Introduction to continuous optimization / / Roman A. Polyak

• Record Nr.: UNINA9910484600303321
• Autore: Polyak Roman A.
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-68713-9
• Descrizione fisica: 1 online resource (552 pages)
• Collana: Springer optimization and its applications ; ; 172
• Disciplina: 519.3
• Soggetti: Mathematical optimization; Optimització matemàtica; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Introduction -- 2 Elements of Calculus and Convex Analysis -- 2.0 Introduction -- 2.1 Elements of Calculus -- 2.1.1 Differentiation of Scalar Functions -- 2.1.2 Differentiation of Vector Functions -- 2.1.3 Second Derivatives -- 2.1.4 Convex Functions in Rn -- 2.1.5 Strictly and Strongly Convex Functions in Rn -- 2.2 Convex Sets -- 2.2.1 Open and Closed Sets -- 2.2.2 Convex Sets -- 2.2.3 Affine Sets -- 2.2.4 Cones -- 2.2.5 Recession Cones -- 2.2.6 Polyhedrons and Polytopes -- 2.3 Closed Convex Functions -- 2.3.1 Operations on Closed Convex Functions -- 2.3.2 Projection on a Closed Convex Set -- 2.3.3 Separation Theorems -- 2.3.4 Some Properties of Convex Functions -- 2.3.4.1 Continuity of Convex Functions -- 2.3.4.2 Differentiability of Convex Functions -- 2.3.5 Subgradients -- 2.3.6 Support Functions -- 2.4 The Legendre-Fenchel Transformation -- 2.4.1 Basic LF Transformation Property -- 2.4.2 The LF Identity and the LF Invariant -- 3 Few Topics in Unconstrained Optimization -- 3.0 Introduction -- 3.1 Optimality Conditions -- 3.1.1 First-Order Necessary Condition -- 3.1.2 Second-Order Necessary Condition -- 3.1.3 Second-Order Sufficient Condition -- 3.2 Nondifferentiable Unconstrained Minimization -- 3.2.1 Subgradient Method -- 3.3 Gradient Methods -- 3.3.1 Gradient Method -- 3.3.2 Fast Gradient Method -- 3.3.3 Gradient Method for Strongly ; Convex Functions -- 3.4 Method Regularization -- 3.5 Proximal Point Method -- 3.6 Newton's Method and Regularized Newton Method -- 3.6.1 Introduction -- 3.6.2 Newton's Method -- 3.6.3 Local Quadratic Convergence of Newton's Method -- 3.6.4 Damped Newton Method -- 3.6.5 Global Convergence of DNM and Its Complexity -- 3.6.6 Regularized Newton Method -- 3.6.7 Local Quadratic Convergence Rate of RNM -- 3.6.8 Damped Regularized Newton Method -- 3.6.9 The Complexity of DRNM.; 3.6.10 Newton's Method as an Affine Invariant -- 4 Optimization with Equality Constraints -- 4.0 Introduction -- 4.1 Lagrangian and First-Order Optimality Condition -- 4.2 Second-Order Necessary and Sufficient Optimality Condition -- 4.3 Optimality Condition for Constrained Optimization Problems with Both Inequality Constraints and Equations -- 4.4 Duality for Equality-Constrained Optimization -- 4.5 Courant's Penalty Method as Tikhonov's Regularization for the Dual Problem -- 4.6 Gradient Methods for ECO -- 4.7 Newton's Method for Nonlinear System of Equations -- 4.8 Newton's Method for ECO -- 4.9 Augmented Lagrangian -- 4.10 The Multipliers Method and the Dual Quadratic Prox -- 4.11 Primal-Dual AL Method for ECO -- 5 Basics in Linear and Convex Optimization -- 5.0 Introduction -- 5.1 Linear Programming -- 5.1.1 Primal and Dual LP Problems -- 5.1.2 Optimality Condition for LP Problem -- 5.1.3 Farkas Lemma -- 5.2 The Karush-Kuhn-Tucker's Theorem -- 5.3 The KKT's Theorem for Convex Optimization with Linear Constraints -- 5.4 Duality in Convex Optimization -- 5.5 Wolfe's Duality -- 5.6 LP Duality -- 5.7 Some Structural LP Properties -- 5.8 Simplex Method -- 5.9 Interior Point Methods -- 5.9.1 Newton Log- Barrier Method for LP -- 5.9.2 Primal-Dual Interior Point Method -- 5.9.3 Affine Scaling Method -- 5.10 SUMT as Dual Interior Regularization -- 5.10.1 Log-Barrier Method and Its Dual Equivalent -- 5.10.2 Hyperbolic Barrier as Dual Parabolic Regularization -- 5.10.3 Exponential Penalty as Dual Regularization with Shannon's Entropy Function -- 5.10.4 Log-Sigmoid Method as Dual Regularization with Fermi-Dirac's Entropy Function -- 5.10.5 Interior Distance Functions -- 5.11 Primal-Dual IPM for Convex Optimization -- 5.12 Gradient Projection Method -- 5.12.1 Convergence of the GP Method -- 5.12.2 Fast GP Method.; 5.12.3 GP Method for Strongly Convex Function -- 5.13 Quadratic Programming -- 5.13.1 Dual GP Method for QP -- 5.13.2 Dual Fast Gradient Projection Method -- 5.14 Quadratic Programming Problems with Quadratic Constraints -- 5.15 Conditional Gradient Method -- 5.16 Primal-Dual Feasible Direction Method -- 6 Self-Concordant Functions and IPM Complexity -- 6.0 Introduction -- 6.1 LF Invariant and SC Functions -- 6.2 Basic Properties of SC Functions -- 6.3 Newton's Method for Minimization of SC Functions -- 6.4 SC Barrier -- 6.5 Path-Following Method -- 6.6 Applications of ν-SC Barrier. IPM Complexity for LP and QP -- 6.6.1 Linear and Quadratic Optimization -- 6.6.2 The Lorentz Cone -- 6.6.3 Semidefinite Optimization -- 6.7 Primal-Dual Predictor-Corrector for LP -- 7 Nonlinear Rescaling: Theory and Methods -- 7.0 Introduction -- 7.1 Nonlinear Rescaling -- 7.1.1 Preliminaries -- 7.1.2 Constraints Transformation and Lagrangian for the Equivalent Problem: Local Properties -- 7.1.3 Primal Transformations and Dual Kernels -- 7.1.4 NR Method and Dual Prox with -Divergence Distance -- 7.1.5 Q-Linear Convergence Rate -- 7.1.6 Stopping Criteria -- 7.1.7 Newton NR Method and ``Hot'' Start Phenomenon -- 7.2 NR with ``Dynamic'' Scaling Parameters -- 7.2.0 Introduction -- 7.2.1 Nonlinear Rescaling as Interior Quadratic Prox -- 7.2.2 Convergence of the NR Method -- 7.2.3 Rate of Convergence -- 7.2.4 Nonlinear Rescaling for LP -- 7.3 Primal-Dual NR Method for ; Convex Optimization -- 7.3.0 Introduction -- 7.3.1 Local Convergence of the PDNR -- 7.3.2 Global Convergence of the PDNR -- 7.4 Nonlinear Rescaling and Augmented Lagrangian -- 7.4.0 Introduction -- 7.4.1 Problem Formulation and Basic Assumptions -- 7.4.2 Lagrangian for the Equivalent Problem -- 7.4.3 Multipliers Method -- 7.4.4 NRAL and the Dual Prox -- 8 Realizations of the NR Principle -- 8.0 Introduction.; 8.1 Modified Barrier Functions -- 8.1.0 Introduction -- 8.1.1 Logarithmic MBF -- 8.1.2 Convergence of the Logarithmic MBF Method -- 8.1.3 Convergence Rate -- 8.1.4 MBF and Duality Issues -- 8.2 Exterior Distance Functions -- 8.2.0 Introduction -- 8.2.1 Exterior Distance -- 8.2.2 Exterior Point Method: Convergence and Convergence Rate -- 8.2.3 Stopping Criteria -- 8.2.4 Modified Interior Distance Functions -- 8.2.5 Local MIDF Properties -- 8.2.6 Modified Center Method -- 8.2.7 Basic Theorem -- 8.3 Nonlinear Rescaling vs. Smoothing Technique -- 8.3.0 Introduction -- 8.3.1 Log-Sigmoid Transformation and Its Modification -- 8.3.2 Equivalent Problem and LS Lagrangian -- 8.3.3 LS Multipliers Method as Interior Prox with Fermi-Dirac Entropy Distance -- 8.3.4 Convergence of the LS Multipliers Method -- 8.3.5 The Upper Bound for the Number of Steps -- 8.3.6 Asymptotic Convergence Rate -- 8.3.7 Generalization and Extension -- 8.3.8 LS Multipliers Method for Linear Programming -- 9 Lagrangian Transformation and Interior Ellipsoid Methods -- 9.0 Introduction -- 9.1 Lagrangian Transformation -- 9.2 Bregman's Distance -- 9.3 Primal LT and Dual Interior Quadratic Prox -- 9.4 Convergence Analysis -- 9.5 LT with Truncated MBF and Interior Ellipsoid Method -- 9.6 Lagrangian Transformation and Dual Affine Scaling Method for LP -- 10 Finding Nonlinear Equilibrium -- 10.0 Introduction -- 10.1 General NE Problem and the Equivalent VI -- 10.2 Problems Leading to NE -- 10.2.1 Convex Optimization -- 10.2.2 Finding a Saddle Point -- 10.2.3 Matrix Game -- 10.2.4 J. Nash Equilibrium in n-Person Concave Game -- 10.2.5 Walras-Wald Equilibrium -- 10.3 NE for Optimal Resource Allocation -- 10.3.1 Introduction -- 10.3.2 Problem Formulation -- 10.3.3 NE as a VI -- 10.3.4 Existence and Uniqueness of the NE -- 10.4 Nonlinear Input-Output Equilibrium -- 10.4.1 Introduction.; 10.4.2 Preliminaries -- 10.4.3 Problem Formulation -- 10.4.4 NIOE as a VI -- 10.4.5 Existence and Uniqueness of the NIOE -- 10.5 Finding NE for Optimal Resource Allocation -- 10.5.1 Introduction -- 10.5.2 Basic Assumptions -- 10.5.3 Pseudo-gradient Projection Method -- 10.5.4 Extra Pseudo-gradient Method for Finding NE -- 10.5.5 Convergence Rate -- 10.5.6 Bound for the Lipschitz Constant -- 10.5.7 Finding NE as a Pricing Mechanizm -- 10.6 Finding Nonlinear Input-Output Equilibrium -- 10.6.1 Introduction -- 10.6.2 Basic Assumptions -- 10.6.3 PGP Method for Finding NIOE -- 10.6.4 EPG Method for Finding NIOE -- 10.6.5 Convergence Rate and Complexity of the EPG Method -- 10.6.6 Lipschitz Constant -- 10.7 Finding J. Nash Equilibrium in n-Person Concave Game -- 10.7.1 Projection Onto Probability Simplex -- 10.7.2 Algorithm for Projection onto PS -- 11 Applications and Numerical Results -- 11.0 Introduction -- 11.1 Truss Topology Design -- 11.2 Intensity-Modulated Radiation Therapy Planning -- 11.3 QP and Its Applications -- 11.3.1 Non-negative Least Squares -- 11.3.2 Support Vector Machines -- 11.3.3 Fast Gradient Projection for Dual QP. Numerical Results -- 11.4 Finding Nonlinear Equilibrium -- 11.5 The ``Hot'' Start Phenomenon -- Concluding Remarks -- Appendix -- References.

Historical dictionary of the Reformation and Counter-Reformation / / Hans J. Hillerbrand

• Record Nr.: UNINA9910962851803321
• Autore: Hillerbrand Hans J. <1931-2020, >
• Pubbl/distr/stampa: Chicago, : Fitzroy Dearborn Publishers, 2000; Chicago ; ; London : , : Fitzroy Dearborn, , 2000
• ISBN: 9781136596773; 1136596771; 9780203058688; 0203058682; 9781299459250; 1299459250; 9781136596704; 1136596704
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (300 p.)
• Disciplina: 270.6/03
• Soggetti: Reformation; Counter-Reformation
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Cover; Title Page; Copyright Page; Table of Contents; Editor's Foreword; Preface; Introduction; Chronology; The Dictionary; Bibliography; About the Author
• Sommario/riassunto: The Reformation of the 16th century has always been seen as one of the pivotal events in European history. Lord Acton, the famous 19th-century British historian, compared the importance of Martin Luther's speech at the diet at Worms in 1521 with Napoleon's defeat at the Battle of Waterloo in 1813. Lord Acton's may or may not be an extravagant claim, but it is certainly true that the events of the 16th and 17th centuries, now called the Reformation and Counter-Reformation, changed forever the religious and political history of the West.The Historical Dictionary of the Reformation an