Optimization under stochastic uncertainty: methods, control and random search methods
Gespeichert in:
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| Format: | Buch |
| Sprache: | English |
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Cham
Springer
2020
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| Schriftenreihe: | International Series in Operations Research & Management Science
296 |
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| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xiv, 393 Seiten Illustrationen |
| ISBN: | 9783030556617 |
| ISSN: | 0884-8289 |
Internformat
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| 650 | 4 | |a Operations research | |
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Datensatz im Suchindex
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| adam_text | Contents Part I 1 2 Stochastic Optimization Methods Optimal Control Under Stochastic Uncertainty ................................. 1.1 Stochastic Control Systems........................................................... 1.1.1 Differential and Integral Equations Under Stochastic Uncertainty........................................................................ 1.1.2 Objective Function............................................................. 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................................................................................................. 3 3 27 30 31 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.................................................. 2.4 Calculation of the Tracking Error Rates (Sensitivities)................. 2.4.1 Partial Derivative with Respect to Δρο............................. 2.4.2 Partial Derivative with Respect to Ac/o ............................. 2.4.3 Partial Derivative with Respect to Αήο............................. 2.4.4 Partial Derivative with Respect to eo................................. 2.5 The Approximate Regulator Optimization Problem...................... 33 33 35 40 42 46 47 48 48 49 50 4 9 13 16 17 19 21 23 25 ix
x Contents 2.6 Active Structural Control Under Stochastic Uncertainty................ 2.6.1 Example................................................................................. References....................................................................................................... 3 Optimal Open-Loop Control of Dynamic Systems Under Stochastic Uncertainty................................................................................ 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........................................................................................................ 52 53 58 61 62 62 66 68 69 4 Construction of Feedback Control by Means of HomotopyMethods References....................................................................................................... 71 77 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........ 5.4.2 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................................................................................. 5.5.4 Discretization of Control Systems ...................................... 5.5.5 Reliability-Based Optimal Control...................................... References....................................................................................................... 79 79 80 84 89 91 91 Part II 6 96 97 97 98 98 99 100 102 110 118 Optimization by Stochastic Methods: Foundations and Optimal Control/Acceleration of Random Search Methods (RSM) Random Search Procedures for Global Optimization.......................... 123 6.1 Introduction......................................................................................... 123 6.2 The Convergence of the Basic Random Search Procedure............ 125 6.2.1 Discrete Optimization Problems.......................................... 128 6.3
Adaptive Random Search Methods................................................... 129 6.3.1 Infinite-Stage Search Processes........................................... 134 6.4 Convex Problems................................................................................ 135 References....................................................................................................... 137
xi Contents 7 Controlled Random Search Under Uncertainty..................................... 139 7.1 The Controlled (or Adaptive) Random Search Method................. 139 7.1.1 The Convergence of the Controlled Random Search Procedure................................................................................ 143 7.1.2 A Stopping Rule ................................................................... 145 7.2 Computation of the Conditional Distribution of F Given the Process History: Information Processing......................................... 146 References....................................................................................................... 150 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 9 10 151 151 153 155 156 160 161 167 Random Search Methods (RSM): Convergence and Convergence Rates Mathematical Model of Random Search Methods and Elementary
Properties................................................................................ References....................................................................................................... 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....................................................................................................... 171 177 179 179 180 180 185 11 Accessibility Theorems................................................................................ 187 References....................................................................................................... 194 12 Convergence Theorems.............................................................................. 195 12.1 Convergence of Random Search Methods with an Absolutely Continuous Mutation Sequence..................................... 195 12.2 Convergence of Random Direction Methods.................................. 198 References....................................................................................................... 205 13 Convergence of Stationary Random Search Methods for Positive Success Probability......................................................................
Reference........................................................................................................ 14 207 211 Random Search Methods of Convergence Order 0(n~a).................. 213 References....................................................................................................... 232
xii Contents 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....................... 279 References....................................................................................................... 325 17 Hybrid Methods........................................................................................... 327 References....................................................................................................... 336 Part IV 18 233 233 256 272 278 Optimization Under Stochastic Uncertainty by Random Search Methods (RSM) 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., at,rt .................................................................... References....................................................................................................... 345 349 A Properties of the Uniform
Distribution on the Unit Sphere................. 351 В Analytical Tools............................................................................................. 357 C Probabilistic Tools........................................................................................ Index 341 341 342 381 391
International Series in Operations Research Management Science Kurt Marti Optimization Under Stochastic Uncertainty Methods, Control and Random Search Methods This book examines application and methods to incorporating stochastic parameter variations into the optimization process to decrease expense in corrective measures. Basic tvpes ot deterministic substitute problems occurring mostly in practice involve i) minimization ot the expected priman՜ costs subiect to expected recourse cost constraints (rehabilite՛ constraints) and remaining deterministic constraints, e.g. box constraints, as well as ii ) minimization ot the expected total costs (costs ot construction, design, recourse costs, etc.) subiect o the remaining deterministic constraints. Alter an introduction into the theon ot dynamic control systems with random parameters, the maior control kms are described, as open loop control, closed-loop, teedback control and open loop feedback control, used tor iterative construction of feedback controls, for approximate solution of optimization and control problems with random parameters and involving expected cost doss-type objective, constraint functions. la !or expansion procedures, and Ilomotopv methods are considered, f camples and applications to stochastic optimization ot regulators arc given. Moreover, for lehahdin based analysis and optimal design problems, corresponding opt!ini֊íitio4 hated limit state functions are constructed. Because ot the complexity ot concrete optimization control problems and their lack ot the mathematic al regularity as required of
Mathematical Programming (MF) techniques, othei optimization techniques, like random search methods íRSM ! became increasingly important. Basic results on the convergence and convergence rates ot random search methods arc presented. .Moreover, tor the improvement ot the - sometimes verv low convergence rate of Rs.M. search methods based on optimal stochastic decision processes are presented. In ordet to improv e the convergence behavior ot RS.M. the random search procedure՛ is embedded into a stochastic decision process tor an optimal control ot the probabihtv distributions of the search variates (mutation random variables!.
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| adam_txt |
Contents Part I 1 2 Stochastic Optimization Methods Optimal Control Under Stochastic Uncertainty . 1.1 Stochastic Control Systems. 1.1.1 Differential and Integral Equations Under Stochastic Uncertainty. 1.1.2 Objective Function. 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. 3 3 27 30 31 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. 2.4 Calculation of the Tracking Error Rates (Sensitivities). 2.4.1 Partial Derivative with Respect to Δρο. 2.4.2 Partial Derivative with Respect to Ac/o . 2.4.3 Partial Derivative with Respect to Αήο. 2.4.4 Partial Derivative with Respect to eo. 2.5 The Approximate Regulator Optimization Problem. 33 33 35 40 42 46 47 48 48 49 50 4 9 13 16 17 19 21 23 25 ix
x Contents 2.6 Active Structural Control Under Stochastic Uncertainty. 2.6.1 Example. References. 3 Optimal Open-Loop Control of Dynamic Systems Under Stochastic Uncertainty. 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. 52 53 58 61 62 62 66 68 69 4 Construction of Feedback Control by Means of HomotopyMethods References. 71 77 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. 5.4.2 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. 5.5.4 Discretization of Control Systems . 5.5.5 Reliability-Based Optimal Control. References. 79 79 80 84 89 91 91 Part II 6 96 97 97 98 98 99 100 102 110 118 Optimization by Stochastic Methods: Foundations and Optimal Control/Acceleration of Random Search Methods (RSM) Random Search Procedures for Global Optimization. 123 6.1 Introduction. 123 6.2 The Convergence of the Basic Random Search Procedure. 125 6.2.1 Discrete Optimization Problems. 128 6.3
Adaptive Random Search Methods. 129 6.3.1 Infinite-Stage Search Processes. 134 6.4 Convex Problems. 135 References. 137
xi Contents 7 Controlled Random Search Under Uncertainty. 139 7.1 The Controlled (or Adaptive) Random Search Method. 139 7.1.1 The Convergence of the Controlled Random Search Procedure. 143 7.1.2 A Stopping Rule . 145 7.2 Computation of the Conditional Distribution of F Given the Process History: Information Processing. 146 References. 150 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 9 10 151 151 153 155 156 160 161 167 Random Search Methods (RSM): Convergence and Convergence Rates Mathematical Model of Random Search Methods and Elementary
Properties. References. 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. 171 177 179 179 180 180 185 11 Accessibility Theorems. 187 References. 194 12 Convergence Theorems. 195 12.1 Convergence of Random Search Methods with an Absolutely Continuous Mutation Sequence. 195 12.2 Convergence of Random Direction Methods. 198 References. 205 13 Convergence of Stationary Random Search Methods for Positive Success Probability.
Reference. 14 207 211 Random Search Methods of Convergence Order 0(n~a). 213 References. 232
xii Contents 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. 279 References. 325 17 Hybrid Methods. 327 References. 336 Part IV 18 233 233 256 272 278 Optimization Under Stochastic Uncertainty by Random Search Methods (RSM) 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., at,rt . References. 345 349 A Properties of the Uniform
Distribution on the Unit Sphere. 351 В Analytical Tools. 357 C Probabilistic Tools. Index 341 341 342 381 391
International Series in Operations Research Management Science Kurt Marti Optimization Under Stochastic Uncertainty Methods, Control and Random Search Methods This book examines application and methods to incorporating stochastic parameter variations into the optimization process to decrease expense in corrective measures. Basic tvpes ot deterministic substitute problems occurring mostly in practice involve i) minimization ot the expected priman՜ costs subiect to expected recourse cost constraints (rehabilite՛ constraints) and remaining deterministic constraints, e.g. box constraints, as well as ii ) minimization ot the expected total costs (costs ot construction, design, recourse costs, etc.) subiect 'o the remaining deterministic constraints. Alter an introduction into the theon ot dynamic control systems with random parameters, the maior control kms are described, as open loop control, closed-loop, teedback control and open loop feedback control, used tor iterative construction of feedback controls, for approximate solution of optimization and control problems with random parameters and involving expected cost doss-type objective, constraint functions. la\!or expansion procedures, and Ilomotopv methods are considered, f camples and applications to stochastic optimization ot regulators arc given. Moreover, for lehahdin based analysis and optimal design problems, corresponding opt!ini֊íitio4 hated limit state functions are constructed. Because ot the complexity ot concrete optimization control problems and their lack ot the mathematic al regularity as required of
Mathematical Programming (MF) techniques, othei optimization techniques, like random search methods íRSM ! became increasingly important. Basic results on the convergence and convergence rates ot random search methods arc presented. .Moreover, tor the improvement ot the - sometimes verv low convergence rate of Rs.M. search methods based on optimal stochastic decision processes are presented. In ordet to improv e the convergence behavior ot RS.M. the random search procedure՛ is embedded into a stochastic decision process tor an optimal control ot the probabihtv distributions of the search variates (mutation random variables!. |
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| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
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| index_date | 2024-07-03T16:28:44Z |
| indexdate | 2024-07-10T09:03:09Z |
| institution | BVB |
| isbn | 9783030556617 |
| issn | 0884-8289 |
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| series | International Series in Operations Research & Management Science |
| series2 | International Series in Operations Research & Management Science |
| spelling | Marti, Kurt 1943- Verfasser (DE-588)115845690 aut Optimization under stochastic uncertainty methods, control and random search methods Kurt Marti Cham Springer 2020 xiv, 393 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier International Series in Operations Research & Management Science 296 0884-8289 Operations Research/Decision Theory Probability Theory and Stochastic Processes Computer Science, general Operations research Decision making Probabilities Computer science Stochastische Optimierung (DE-588)4057625-5 gnd rswk-swf Entscheidung bei Unsicherheit (DE-588)4070864-0 gnd rswk-swf Stochastische Optimierung (DE-588)4057625-5 s Entscheidung bei Unsicherheit (DE-588)4070864-0 s DE-604 Erscheint auch als Online-Ausgabe 978-3-030-55662-4 International Series in Operations Research & Management Science 296 (DE-604)BV035421305 296 Digitalisierung UB Augsburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032525478&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Augsburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032525478&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Marti, Kurt 1943- Optimization under stochastic uncertainty methods, control and random search methods International Series in Operations Research & Management Science Operations Research/Decision Theory Probability Theory and Stochastic Processes Computer Science, general Operations research Decision making Probabilities Computer science Stochastische Optimierung (DE-588)4057625-5 gnd Entscheidung bei Unsicherheit (DE-588)4070864-0 gnd |
| subject_GND | (DE-588)4057625-5 (DE-588)4070864-0 |
| title | Optimization under stochastic uncertainty methods, control and random search methods |
| title_auth | Optimization under stochastic uncertainty methods, control and random search methods |
| title_exact_search | Optimization under stochastic uncertainty methods, control and random search methods |
| title_exact_search_txtP | Optimization under stochastic uncertainty methods, control and random search methods |
| title_full | Optimization under stochastic uncertainty methods, control and random search methods Kurt Marti |
| title_fullStr | Optimization under stochastic uncertainty methods, control and random search methods Kurt Marti |
| title_full_unstemmed | Optimization under stochastic uncertainty methods, control and random search methods Kurt Marti |
| title_short | Optimization under stochastic uncertainty |
| title_sort | optimization under stochastic uncertainty methods control and random search methods |
| title_sub | methods, control and random search methods |
| topic | Operations Research/Decision Theory Probability Theory and Stochastic Processes Computer Science, general Operations research Decision making Probabilities Computer science Stochastische Optimierung (DE-588)4057625-5 gnd Entscheidung bei Unsicherheit (DE-588)4070864-0 gnd |
| topic_facet | Operations Research/Decision Theory Probability Theory and Stochastic Processes Computer Science, general Operations research Decision making Probabilities Computer science Stochastische Optimierung Entscheidung bei Unsicherheit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032525478&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032525478&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035421305 |
| work_keys_str_mv | AT martikurt optimizationunderstochasticuncertaintymethodscontrolandrandomsearchmethods |