Problems and the Principle of Optimality -- 29.3. The Functional Equation -- 29.4. Cargo Loading Problem -- 29.5. Forward and Backward Computations, Sensitivity Analysis -- 29.6. Shortest Route Problem -- 29.7. Investment Planning -- 29.8. Inventory Problem -- 29.9. Reliability Problem -- 29.10. Cases where Decision Variables are Continuous -- 29.11. The Problem of Dimensionality -- 29.12. Reduction in Dimensionality -- 29.13. Stochastic Dynamic Programming -- 29.14. Infinite Stage Process -- 29.15. Exercises -- Bibliography -- Index
  • 21.4. Initial Basic Feasible Solution -- 21.5. The Case of Unbounded Sj -- 21.6. Remarks on Methods of Decomposition -- 21.7. Example -- 21.8. Exercises -- Chapter 22. Polynomial Time Algorithms for Linear Programming -- 22.1. Introduction -- 22.2. Computational Complexity of Linear Programs -- 22.3. Khachiyan's Ellipsoid Method -- 22.4. Solving Linear Programming Problems by the Ellipsoid Method -- 22.5. Karmarkar's Polynomial-Time Algorithm -- 22.6. Convergence and Complexity of Karmarkar's Algorithm -- 22.7. Conversion of a General Linear Program into Karmarkar's Form -- 22.8. Exercises -- PART 3: NONLINEAR AND DYNAMIC PROGRAMMING -- Chapter 23. Nonlinear Programming -- 23.1. Introduction -- 23.2. Unconstrained Optimization -- 23.3. Constrained Optimization -- 23.4. Kuhn-Tucker Optimality Conditions -- 23.5. Kuhn-Tucker Constraint Qualification -- 23.6. Other Constraint Qualifications -- 23.7. Lagrange Saddle Point Problem and Kuhn-Tucker Conditions -- 23.8. Exercises -- Chapter 24. Quadratic Programming -- 24.1 Introduction -- 24.2. Wolfe's Method -- 24.3. Dantzig's Method -- 24.4. Beale's Method -- 24.5. Lemke's complementary Pivoting Algorthm -- 24.6. Exercises -- Chapter 25. Methods of Nonlinear Programming -- 25.1. Separable Programming -- 25.2. Kelley's Cutting Plane Method -- 25.3. Zoutendijk's Method of Feasible Directions -- 25.4. Rosen's Gradient Projection Method -- 25.5. Wolfe's Reduced Gradient Method -- 25.6. Zangwill's Convex Simplex Method -- 25.7. Dantzig's Method for Convex Programs -- 25.8. Exercises -- Chapter 26. Duality in Nonlinear Programming -- 26.1. Introduction -- 26.2. Duality Theorems -- 26.3. Special Cases -- Chapter 27. Stochastic Programming -- 27.1. Introduction -- 27.2. General Stochastic Linear Program [422, 423] -- 27.3. The Sochastic Objective Function -- 27.4. The General Case -- 27.5. Exercises