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# Linear and nonlinear programming / David G. Luenberger, Yinyu Ye.

Material type: TextSeries: International series in operations research & management science ; volume 228.Publisher: Cham : Springer, [2016]Edition: Fourth editionDescription: xiii, 546 pages : illustrations (black and white) ; 25 cmContent type: text Media type: unmediated Carrier type: volumeISBN: 9783319188416; 3319188410; 3319188429; 9783319188423DDC classification: 519.72 LOC classification: T57.7 | .L84 2016Online resources: Click here to access online Also issued online.
Contents:
Machine generated contents note: 1. Introduction -- 1.1. Optimization -- 1.2. Types of Problems -- 1.3. Size of Problems -- 1.4. Iterative Algorithms and Convergence -- pt. I Linear Programming -- 2. Basic Properties of Linear Programs -- 2.1. Introduction -- 2.2. Examples of Linear Programming Problems -- 2.3. Basic Solutions -- 2.4. The Fundamental Theorem of Linear Programming -- 2.5. Relations to Convexity -- 2.6. Exercises -- 3. The Simplex Method -- 3.1. Pivots -- 3.2. Adjacent Extreme Points -- 3.3. Determining a Minimum Feasible Solution -- 3.4.Computational Procedure: Simplex Method -- 3.5. Finding a Basic Feasible Solution -- 3.6. Matrix Form of the Simplex Method -- 3.7. Simplex Method for Transportation Problems -- 3.8. Decomposition -- 3.9. Summary -- 3.10. Exercises -- 4. Duality and Complementarity -- 4.1. Dual Linear Programs -- 4.2. The Duality Theorem -- 4.3. Relations to the Simplex Procedure -- 4.4. Sensitivity and Complementary Slackness -- 4.5. Max Flow -- Min Cut Theorem.
Note continued: 4.6. The Dual Simplex Method -- 4.7.*The Primal-Dual Algorithm -- 4.8. Summary -- 4.9. Exercises -- 5. Interior-Point Methods -- 5.1. Elements of Complexity Theory -- 5.2.*The Simplex Method Is Not Polynomial-Time -- 5.3.*The Ellipsoid Method -- 5.4. The Analytic Center -- 5.5. The Central Path -- 5.6. Solution Strategies -- 5.7. Termination and Initialization -- 5.8. Summary -- 5.9. Exercises -- 6. Conic Linear Programming -- 6.1. Convex Cones -- 6.2. Conic Linear Programming Problem -- 6.3. Farkas' Lemma for Conic Linear Programming -- 6.4. Conic Linear Programming Duality -- 6.5.Complementarity and Solution Rank of SDP -- 6.6. Interior-Point Algorithms for Conic Linear Programming -- 6.7. Summary -- 6.8. Exercises -- pt. II Unconstrained Problems -- 7. Basic Properties of Solutions and Algorithms -- 7.1. First-Order Necessary Conditions -- 7.2. Examples of Unconstrained Problems -- 7.3. Second-Order Conditions -- 7.4. Convex and Concave Functions.
Note continued: 7.5. Minimization and Maximization of Convex Functions -- 7.6.*Zero-Order Conditions -- 7.7. Global Convergence of Descent Algorithms -- 7.8. Speed of Convergence -- 7.9. Summary -- 7.10. Exercises -- 8. Basic Descent Methods -- 8.1. Line Search Algorithms -- 8.2. The Method of Steepest Descent -- 8.3. Applications of the Convergence Theory -- 8.4. Accelerated Steepest Descent -- 8.5. Newton's Method -- 8.6. Coordinate Descent Methods -- 8.7. Summary -- 8.8. Exercises -- 9. Conjugate Direction Methods -- 9.1. Conjugate Directions -- 9.2. Descent Properties of the Conjugate Direction Method -- 9.3. The Conjugate Gradient Method -- 9.4. The C -- G Method as an Optimal Process -- 9.5. The Partial Conjugate Gradient Method -- 9.6. Extension to Nonquadratic Problems -- 9.7.*Parallel Tangents -- 9.8. Exercises -- 10. Quasi-Newton Methods -- 10.1. Modified Newton Method -- 10.2. Construction of the Inverse -- 10.3. Davidon-Fletcher-Powell Method -- 10.4. The Broyden Family.
Note continued: 10.5. Convergence Properties -- 10.6. Scaling -- 10.7. Memoryless Quasi-Newton Methods -- 10.8.*Combination of Steepest Descent and Newton's Method -- 10.9. Summary -- 10.10. Exercises -- pt. III Constrained Minimization -- 11. Constrained Minimization Conditions -- 11.1. Constraints -- 11.2. Tangent Plane -- 11.3. First-Order Necessary Conditions (Equality Constraints) -- 11.4. Examples -- 11.5. Second-Order Conditions -- 11.6. Eigenvalues in Tangent Subspace -- 11.7. Sensitivity -- 11.8. Inequality Constraints -- 11.9. Zero-Order Conditions and Lagrangian Relaxation -- 11.10. Summary -- 11.11. Exercises -- 12. Primal Methods -- 12.1. Advantage of Primal Methods -- 12.2. Feasible Direction Methods -- 12.3. Active Set Methods -- 12.4. The Gradient Projection Method -- 12.5. Convergence Rate of the Gradient Projection Method -- 12.6. The Reduced Gradient Method -- 12.7. Convergence Rate of the Reduced Gradient Method -- 12.8.*Variations -- 12.9. Summary -- 12.10. Exercises.
Note continued: 13. Penalty and Barrier Methods -- 13.1. Penalty Methods -- 13.2. Barrier Methods -- 13.3. Properties of Penalty and Barrier Functions -- 13.4. Newton's Method and Penalty Functions -- 13.5. Conjugate Gradients and Penalty Methods -- 13.6. Normalization of Penalty Functions -- 13.7. Penalty Functions and Gradient Projection -- 13.8.*Exact Penalty Functions -- 13.9. Summary -- 13.10. Exercises -- 14. Duality and Dual Methods -- 14.1. Global Duality -- 14.2. Local Duality -- 14.3. Canonical Convergence Rate of Dual Steepest Ascent -- 14.4. Separable Problems and Their Duals -- 14.5. Augmented Lagrangian -- 14.6. The Method of Multipliers -- 14.7. The Alternating Direction Method of Multipliers -- 14.8.̀Cutting Plane Methods -- 14.9. Exercises -- 15. Primal-Dual Methods -- 15.1. The Standard Problem -- 15.2.A Simple Merit Function -- 15.3. Basic Primal-Dual Methods -- 15.4. Modified Newton Methods -- 15.5. Descent Properties -- 15.6.̀Rate of Convergence.
Note continued: 15.7. Primal-Dual Interior Point Methods -- 15.8. Summary -- 15.9. Exercises -- A. Mathematical Review -- A.1. Sets -- A.2. Matrix Notation -- A.3. Spaces -- A.4. Eigenvalues and Quadratic Forms -- A.5. Topological Concepts -- A.6. Functions -- B. Convex Sets -- B.1. Basic Definitions -- B.2. Hyperplanes and Polytopes -- B.3. Separating and Supporting Hyperplanes -- B.4. Extreme Points -- C. Gaussian Elimination -- D. Basic Network Concepts -- D.1. Flows in Networks -- D.2. Tree Procedure -- D.3. Capacitated Networks.
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"ISSN 2214-7934 (electronic)"--Title page verso.

Includes bibliographical references and index.

Machine generated contents note: 1. Introduction -- 1.1. Optimization -- 1.2. Types of Problems -- 1.3. Size of Problems -- 1.4. Iterative Algorithms and Convergence -- pt. I Linear Programming -- 2. Basic Properties of Linear Programs -- 2.1. Introduction -- 2.2. Examples of Linear Programming Problems -- 2.3. Basic Solutions -- 2.4. The Fundamental Theorem of Linear Programming -- 2.5. Relations to Convexity -- 2.6. Exercises -- 3. The Simplex Method -- 3.1. Pivots -- 3.2. Adjacent Extreme Points -- 3.3. Determining a Minimum Feasible Solution -- 3.4.Computational Procedure: Simplex Method -- 3.5. Finding a Basic Feasible Solution -- 3.6. Matrix Form of the Simplex Method -- 3.7. Simplex Method for Transportation Problems -- 3.8. Decomposition -- 3.9. Summary -- 3.10. Exercises -- 4. Duality and Complementarity -- 4.1. Dual Linear Programs -- 4.2. The Duality Theorem -- 4.3. Relations to the Simplex Procedure -- 4.4. Sensitivity and Complementary Slackness -- 4.5. Max Flow -- Min Cut Theorem.

Note continued: 4.6. The Dual Simplex Method -- 4.7.*The Primal-Dual Algorithm -- 4.8. Summary -- 4.9. Exercises -- 5. Interior-Point Methods -- 5.1. Elements of Complexity Theory -- 5.2.*The Simplex Method Is Not Polynomial-Time -- 5.3.*The Ellipsoid Method -- 5.4. The Analytic Center -- 5.5. The Central Path -- 5.6. Solution Strategies -- 5.7. Termination and Initialization -- 5.8. Summary -- 5.9. Exercises -- 6. Conic Linear Programming -- 6.1. Convex Cones -- 6.2. Conic Linear Programming Problem -- 6.3. Farkas' Lemma for Conic Linear Programming -- 6.4. Conic Linear Programming Duality -- 6.5.Complementarity and Solution Rank of SDP -- 6.6. Interior-Point Algorithms for Conic Linear Programming -- 6.7. Summary -- 6.8. Exercises -- pt. II Unconstrained Problems -- 7. Basic Properties of Solutions and Algorithms -- 7.1. First-Order Necessary Conditions -- 7.2. Examples of Unconstrained Problems -- 7.3. Second-Order Conditions -- 7.4. Convex and Concave Functions.

Note continued: 7.5. Minimization and Maximization of Convex Functions -- 7.6.*Zero-Order Conditions -- 7.7. Global Convergence of Descent Algorithms -- 7.8. Speed of Convergence -- 7.9. Summary -- 7.10. Exercises -- 8. Basic Descent Methods -- 8.1. Line Search Algorithms -- 8.2. The Method of Steepest Descent -- 8.3. Applications of the Convergence Theory -- 8.4. Accelerated Steepest Descent -- 8.5. Newton's Method -- 8.6. Coordinate Descent Methods -- 8.7. Summary -- 8.8. Exercises -- 9. Conjugate Direction Methods -- 9.1. Conjugate Directions -- 9.2. Descent Properties of the Conjugate Direction Method -- 9.3. The Conjugate Gradient Method -- 9.4. The C -- G Method as an Optimal Process -- 9.5. The Partial Conjugate Gradient Method -- 9.6. Extension to Nonquadratic Problems -- 9.7.*Parallel Tangents -- 9.8. Exercises -- 10. Quasi-Newton Methods -- 10.1. Modified Newton Method -- 10.2. Construction of the Inverse -- 10.3. Davidon-Fletcher-Powell Method -- 10.4. The Broyden Family.

Note continued: 10.5. Convergence Properties -- 10.6. Scaling -- 10.7. Memoryless Quasi-Newton Methods -- 10.8.*Combination of Steepest Descent and Newton's Method -- 10.9. Summary -- 10.10. Exercises -- pt. III Constrained Minimization -- 11. Constrained Minimization Conditions -- 11.1. Constraints -- 11.2. Tangent Plane -- 11.3. First-Order Necessary Conditions (Equality Constraints) -- 11.4. Examples -- 11.5. Second-Order Conditions -- 11.6. Eigenvalues in Tangent Subspace -- 11.7. Sensitivity -- 11.8. Inequality Constraints -- 11.9. Zero-Order Conditions and Lagrangian Relaxation -- 11.10. Summary -- 11.11. Exercises -- 12. Primal Methods -- 12.1. Advantage of Primal Methods -- 12.2. Feasible Direction Methods -- 12.3. Active Set Methods -- 12.4. The Gradient Projection Method -- 12.5. Convergence Rate of the Gradient Projection Method -- 12.6. The Reduced Gradient Method -- 12.7. Convergence Rate of the Reduced Gradient Method -- 12.8.*Variations -- 12.9. Summary -- 12.10. Exercises.

Note continued: 13. Penalty and Barrier Methods -- 13.1. Penalty Methods -- 13.2. Barrier Methods -- 13.3. Properties of Penalty and Barrier Functions -- 13.4. Newton's Method and Penalty Functions -- 13.5. Conjugate Gradients and Penalty Methods -- 13.6. Normalization of Penalty Functions -- 13.7. Penalty Functions and Gradient Projection -- 13.8.*Exact Penalty Functions -- 13.9. Summary -- 13.10. Exercises -- 14. Duality and Dual Methods -- 14.1. Global Duality -- 14.2. Local Duality -- 14.3. Canonical Convergence Rate of Dual Steepest Ascent -- 14.4. Separable Problems and Their Duals -- 14.5. Augmented Lagrangian -- 14.6. The Method of Multipliers -- 14.7. The Alternating Direction Method of Multipliers -- 14.8.̀Cutting Plane Methods -- 14.9. Exercises -- 15. Primal-Dual Methods -- 15.1. The Standard Problem -- 15.2.A Simple Merit Function -- 15.3. Basic Primal-Dual Methods -- 15.4. Modified Newton Methods -- 15.5. Descent Properties -- 15.6.̀Rate of Convergence.

Note continued: 15.7. Primal-Dual Interior Point Methods -- 15.8. Summary -- 15.9. Exercises -- A. Mathematical Review -- A.1. Sets -- A.2. Matrix Notation -- A.3. Spaces -- A.4. Eigenvalues and Quadratic Forms -- A.5. Topological Concepts -- A.6. Functions -- B. Convex Sets -- B.1. Basic Definitions -- B.2. Hyperplanes and Polytopes -- B.3. Separating and Supporting Hyperplanes -- B.4. Extreme Points -- C. Gaussian Elimination -- D. Basic Network Concepts -- D.1. Flows in Networks -- D.2. Tree Procedure -- D.3. Capacitated Networks.

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