Stochastic Optimization Methods: Applications in Engineering and Operations Research

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