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Stochastic Optimization Methods: Applications in Engineering and Operations Research 3rd ed. 2015 [Kõva köide]

  • Formaat: Hardback, 368 pages, kõrgus x laius: 235x155 mm, kaal: 7096 g, 23 Illustrations, black and white; XXIV, 368 p. 23 illus., 1 Hardback
  • Ilmumisaeg: 23-Mar-2015
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3662462133
  • ISBN-13: 9783662462133
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Stochastic Optimization Methods: Applications in Engineering and Operations Research 3rd ed. 2015Suurem pilt
  • Formaat: Hardback, 368 pages, kõrgus x laius: 235x155 mm, kaal: 7096 g, 23 Illustrations, black and white; XXIV, 368 p. 23 illus., 1 Hardback
  • Ilmumisaeg: 23-Mar-2015
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3662462133
  • ISBN-13: 9783662462133
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783662462133.html
Märksõnad:
This book examines optimization problems that in practice involve random model parameters. It details the computation of robust optimal solutions, i.e., optimal solutions that are insensitive with respect to random parameter variations, where appropriate deterministic substitute problems are needed. Based on the probability distribution of the random data and using decision theoretical concepts, optimization problems under stochastic uncertainty are converted into appropriate deterministic substitute problems.

Due to the probabilities and expectations involved, the book also shows how to apply approximative solution techniques. Several deterministic and stochastic approximation methods are provided: Taylor expansion methods, regression and response surface methods (RSM), probability inequalities, multiple linearization of survival/failure domains, discretization methods, convex approximation/deterministic descent directions/efficient points, stochastic approximation and gradient procedures and differentiation formulas for probabilities and expectations.

In the third edition, this book further develops stochastic optimization methods. In particular, it now shows how to apply stochastic optimization methods to the approximate solution of important concrete problems arising in engineering, economics and operations research.

Arvustused

The considered book presents a mathematical analysis of the stochastic models of important applied optimization problems. presents detailed methods to solve these problems, rigorously proves their properties, and uses examples to illustrate the proposed methods. This book would be particularly beneficial to mathematicians working in the field of stochastic control and mechanical design. (Antanas Zilinskas, Interfaces, Vol. 45 (6), 2015)

1 Stochastic Optimization Methods
1(36)
1.1 Introduction
1(2)
1.2 Deterministic Substitute Problems: Basic Formulation
3(4)
1.2.1 Minimum or Bounded Expected Costs
4(2)
1.2.2 Minimum or Bounded Maximum Costs (Worst Case)
6(1)
1.3 Optimal Decision/Design Problems with Random Parameters
7(5)
1.4 Deterministic Substitute Problems in Optimal Decision/Design
12(4)
1.4.1 Expected Cost or Loss Functions
13(3)
1.5 Basic Properties of Deterministic Substitute Problems
16(2)
1.6 Approximations of Deterministic Substitute Problems in Optimal Design/Decision
18(10)
1.6.1 Approximation of the Loss Function
18(3)
1.6.2 Approximation of State (Performance) Functions
21(3)
1.6.3 Taylor Expansion Methods
24(4)
1.7 Approximation of Probabilities: Probability Inequalities
28(9)
1.7.1 Bonferroni-Type Inequalities
28(2)
1.7.2 Tschebyscheff-Type Inequalities
30(7)
2 Optimal Control Under Stochastic Uncertainty
37(42)
2.1 Stochastic Control Systems
37(12)
2.1.1 Random Differential and Integral Equations
39(6)
2.1.2 Objective Function
45(4)
2.2 Control Laws
49(3)
2.3 Convex Approximation by Inner Linearization
52(5)
2.4 Computation of Directional Derivatives
57(10)
2.5 Canonical (Hamiltonian) System of Differential Equations/Two-Point Boundary Value Problem
67(2)
2.6 Stationary Controls
69(2)
2.7 Canonical (Hamiltonian) System of Differential
71(2)
2.8 Computation of Expectations by Means of Taylor Expansions
73(6)
2.8.1 Complete Taylor Expansion
74(1)
2.8.2 Inner or Partial Taylor Expansion
75(4)
3 Stochastic Optimal Open-Loop Feedback Control
79(40)
3.1 Dynamic Structural Systems Under Stochastic Uncertainty
79(5)
3.1.1 Stochastic Optimal Structural Control: Active Control
79(2)
3.1.2 Stochastic Optimal Design of Regulators
81(1)
3.1.3 Robust (Optimal) Open-Loop Feedback Control
82(1)
3.1.4 Stochastic Optimal Open-Loop Feedback Control
82(2)
3.2 Expected Total Cost Function
84(1)
3.3 Open-Loop Control Problem on the Remaining Time Interval [ tb, t f]
85(1)
3.4 The Stochastic Hamiltonian of (3.7a--d)
85(2)
3.4.1 Expected Hamiltonian (with Respect to the Time Interval [ tb, t f] and Information tb)
86(1)
3.4.2 H-Minimal Control on [ tb, t f]
86(1)
3.5 Canonical (Hamiltonian) System
87(1)
3.6 Minimum Energy Control
88(10)
3.6.1 Endpoint Control
89(4)
3.6.2 Endpoint Control with Different Cost Functions
93(2)
3.6.3 Weighted Quadratic Terminal Costs
95(3)
3.7 Nonzero Costs for Displacements
98(5)
3.7.1 Quadratic Control and Terminal Costs
100(3)
3.8 Stochastic Weight Matrix Q = Q(t, ω)
103(4)
3.9 Uniformly Bounded Sets of Controls Dt, t0 ≤ t ≤ t f
107(5)
3.10 Approximate Solution of the Two-Point Boundary Value Problem (BVP)
112(3)
3.11 Example
115(4)
4 Adaptive Optimal Stochastic Trajectory Planning and Control (AOSTPC)
119(76)
4.1 Introduction
119(2)
4.2 Optimal Trajectory Planning for Robots
121(3)
4.3 Problem Transformation
124(5)
4.3.1 Transformation of the Dynamic Equation
126(1)
4.3.2 Transformation of the Control Constraints
127(1)
4.3.3 Transformation of the State Constraints
128(1)
4.3.4 Transformation of the Objective Function
129(1)
4.4 OSTP: Optimal Stochastic Trajectory Planning
129(13)
4.4.1 Computational Aspects
137(4)
4.4.2 Optimal Reference Trajectory, Optimal Feedforward Control
141(1)
4.5 AOSTP: Adaptive Optimal Stochastic Trajectory Planning
142(15)
4.5.1 (OSTP)-Transformation
146(2)
4.5.2 The Reference Variational Problem
148(2)
4.5.3 Numerical Solutions of (OSTP) in Real-Time
150(7)
4.6 Online Control Corrections: PD-Controller
157(16)
4.6.1 Basic Properties of the Embedding q(t, e)
158(3)
4.6.2 The 1st Order Differential dq
161(7)
4.6.3 The 2nd Order Differential d2q
168(4)
4.6.4 Third and Higher Order Differentials
172(1)
4.7 Online Control Corrections: PID Controllers
173(22)
4.7.1 Basic Properties of the Embedding q(t, ε)
176(1)
4.7.2 Taylor Expansion with Respect to ε
177(1)
4.7.3 The 1st Order Differential dq
178(17)
5 Optimal Design of Regulators
195(58)
5.1 Tracking Error
197(3)
5.1.1 Optimal PD-Regulator
198(2)
5.2 Parametric Regulator Models
200(3)
5.2.1 Linear Regulator
200(1)
5.2.2 Explicit Representation of Polynomial Regulators
201(1)
5.2.3 Remarks to the Linear Regulator
202(1)
5.3 Computation of the Expected Total Costs of the Optimal Regulator Design
203(4)
5.3.1 Computation of Conditional Expectations by Taylor Expansion
204(2)
5.3.2 Quadratic Cost Functions
206(1)
5.4 Approximation of the Stochastic Regulator Optimization Problem
207(9)
5.4.1 Approximation of the Expected Costs: Expansions of 1st Order
209(7)
5.5 Computation of the Derivatives of the Tracking Error
216(14)
5.5.1 Derivatives with Respect to Dynamic Parameters at Stage j
217(4)
5.5.2 Derivatives with Respect to the Initial Values at Stage j
221(3)
5.5.3 Solution of the Perturbation Equation
224(6)
5.6 Computation of the Objective Function
230(3)
5.7 Optimal PID-Regulator
233(20)
5.7.1 Quadratic Cost Functions
235(15)
5.7.2 The Approximate Regulator Optimization Problem
250(3)
6 Expected Total Cost Minimum Design of Plane Frames
253(36)
6.1 Introduction
253(1)
6.2 Stochastic Linear Programming Techniques
254(35)
6.2.1 Limit (Collapse) Load Analysis of Structures as a Linear Programming Problem
254(3)
6.2.2 Plane Frames
257(6)
6.2.3 Yield Condition in Case of M---N-Interaction
263(7)
6.2.4 Approximation of the Yield Condition by Using Reference Capacities
270(3)
6.2.5 Asymmetric Yield Stresses
273(6)
6.2.6 Violation of the Yield Condition
279(1)
6.2.7 Cost Function
280(9)
7 Stochastic Structural Optimization with Quadratic Loss Functions
289(34)
7.1 Introduction
289(3)
7.2 State and Cost Functions
292(7)
7.2.1 Cost Functions
296(3)
7.3 Minimum Expected Quadratic Costs
299(5)
7.4 Deterministic Substitute Problems
304(4)
7.4.1 Weight (Volume)-Minimization Subject to Expected Cost Constraints
304(2)
7.4.2 Minimum Expected Total Costs
306(2)
7.5 Stochastic Nonlinear Programming
308(7)
7.5.1 Symmetric, Non Uniform Yield Stresses
311(1)
7.5.2 Non Symmetric Yield Stresses
312(3)
7.6 Reliability Analysis
315(3)
7.7 Numerical Example: 12-Bar Truss
318(5)
7.7.1 Numerical Results: MEC
320(2)
7.7.2 Numerical Results: ECBO
322(1)
8 Maximum Entropy Techniques
323(34)
8.1 Uncertainty Functions Based on Decision Problems
323(8)
8.1.1 Optimal Decisions Based on the Two-Stage Hypothesis Finding (Estimation) and Decision Making Procedure
323(5)
8.1.2 Stability/Instability Properties
328(3)
8.2 The Generalized Inaccuracy Function H(λ, β)
331(15)
8.2.1 Special Loss Sets V
334(8)
8.2.2 Representation of Hε(λ, β) and H(λ, β) by Means of Lagrange Duality
342(4)
8.3 Generalized Divergence and Generalized Minimum Discrimination Information
346(11)
8.3.1 Generalized Divergence
346(6)
8.3.2 I-, J-Projections
352(1)
8.3.3 Minimum Discrimination Information
353(4)
References 357(8)
Index 365
Dr. Kurt Marti is a full Professor of Engineering Mathematics at the "Federal Armed Forces University of Munich. He is Chairman of the IFIP-Working Group 7.7 on Stochastic Optimization and has been Chairman of the GAMM-Special Interest Group Applied Stochastics and Optimization. Professor Marti has published several books, both in German and in English and he is author of more than 160 papers in refereed journals.