Format:
1 Online-Ressource (390 Seiten)
ISBN:
9783030556624
Series Statement:
International Series in Operations Research and Management Science Ser. v.296
Note:
Description based on publisher supplied metadata and other sources
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Intro -- Preface -- Contents -- Symbols and Abbreviations -- Part I Stochastic Optimization Methods -- 1 Optimal Control Under Stochastic Uncertainty -- 1.1 Stochastic Control Systems -- 1.1.1 Differential and Integral Equations Under Stochastic Uncertainty -- 1.1.1.1 Parametric Representation of the Differential/Integral Equation Under Stochastic Uncertainty -- 1.1.2 Objective Function -- 1.1.2.1 Optimal Control Under Stochastic Uncertainty -- 1.2 Control Laws -- 1.3 Computation of Expectations by Means of Taylor Expansions -- 1.3.1 Complete Taylor Expansion -- 1.3.2 Inner or Partial Taylor Expansion -- 1.4 Taylor Approximation of Control Problems Under Stochastic Uncertainty: General Procedure -- 1.5 Control Problems with Linear and Sublinear Cost Functions -- 1.6 Stochastic Optimal Open-Loop Feedback Control of Tracking Systems -- 1.6.1 Approximation of the Expected Costs: Expansions of 1st Order -- 1.6.2 Approximate Computation of the Fundamental Matrix -- References -- 2 Stochastic Optimization of Regulators -- 2.1 Introduction -- 2.2 Regulator Design Under Stochastic Uncertainty -- 2.3 Optimal Feedback Functions Under Stochastic Uncertainty -- 2.3.1 Quadratic Cost Functions -- 2.3.1.1 Computation of the Expectation by Taylor Expansion -- 2.3.1.2 Approximation of the Expectation of the Total Cost Function -- 2.4 Calculation of the Tracking Error Rates (Sensitivities) -- 2.4.1 Partial Derivative with Respect to pD -- 2.4.2 Partial Derivative with Respect to q0 -- 2.4.3 Partial Derivative with Respect to 0 -- 2.4.4 Partial Derivative with Respect to e0 -- 2.4.4.1 Partial Derivative with Respect to eq -- 2.5 The Approximate Regulator Optimization Problem -- 2.6 Active Structural Control Under Stochastic Uncertainty -- 2.6.1 Example -- References -- 3 Optimal Open-Loop Control of Dynamic Systems Under Stochastic Uncertainty
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3.1 Optimal Control Problems Under Stochastic Uncertainty -- 3.1.1 Computation of the Expectation of the Cost Functions L, G -- 3.2 Solution of the Substitute Control Problem -- 3.3 More General Dynamic Control Systems -- Reference -- 4 Construction of Feedback Control by Means of Homotopy Methods -- References -- 5 Constructions of Limit State Functions -- 5.1 Introduction -- 5.2 Optimization-Based Construction of Limit State Functions -- 5.3 The (Limit) State Function s -- 5.3.1 Characterization of Safe States -- 5.4 Computation of the State Function for Concrete Cases -- 5.4.1 Mechanical Structures Under Stochastic Uncertainty -- 5.4.1.1 Trusses -- 5.4.1.2 Elastic-Plastic Mechanical Structures -- 5.4.2 Linear-Quadratic Problems with Scalar Response Function -- 5.4.3 Approximation of the General Operating Condition -- 5.4.4 Two-Sided Constraints for the Response Functions -- 5.5 Systems/Structures with Parameter-Dependent States -- 5.5.1 Dynamic Control Systems -- 5.5.2 Variational Problems -- 5.5.3 Example to Systems with Control and Variational Problems -- 5.5.3.1 Control Problems -- 5.5.3.2 Variational Problems -- 5.5.3.3 Transformation of Control Problems into Variational Problems -- 5.5.4 Discretization of Control Systems -- 5.5.4.1 Control Problems with Quadratic Objective Functions -- 5.5.4.2 Tracking Problems -- 5.5.4.3 Endpoint Control -- 5.5.4.4 Control Problems with Sublinear Objective Functions -- 5.5.5 Reliability-Based Optimal Control -- 5.5.5.1 Computation of the Probability of Survival -- 5.5.5.2 Tracking Problems with Quadratic Cost Function -- 5.5.5.3 Endpoint Control in Case of Sublinear Cost Function -- 5.5.5.4 Further Lower Bound for psD -- 5.5.5.5 Reliability Computation by Using Copulas -- 5.5.5.6 Approximations of PoS -- References
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Part II Optimization by Stochastic Methods: Foundations and Optimal Control/Acceleration of Random Search Methods (RSM) -- 6 Random Search Procedures for Global Optimization -- 6.1 Introduction -- 6.2 The Convergence of the Basic Random Search Procedure -- 6.2.1 Discrete Optimization Problems -- 6.3 Adaptive Random Search Methods -- 6.3.1 Infinite-Stage Search Processes -- 6.4 Convex Problems -- References -- 7 Controlled Random Search Under Uncertainty -- 7.1 The Controlled (or Adaptive) Random Search Method -- 7.1.1 The Convergence of the Controlled Random Search Procedure -- 7.1.2 A Stopping Rule -- 7.2 Computation of the Conditional Distribution of F Given the Process History: Information Processing -- References -- 8 Controlled Random Search Procedures for Global Optimization -- 8.1 Introduction -- 8.2 Convergence of the Random Search Procedure -- 8.3 Controlled Random Search Methods -- 8.4 Computation of Optimal Controls -- 8.5 Convergence Rates of Controlled Random Search Procedures -- 8.6 Numerical Realizations of Optimal Control Laws -- References -- Part III Random Search Methods (RSM): Convergence and Convergence Rates -- 9 Mathematical Model of Random Search Methods and Elementary Properties -- References -- 10 Special Random Search Methods -- 10.1 R-S-M with Absolutely Continuous Mutation Sequence -- 10.2 Random Direction Methods -- 10.3 Relationships Between Random Direction Methods and Methods with an Absolutely Continuous Mutation Sequence -- References -- 11 Accessibility Theorems -- References -- 12 Convergence Theorems -- 12.1 Convergence of Random Search Methods with an Absolutely Continuous Mutation Sequence -- 12.2 Convergence of Random Direction Methods -- References -- 13 Convergence of Stationary Random Search Methods for Positive Success Probability -- Reference -- 14 Random Search Methods of Convergence Order O(n-α)
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References -- 15 Random Search Methods with a Linear Rate of Convergence -- 15.1 Methods with a Rate of Convergence that Is at Least Linear -- 15.2 Methods with a Rate of Convergence that Is at Most Linear -- 15.3 Linear Convergence for Positive Probability of Success -- References -- 16 Success/Failure-Driven Random Direction Procedures -- References -- 17 Hybrid Methods -- References -- Part IV Optimization Under Stochastic Uncertainty by Random Search Methods (RSM) -- 18 Solving Optimization Problems Under Stochastic Uncertainty by Random Search Methods (RSM) -- 18.1 Introduction -- 18.2 Convergence of the Search Process (Xt) -- 18.3 Estimation of the Minimum, Maximum Entry, Leaving Probability, Resp., αt, rt -- References -- A Properties of the Uniform Distribution on the Unit Sphere -- B Analytical Tools -- C Probabilistic Tools -- Index
Additional Edition:
Erscheint auch als Druck-Ausgabe Marti, Kurt Optimization under Stochastic Uncertainty Cham : Springer International Publishing AG,c2020 ISBN 9783030556617
Language:
English
Subjects:
Economics
,
Mathematics
Keywords:
Stochastische Optimierung
;
Entscheidung bei Unsicherheit
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