Stochastic Optimization Methods: Applications in Engineering and Operations Research

✍ Scribed by Kurt Marti

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 389
• Edition: 4
• Category: Library

✦ Synopsis

This book examines optimization problems that in practice involve random model parameters. It outlines the computation of robust optimal solutions, i.e., optimal solutions that are insensitive 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 corresponding deterministic 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.

The fourth edition of this classic text has been carefully and thoroughly revised. It includes new chapters on the solution of stochastic linear programs by discretization of the underlying probability distribution, and on solving deterministic optimization problems by means of controlled random search methods and multiple random search procedures. It also presents a new application of stochastic optimization methods to machine learning problems with different loss functions. For the computation of optimal feedback controls under stochastic uncertainty, besides the open-loop feedback procedures, a new method based on Taylor expansions with respect to the gain parameters is presented.

The book is intended for researchers and graduate students who are interested in stochastics, stochastic optimization, and control. It will also benefit professionals and practitioners whose work involves technical, economicand/or operations research problems under stochastic uncertainty.

✦ Table of Contents

Preface
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
1.6.3 Taylor Expansion Methods
1.7 Approximation of Probabilities—Probability Inequalities
1.7.1 Bonferroni-Type Inequalities
1.7.2 Tschebyscheff-Type Inequalities
References
2 Solution of Stochastic Linear Programs by Discretization Methods
2.1 A Priori Error Bounds
2.2 Disretization and Error Bounds
2.2.1 Special Representations of the Random Matrix ( T ( cdot) , h ( cdot) )
2.3 Approximations of F with a Given Error Level ε
2.4 Norm Bounds for Optimal Solutions of (2.2a)–(2.2c)
2.5 Invariant Discretizations
References
3 Optimal Control Under Stochastic Uncertainty
3.1 Stochastic Control Systems
3.1.1 Random Differential and Integral Equations
3.1.2 Objective Function
3.2 Control Laws
3.3 Convex Approximation by Inner Linearization
3.4 Computation of Directional Derivatives
3.5 Canonical (Hamiltonian) System of Differential Equations/Two-Point Boundary Value Problem
3.6 Stationary Controls
3.7 Canonical (Hamiltonian) System of Differential
3.8 Computation of Expectations by Means of Taylor Expansions
3.8.1 Complete Taylor Expansion
3.8.2 Inner or Partial Taylor Expansion
References
4 Random Search Methods for Global Optimization—Basics
4.1 Introduction
4.2 The Convergence of the Basic Random Search Procedure
4.2.1 Discrete Optimization Problems
4.3 Adaptive Random Search Methods
4.3.1 Infinite-Stage Search Processes
4.4 Convex Problems
References
5 Controlled Random Search Methods as a Stochastic Decision Process
5.1 The Controlled (or Adaptive) Random Search Method
5.1.1 The Convergence of the Controlled Random Search Procedure
5.1.2 A Stopping Rule
5.2 Computation of the Conditional Distribution of upper FF Given the Process History: Information Processing
References
6 Applications to Random Search Methods with Joint Normal Search Variates
6.1 Introduction
6.2 Convergence of the Random Search Procedure (6.2)
6.3 Controlled Random Search Methods
6.4 Computation of Optimal Controls
6.5 Convergence Rates of Controlled Random Search Procedures
6.6 Numerical Realizations of Optimal Control Laws
References
7 Random Search Methods with Multiple Search Points
7.1 Standard RSM
7.2 Multiple RSM
7.3 Probability of Failure, Probability of Success
7.3.1 Monotonicity of the Probability Functions pf, ps
7.3.2 Asymptotic Behavior in Case of i.i.d. Search Variables
7.3.3 Estimation of pf and ps in Case of Arbitrary Stochastically Independent Search Variables Yt,j=Yj
7.4 Reachability Results Multiple RSM
7.5 Optimal Search Point Among Multiple Search Variables
7.5.1 The Optimized Search Process
7.5.2 Probability of Reaching Bε from the Outside
References
8 Approximation of Feedback Control Systems
8.1 Introduction
8.2 Control Laws
8.3 Linear State-Feedback Control Systems
8.3.1 Taylor Expansion of the Feedback Control System with Respect to the Gain Matrix G=(gij)
8.3.2 Time-Dependent Gain Matrices
8.4 Optimal Feedback Control Problem
8.4.1 Stepwise Optimization of u0(cdot),G
8.5 Approximation of Nonlinear Feedback Control Systems
8.6 Approximation Error
8.7 Extensions
8.7.1 Special Representations of the Open-Loop (Prior) Control Function u0(cdot)
8.7.2 Nonlinear Feedback Function
References
9 Stochastic Optimal Open-Loop Feedback Control
9.1 Dynamic Structural Systems Under Stochastic Uncertainty
9.1.1 Stochastic Optimal Structural Control: Active Control
9.1.2 Stochastic Optimal Design of Regulators
9.1.3 Robust (Optimal) Open-Loop Feedback Control
9.1.4 Stochastic Optimal Open-Loop Feedback Control
9.2 Expected Total Cost Function
9.3 Open-Loop Control Problem on the Remaining Time Interval [tb,tf]
9.4 The Stochastic Hamiltonian of (9.7a)–(9.7d)
9.4.1 Expected Hamiltonian (with Respect to the Time Interval [tb,tf] and Information mathfrakAtb)
9.4.2 H-Minimal Control on [tb,tf]
9.5 Canonical (Hamiltonian) System
9.6 Minimum-Energy Control
9.6.1 Endpoint Control
9.6.2 Endpoint Control with Different Cost Functions
9.6.3 Weighted Quadratic Terminal Costs
9.7 Nonzero Costs for Displacements
9.7.1 Quadratic Control and Terminal Costs
9.8 Stochastic Weight Matrix Q=Q(t,ω)
9.9 Uniformly Bounded Sets of Controls Dt, t0 leqt leqtf
9.10 Approximate Solution of the Two-Point Boundary Value Problem (BVP)
9.10.1 Approximate Solution of the Fixed Point Eq. (9.75)
9.11 Example
References
10 Adaptive Optimal Stochastic Trajectory Planning and Control (AOSTPC)
10.1 Introduction
10.2 Optimal Trajectory Planning for Robots
10.3 Problem Transformation
10.3.1 Transformation of the Dynamic Equation
10.3.2 Transformation of the Control Constraints
10.3.3 Transformation of the State Constraints
10.3.4 Transformation of the Objective Function
10.4 OSTP—Optimal Stochastic Trajectory Planning
10.4.1 Computational Aspects
10.4.2 Optimal Reference Trajectory, Optimal Feedforward Control
10.5 AOSTP—Adaptive Optimal Stochastic Trajectory Planning
10.5.1 (OSTP)-Transformation
10.5.2 The Reference Variational Problem
10.5.3 Numerical Solutions of (OSTP) in Real-Time
10.6 Online Control Corrections: PD-Controller
10.6.1 Basic Properties of the Embedding q(t, ε)
10.6.2 The First-Order Differential dq
10.6.3 The Second-Order Differential d2q
10.6.4 Third and Higher Order Differentials
10.7 Online Control Corrections: PID Controllers
10.7.1 Basic Properties of the Embedding q (t, ε)
10.7.2 Taylor Expansion with Respect to epsilonε
10.7.3 The First-Order Differential dq
References
11 Machine Learning Under Stochastic Uncertainty
11.1 Foundations
11.2 Stochastic Optimization Methods in Machine Learning
11.2.1 Least Squares Estimation of the Parameter Vector
11.3 Estimation with Sublinear Loss Function q=q(z)
11.3.1 Representation by a Stochastic Linear Optimization Problem (SLOP)
11.3.2 Numerical Solution of the (SLOP)
11.3.3 Two-Stage Stochastic Linear Programs (SLP)
11.4 Two and Multiple Group Classification Under Stochastic Uncertainty
11.4.1 Two Classes (J=2, L=1)
11.5 Multi-classification
11.5.1 Reduction of a Multi-classifier to Several Two-Class Classifiers
References
12 Stochastic Structural Optimization with Quadratic Loss Functions
12.1 Introduction
12.2 State and Cost Functions
12.2.1 Cost Functions
12.3 Minimum Expected Quadratic Costs
12.4 Deterministic Substitute Problems
12.4.1 Weight (Volume)-Minimization Subject to Expected Cost Constraints
12.4.2 Minimum Expected Total Costs
12.5 Stochastic Nonlinear Programming
12.5.1 Symmetric, Non-uniform Yield Stresses
12.5.2 Non Symmetric Yield Stresses
12.6 Reliability Analysis
12.7 Numerical Example: 12-Bar Truss
12.7.1 Numerical Results: MEC
12.7.2 Numerical Results: ECBO
References
13 Maximum Entropy Techniques
13.1 Uncertainty Functions Based on Decision Problems
13.1.1 Optimal Decisions Based on the Two-Stage Hypothesis Finding (Estimation) and Decision-Making Procedure
13.1.2 Stability/Instability Properties
13.2 The Generalized Inaccuracy Function H(λ,β)
13.2.1 Special Loss Sets V
13.2.2 Representation of Hε(λ,β) and H(λ,β) by Means of Lagrange Duality
13.3 Generalized Divergence and Generalized Minimum Discrimination Information
13.3.1 Generalized Divergence
13.3.2 I-, J-Projections
13.3.3 Minimum Discrimination Information
References
Index