Features optimization problems that in practice involve random model parameters
Provides applications from the fields of robust optimal control / design in case of stochastic uncertainty
Includes numerous references to stochastic optimization, stochastic programming and its applications to engineering, operations research and economics
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.
Content Level » Research
Keywords » calculus - model - optimization problems - regression - response surface methodology - stochastic approximation - stochastic optimization
Related subjects » Computational Intelligence and Complexity - Mathematics - Operations Research & Decision Theory
Author(s): Kurt Marti
Edition: 3
Publisher: Springer
Year: 2015
Language: English
Pages: C, XIV, 368
Cover
S Title
Stochastic Optimization Methods
Copyright
© Springer-Verlag Berlin Heidelberg 2015
ISBN 978-3-662-46213-3
ISBN 978-3-662-46214-0 (eBook)
DOI 10.1007/978-3-662-46214-0
Library of Congress Control Number: 2015933010
Preface
Preface to the First Edition
Preface to the Second Edition
Outline of the 3rd Edition
Contents
1 Stochastic Optimization Methods
1.1 Introduction
1.2 Deterministic Substitute Problems: Basic Formulation
1.2.1 Minimum or Bounded Expected Costs
1.2.2 Minimum or Bounded Maximum Costs (Worst Case)
1.3 Optimal Decision/Design Problems with Random Parameters
1.4 Deterministic Substitute Problems in Optimal Decision/Design
1.4.1 Expected Cost or Loss Functions
1.5 Basic Properties of Deterministic Substitute Problems
1.6 Approximations of Deterministic Substitute Problems in Optimal Design/Decision
1.6.1 Approximation of the Loss Function
1.6.2 Approximation of State (Performance) Functions
Approximation of Expected Loss Functions
1.6.3 Taylor Expansion Methods
(Complete) Expansion with Respect to a
Inner (Partial) Expansions with Respect to a
1.7 Approximation of Probabilities: Probability Inequalities
1.7.1 Bonferroni-Type Inequalities
1.7.2 Tschebyscheff-Type Inequalities
Two-Sided Constraints
One-Sided Inequalities
2 Optimal Control Under Stochastic Uncertainty
2.1 Stochastic Control Systems
2.1.1 Random Differential and Integral Equations
Parametric Representation of the Random Differential/Integral Equation
Stochastic Differential Equations
2.1.2 Objective Function
Optimal Control Under Stochastic Uncertainty
2.2 Control Laws
2.3 Convex Approximation by Inner Linearization
2.4 Computation of Directional Derivatives
2.5 Canonical (Hamiltonian) System of Differential Equations/Two-Point Boundary Value Problem
2.6 Stationary Controls
2.7 Canonical (Hamiltonian) System of Differential
2.8 Computation of Expectations by Means of Taylor Expansions
2.8.1 Complete Taylor Expansion
2.8.2 Inner or Partial Taylor Expansion
3 Stochastic Optimal Open-Loop Feedback Control
3.1 Dynamic Structural Systems Under Stochastic Uncertainty
3.1.1 Stochastic Optimal Structural Control: Active Control
3.1.2 Stochastic Optimal Design of Regulators
3.1.3 Robust (Optimal) Open-Loop Feedback Control
3.1.4 Stochastic Optimal Open-Loop Feedback Control
3.2 Expected Total Cost Function
3.3 Open-Loop Control Problem on the Remaining Time Interval [tb,tf]
3.4 The Stochastic Hamiltonian of (3.7a–d)
3.4.1 Expected Hamiltonian (with Respect to the Time Interval [tb,tf] and Information Atb)
3.4.2 H-Minimal Control on [tb;tf]
Strictly Convex Cost Function; No Control Constraints
3.5 Canonical (Hamiltonian) System
3.6 Minimum Energy Control
3.6.1 Endpoint Control
Quadratic Control Costs
3.6.2 Endpoint Control with Different Cost Functions
3.6.3 Weighted Quadratic Terminal Costs
Quadratic Control Costs
3.7 Nonzero Costs for Displacements
3.7.1 Quadratic Control and Terminal Costs
3.8 Stochastic Weight Matrix Q=Q(t;.)
3.9 Uniformly Bounded Sets of Controls Dt, t0 =t =tf
3.10 Approximate Solution of the Two-Point Boundary Value Problem (BVP)
3.11 Example
4 Adaptive Optimal Stochastic Trajectory Planning and Control (AOSTPC)
4.1 Introduction
4.2 Optimal Trajectory Planning for Robots
4.3 Problem Transformation
4.3.1 Transformation of the Dynamic Equation
4.3.2 Transformation of the Control Constraints
4.3.3 Transformation of the State Constraints
4.3.4 Transformation of the Objective Function
4.4 OSTP: Optimal Stochastic Trajectory Planning
4.4.1 Computational Aspects
4.4.2 Optimal Reference Trajectory, Optimal Feedforward Control
4.5 AOSTP: Adaptive Optimal Stochastic Trajectory Planning
4.5.1 (OSTP)-Transformation
4.5.2 The Reference Variational Problem
Transformation of the Initial State Values
4.5.3 Numerical Solutions of (OSTP) in Real-Time
Discretization Techniques
NN-Approximation
Linearization Methods
Combination of Discretization and Linearization
4.6 Online Control Corrections: PD-Controller
4.6.1 Basic Properties of the Embedding q(t, e)
e=e0:= 0
e= e1 := 1
Taylor-Expansion with Respect to e
4.6.2 The 1st Order Differential dq
4.6.3 The 2nd Order Differential d2q
4.6.4 Third and Higher Order Differentials
4.7 Online Control Corrections: PID Controllers
4.7.1 Basic Properties of the Embedding q (t; )
4.7.2 Taylor Expansion with Respect to
4.7.3 The 1st Order Differential dq
Diagonalmatrices Kp, Kd; Ki
Mean Absolute 1st Order Tracking Error
Minimality or Boundedness Properties
5 Optimal Design of Regulators
Stochastic Optimal Design of Regulators
5.1 Tracking Error
5.1.1 Optimal PD-Regulator
5.2 Parametric Regulator Models
5.2.1 Linear Regulator
Nonlinear (Polynomial) Regulator
5.2.2 Explicit Representation of Polynomial Regulators
5.2.3 Remarks to the Linear Regulator
Remark to the Regulator Costs
5.3 Computation of the Expected Total Costs of the Optimal Regulator Design
5.3.1 Computation of Conditional Expectationsby Taylor Expansion
Computation of a (t;a)
5.3.2 Quadratic Cost Functions
5.4 Approximation of the Stochastic Regulator OptimizationProblem
5.4.1 Approximation of the Expected Costs: Expansions of 1st Order
5.5 Computation of the Derivatives of the Tracking Error
5.5.1 Derivatives with Respect to Dynamic Parametersat Stage j
5.5.2 Derivatives with Respect to the Initial Valuesat Stage j
5.5.3 Solution of the Perturbation Equation
Selection of the Matrices Kp,Kd
5.6 Computation of the Objective Function
5.7 Optimal PID-Regulator
5.7.1 Quadratic Cost Functions
Computation of the Expectation by Taylor Expansion
Calculation of the Sensitivities
Partial Derivative with Respect to .pD
Partial Derivative with Respect to .q0
Partial Derivative with Respect to .0
5.7.2 The Approximate Regulator Optimization Problem
6 Expected Total Cost Minimum Design of Plane Frames
6.1 Introduction
6.2 Stochastic Linear Programming Techniques
6.2.1 Limit (Collapse) Load Analysis of Structuresas a Linear Programming Problem
Plastic and Elastic Design of Structures
6.2.2 Plane Frames
6.2.3 Yield Condition in Case of M–N-Interaction
Symmetric Yield Stresses
Piecewise Linearization of K0a
6.2.4 Approximation of the Yield Condition by Using Reference Capacities
6.2.5 Asymmetric Yield Stresses
Formulation of the Yield Condition in the Non Symmetric Case
Use of Reference Capacities
Stochastic Optimization
6.2.6 Violation of the Yield Condition
6.2.7 Cost Function
Choice of the Cost Factors
Total Costs
Discretization Methods
Complete Recourse
Numerical Implementation
7 Stochastic Structural Optimization with Quadratic Loss Functions
7.1 Introduction
7.2 State and Cost Functions
7.2.1 Cost Functions
7.3 Minimum Expected Quadratic Costs
7.4 Deterministic Substitute Problems
7.4.1 Weight (Volume)-Minimization Subjectto Expected Cost Constraints
7.4.2 Minimum Expected Total Costs
7.5 Stochastic Nonlinear Programming
7.5.1 Symmetric, Non Uniform Yield Stresses
7.5.2 Non Symmetric Yield Stresses
7.6 Reliability Analysis
7.7 Numerical Example: 12-Bar Truss
7.7.1 Numerical Results: MEC
7.7.2 Numerical Results: ECBO
8 Maximum Entropy Techniques
8.1 Uncertainty Functions Based on Decision Problems
8.1.1 Optimal Decisions Based on the Two-Stage Hypothesis Finding (Estimation) and Decision Making Procedure
8.1.2 Stability/Instability Properties
8.2 The Generalized Inaccuracy Function H(.,ß)
8.2.1 Special Loss Sets V
8.2.2 Representation of H(.;ß) and H(.;ß)by Means of Lagrange Duality
8.3 Generalized Divergence and Generalized Minimum Discrimination Information
8.3.1 Generalized Divergence
8.3.2 I-, J-Projections
8.3.3 Minimum Discrimination Information
References
Index
