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

• Record Nr.: UNINA9910508437503321
• Autore: Luenberger David G. <1937->
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-85450-7
• Edizione: [5th ed.]
• Descrizione fisica: 1 online resource (609 pages)
• Collana: International Series in Operations Research and Management Science ; ; v.228
• Disciplina: 519.72
• Soggetti: Linear programming
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Introduction -- 1.1 Optimization -- 1.2 Types of Problems -- Linear Programming -- Conic Linear Programming -- Unconstrained Problems -- Constrained Problems -- 1.3 Complexity of Problems -- 1.4 Iterative Algorithms and Convergence -- Part I Linear Programming -- 2 Basic Properties of Linear Programs -- 2.1 Introduction -- 2.2 Examples of Linear Programming Problems -- 2.3 Basic Feasible Solutions -- 2.4 The Fundamental Theorem of Linear Programming -- 2.5 Relations to Convex Geometry -- 2.6 Farkas' Lemma and Alternative Systems -- 2.7 Summary -- 2.8 Exercises -- References -- 3 Duality and Complementarity -- 3.1 Dual Linear Programs and Interpretations -- 3.2 The Duality Theorem -- 3.3 Geometric and Economic Interpretations -- Dual Multipliers-Shadow Prices -- 3.4 Sensitivity and Complementary Slackness -- Sensitivity -- Complementary Slackness -- 3.5 Selected Applications of the Duality -- Robust and Distributionally Robust Optimization -- Online Linear Programming -- 3.6 Max Flow-Min Cut Theorem -- Max Flow Augmenting Algorithm -- Max Flow-Min Cut Theorem -- Relation to Duality -- 3.7 Summary -- 3.8 Exercises -- References -- 4 The Simplex Method -- 4.1 Adjacent Basic Feasible Solutions (Extreme Points) -- Nondegeneracy Assumption -- Determination of Vector to Leave Basis -- Conic Combination Interpretations -- 4.2 The Primal Simplex Method -- Determining an Optimal Feasible Solution -- The Simplex Procedure -- ; Finding an Initial Basic Feasible Solution -- 4.3 The Dual Simplex Method -- The Primal-Dual Algorithm -- 4.4 The Simplex Tableau Method -- Decomposition -- 4.5 The Simplex Method for Transportation Problems -- Finding a Basic Feasible Solution -- The Northwest Corner Rule -- Basis Triangularity -- The Transportation Simplex Method -- Simplex Multipliers -- Cycle of Change.; The Transportation Simplex Algorithm -- 4.6 Efficiency Analysis of the Simplex Method -- 4.7 Summary -- 4.8 Exercises -- References -- 5 Interior-Point Methods -- 5.1 Elements of Complexity Theory -- 5.2 The Simplex Method Is Not Polynomial-Time -- 5.3 The Ellipsoid Method -- Cutting Plane and New Containing Ellipsoid -- Convergence -- Ellipsoid Method for Usual Form of LP -- 5.4 The Analytic Center -- Cutting Plane and Analytic Volume of Reduction -- 5.5 The Central Path -- Dual Central Path -- Primal-Dual Central Path -- Duality Gap -- 5.6 Solution Strategies -- Primal Barrier Method -- Primal-Dual Path-Following -- Primal-Dual Potential Reduction Algorithm -- Iteration Complexity -- 5.7 Termination and Initialization -- Termination -- Initialization -- The HSD Algorithm -- 5.8 Summary -- 5.9 Exercises -- References -- 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 -- Null-Space Rank Reduction -- Gaussian Projection Rank Reduction -- Randomized Binary Rank Reduction -- Objective-Guide Rank Reduction -- 6.6 Interior-Point Algorithms for Conic Linear Programming -- Initialization: The HSD Algorithm -- 6.7 Summary -- 6.8 Exercises -- References -- Part II Unconstrained Problems -- 7 Basic Properties of Solutions and Algorithms -- 7.1 First-Order Necessary Conditions -- Feasible and Descent Directions -- 7.2 Examples of Unconstrained Problems -- 7.3 Second-Order Conditions -- Sufficient Conditions for a Relative Minimum -- 7.4 Convex and Concave Functions -- Properties of Convex Functions -- Properties of Differentiable Convex Functions -- 7.5 Minimization and Maximization of Convex Functions -- 7.6 Global Convergence of Descent Algorithms -- Iterative Algorithms -- Descent.; Closed Mappings -- Global Convergence Theorem -- Spacer Steps -- 7.7 Speed of Convergence -- Order of Convergence -- Linear Convergence -- Arithmetic Convergence -- Average Rates -- Convergence of Vectors -- Complexity -- 7.8 Summary -- 7.9 Exercises -- References -- 8 Basic Descent Methods -- 8.1 Line Search Algorithms -- 0th-Order Method: Golden Section Search and Curve Fitting -- Search by Golden Section -- Quadratic Fit -- 1st-Order Method: Bisection and Curve Fitting Methods -- The Bisection Method -- Quadratic Fit: Method of False Position -- Cubic Fit -- 2nd-Order Method: Newton's Method -- Global Convergence of Curve Fitting -- Closedness of Line Search Algorithms -- Inaccurate Line Search -- Armijo's Rule -- 8.2 The Method of Steepest Descent: First-Order -- The Method -- Global Convergence and Convergence Speed -- The Quadratic Case -- The Nonquadratic Case -- 8.3 Applications of the Convergence Theory and Preconditioning -- Scaling as Preconditioning -- 8.4 Accelerated Steepest Descent -- The Heavy Ball Method -- The Method of False Position -- 8.5 Multiplicative Steepest Descent -- Affine-Scaling Method -- Mirror-Descent Method -- 8.6 Newton's Method: Second-Order -- Order Two Convergence -- Modifications -- Newton's Method and Logarithms -- Self-concordant Functions -- 8.7 Sequential Quadratic Optimization Methods -- Trust Region Method -- A Homotopy or Path-Following Method -- 8.8 Coordinate and Stochastic Gradient Descent Methods -- Global Convergence -- Local ; Convergence Rate -- Convergence Speed of a Randomized Coordinate Descent Method -- Stochastic Gradient Descent (SGD) Method -- 8.9 Summary -- 8.10 Exercises -- References -- 9 Conjugate Direction Methods -- 9.1 Conjugate Directions -- 9.2 Descent Properties of the Conjugate Direction Method -- 9.3 The Conjugate Gradient Method -- Conjugate Gradient Algorithm.; Verification of the Algorithm -- 9.4 The C-G Method as an Optimal Process -- Bounds on Convergence -- 9.5 The Partial Conjugate Gradient Method -- 9.6 Extension to Nonquadratic Problems -- Quadratic Approximation -- Line Search Methods -- Convergence -- Preconditioning and Partial Methods -- 9.7 Parallel Tangents -- 9.8 Exercises -- References -- 10 Quasi-Newton Methods -- 10.1 Modified Newton Method -- Other Modified Newton's Methods -- 10.2 Construction of the Inverse -- Rank One Correction -- 10.3 Davidon-Fletcher-Powell Method -- Positive Definiteness -- Finite Step Convergence -- 10.4 The Broyden Family -- Partial Quasi-Newton Methods -- 10.5 Convergence Properties -- Global Convergence -- Local Convergence -- 10.6 Scaling -- Improvement of Eigenvalue Ratio -- Scale Factors -- A Self-Scaling Quasi-Newton Algorithm -- 10.7 Memoryless Quasi-Newton Methods -- Scaling and Preconditioning -- 10.8 Combination of Steepest Descent and Newton's Method -- 10.9 Summary -- 10.10 Exercises -- References -- Part III Constrained Optimization -- 11 Constrained Optimization Conditions -- 11.1 Constraints and Tangent Plane -- Tangent Plane -- 11.2 First-Order Necessary Conditions (Equality Constraints) -- Sensitivity -- 11.3 Equality Constrained Optimization Examples -- Large-Scale Applications -- 11.4 Second-Order Conditions (Equality Constraints) -- Eigenvalues in Tangent Subspace -- Projected Hessians -- 11.5 Inequality Constraints -- First-Order Necessary Conditions -- The Lagrangian and First-Order Conditions -- Second-Order Conditions -- Sensitivity -- 11.6 Mix-Constrained Optimization Examples -- 11.7 Lagrangian Duality and Zero-Order Conditions -- 11.8 Rules for Constructing the Lagrangian Dual Explicitly -- 11.9 Summary -- 11.10 Exercises -- References -- 12 Primal Methods -- 12.1 Infeasible Direction and the Steepest Descent Projection Method.; 12.2 Feasible Direction Methods: Sequential Linear Programming -- 12.3 The Gradient Projection Method -- Linear Constraints -- Nonlinear Constraints -- 12.4 Convergence Rate of the Gradient Projection Method -- Geodesic Descent -- Geodesics -- Lagrangian and Geodesics -- Rate of Convergence -- Problems with Inequalities -- 12.5 The Reduced Gradient Method -- Linear Constraints -- Global Convergence -- Nonlinear Constraints -- 12.6 Convergence Rate of the Reduced Gradient Method -- 12.7 Sequential Quadratic Optimization Methods -- 12.8 Active Set Methods -- Changes in Working Set -- 12.9 Summary -- 12.10 Exercises -- References -- 13 Penalty and Barrier Methods -- 13.1 Penalty Methods -- The Method -- Convergence -- 13.2 Barrier Methods -- The Method -- Convergence -- 13.3 Lagrange Multipliers in Penalty and Barrier Methods -- Lagrange Multipliers in the Penalty Method -- The Hessian Matrix -- Lagrange Multipliers in the Barrier Method -- 13.4 Newton's Method for the Logarithmic Barrier Optimization -- The KKT Condition System of the Logarithmic Barrier Function -- The KKT System of a ``Shifted'' Barrier -- The Interior Ellipsoidal-Trust Region Method with Barrier -- 13.5 Newton's Method for Equality Constrained Optimization -- Normalization of Penalty Functions -- Inequalities -- 13.6 Conjugate Gradients and Penalty Methods -- 13.7 Penalty Functions and Gradient Projection -- Underlying Concept -- Implementing the First Step -- Inequality Constraints -- 13.8 Summary -- 13.9 Exercises -- References -- 14 ; Local Duality and Dual Methods -- 14.1 Local Duality and the Lagrangian Method -- Inequality Constraints -- Partial Duality -- The Lagrangian Method: Dual Steepest Ascent -- Preconditioning or Scaling -- 14.2 Separable Problems and Their Duals -- Decomposition -- 14.3 The Augmented Lagrangian and Interpretation -- The Penalty Viewpoint.; Geometric Interpretation.

Around the World in 80 Ways : Exploring Our Planet Through Maps and Data / / by Stephen Webb

• Record Nr.: UNINA9910678250303321
• Autore: Webb Stephen
• Pubbl/distr/stampa: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2023
• ISBN: 9783031024405
• Edizione: [1st ed. 2023.]
• Descrizione fisica: 1 online resource (225 pages)
• Disciplina: 526.0285; 526.0285425
• Soggetti: Geography; Cartography; Geophysics; Sociology
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Mapping the World -- The World Itself -- The World of Countries -- The World Perturbed by People -- The World of the Built Environment -- The World of Thoughts, Ideas, and Beliefs -- The World of Wellbeing -- The World of Sport and Leisure -- The World of Economics -- Afterword.
• Sommario/riassunto: Around the World in 80 Ways offers a (sometimes opinionated) discussion of 80 data-driven maps of our planet. Taken together, the maps tell a story about the physical world; about the impact our species is having on the world; and about how people live in the world – or at least how we lived immediately before the emergence of Covid-; 19. The maps lie. All maps lie. But the origins of the deceptions are explained, the data sources are signposted and referenced, and the readers are shown how to create their own maps using freely available software. The reader is thus armed with the tools needed to explore local, national or world data – on topics ranging from science to society; environment to entertainment; wealth to wellbeing – a valuable skill in an age when certain politicians are happy to refer to “alternative facts” and media outlets deliver data visualizations that sometimes mislead as much as inform.