Optimization Theory with Applications 1st edition by Donald Pierre – Ebook PDF Instant Download/Delivery: 0486136957, 9780486136950
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ISBN 10: 0486136957
ISBN 13: 9780486136950
Author: Donald Pierre
Optimization principles are of undisputed importance in modern design and system operation. They can be used for many purposes: optimal design of systems, optimal operation of systems, determination of performance limitations of systems, or simply the solution of sets of equations. While most books on optimization are limited to essentially one approach, this volume offers a broad spectrum of approaches, with emphasis on basic techniques from both classical and modern work. After an introductory chapter introducing those system concepts that prevail throughout optimization problems of all types, the author discusses the classical theory of minima and maxima (Chapter 2). In Chapter 3, necessary and sufficient conditions for relative extrema of functionals are developed from the viewpoint of the Euler-Lagrange formalism of the calculus of variations. Chapter 4 is restricted to linear time-invariant systems for which significant results can be obtained via transform methods with a minimum of computational difficulty. In Chapter 5, emphasis is placed on applied problems which can be converted to a standard problem form for linear programming solutions, with the fundamentals of convex sets and simplex technique for solution given detailed attention. Chapter 6 examines search techniques and nonlinear programming. Chapter 7 covers Bellman’s principle of optimality, and finally, Chapter 8 gives valuable insight into the maximum principle extension of the classical calculus of variations. Designed for use in a first course in optimization for advanced undergraduates, graduate students, practicing engineers, and systems designers, this carefully written text is accessible to anyone with a background in basic differential equation theory and matrix operations. To help students grasp the material, the book contains many detailed examples and problems, and also includes reference sections for additional reading.
Optimization Theory with Applications 1st Table of contents:
1 Introduction
1-1 Optimization in Perspective
1-2 The Concepts of System and State
1-3 Performance Measures
1-4 Constraints
1-5 Optimization Problems
1-6 Conditions for Optimality
1-7 Approaches to Solution
1-8 Forms of Solutions
1-9 Sensitivity and Identification
1-10 Discussion
2 Classical Theory of Minima and Maxima
2-1 Introduction
2-2 Basic Concepts and Notation
2-3 Functions of One Variable
2-4 Functions of Several Variables
2-5 Equality Constraints and a Lagrange Multiplier
2-6 General Case of Equality Constraints
2-7 Inequality Constraints
2-8 Extremization of Integrals
2-9 Sensitivity Analysis
2-10 Conclusion
3 Classical Calculus of Variations
3-1 Introduction
3-2 Preliminary Concepts
a. Continuity, Extrema, and Variations
b. Classes of Problems and Equivalence Relations
3-3 The Problem of Lagrange: Scalar Case
a. Problem Statement and the First Variation
b. Fundamental Lemma
c. First Necessary Condition and First-Variational Curves
d. A Corner Condition
e. The Euler-Lagrange Equation
3-4 Isoperimetric Constraints
3-5 Variable End-Point Conditions
3-6 Corner Conditions
3-7 The Problem of Lagrange: State-Vector Case
3-8 Constraints
a. Isoperimetric Constraints
b. Constraints of the Form g1(x,x,t) = 0
c. Constraints of the Form zi(x,t) = 0
d. Inequality Constraints
3-9 A General Control Problem
3-10 Sufficient Conditions and Additional Necessary Conditions
a. Discussion
b. The Second Variation and the Legendre Condition
c. Fields of Solutions
d. The Jacobi Condition
e. Sufficient Condition for Weak Extrema
f. Green’s Theorem
g. The Weierstrass Condition and Strong Extrema
3-11 Direct Methods
a. Discussion
b. Series Approximations
c. Finite Differences
3-12 Sensitivity Considerations
3-13 Conclusion
4 Wiener-Hopf Spectrum Factorization and Frequency-Domain Optimization
4-1 Introduction
4-2 Filter, Control, and Predictor Problems
a. Description of the System
b. Integral-Square-Error Problems
c. Mean-Square-Error Problems
4-3 A General Optimal Pulse-Shape Problem
a. Description of the System
b. Maximum Peak Output
c. Maximum of the Average Output
4-4 A General Problem and Solution
a. The Problem
b. Initial Steps to Solution
c. Notation for Spectrum Factorization
d. Solution Using Wiener-Hopf Spectrum Factorization
4-5 Solutions to Filter, Control, and Predictor Problems
4-6 Solutions to Optimal Pulse-Shape Problems
4-7 Non-Wiener-Hopf-Type Frequency-Domain Problems
a. General Comments
b. Pulse Shape for Maximum Output Energy
4-8 Sensitivity Considerations in Design
4-9 Conclusion
5 The Simplex Technique and Linear Programming
5-1 Introduction
5-2 The General Problem and Its Standard Form
5-3 Conversion to the Standard Form
5-4 Analytical Basis
a. Prelude
b. Convexity
c. Extreme Point and Verticy Properties
d. Optimal P at a Vertex
5-5 Simplex Algorithm Theory
5-6 Simplex Algorithm Mechanics: The Simplex Tableau
5-7 Initializing and Scaling
a. Avoiding Initial Degeneracy
b. Generating an Initial Basic Feasible Solution
c. Scaling
5-8 Upper-Bounding Algorithm
5-9 Dual Problems
a. Duals in General
b. Symmetric Duals
c. Other Duals
5-10 Sensitivity Analysis
5-11 Analog Solutions
a. Analogies
b. Linear Programming on the General-Purpose Analog Computer
5-12 Applications
a. Problems of Economics
b. Control Problems
c. Communications Problems
d. Circuit Design Applications
e. Field Problems
f. Other Applications
5-13 Conclusion
6 Search Techniques and Nonlinear Programming
6-1 Introduction
6-2 Geometrical Interpretation and Scaling
a. Local Properties
b. Regional Properties
c. Scaling and Change of Variables
d. Noise Considerations
e. Constraint Geometry
6-3 One-Dimensional Search
a. Newton-Raphson Search
b. Cubic-Convergent Search without Second Derivatives
c. Quadratic-Convergent Search without Derivatives
d. Fibonacci Search
e. Search by Golden Section
f. One-Dimensional Search in n-Dimensional Space
6-4 Nonsequential Methods
a. Nonsequential Random Search
b. Nonsequential Factorial Search
6-5 Univariate and Relaxation Search
a. Univariate Search
b. Southwell’s Relaxation Search
c. Southwell-Synge Search
6-6 Basic Gradient Methods
a. Common Features
b. Continuous Steepest Ascent (Descent)
c. Discrete Steepest Ascent (Descent)
d. Newton Search
6-7 Acceleration-Step Search
a. Two-Dimensional Case
b. n-Dimensional Case: PARTAN
6-8 Conjugate-Direction Methods
a. Conjugate Directions
b. Method of Fletcher and Reeves
c. Davidon’s Method via Fletcher and Powell (The DFP Method)
6-9 Other Search Methods
a. Discussion
b. Pattern Search
c. Search by Directed Array
d. Creeping Random Methods
e. Centroid Methods
6-10 Combined Use of Indirect and Direct Methods
a. Equation Solution by Search
b. Reduction of Dimensionality
6-11 Constraints
a. The Nonlinear Programming Problem
b. Outside Penalty Functions for Inequality Constraints
c. Penalty Functions for Equality Constraints
d. Minimization of the Penalized Performance Measure
e. Inside Penalty Functions
f. Equality Constraints and Classical Lagrange Multipliers
g. General Constraints and Lagrange Multipliers
6-12 Comparison of Techniques
6-13 Conclusion
7 A Principle of Optimality and Dynamic Programming
7-1 Introduction
7-2 Allocation Problems
a. Problem Statement and Applications
b. Dynamic Programming Approach to Solution
7-3 Efficiency Comparison
7-4 Redundancy to Improve Reliability
7-5 Minimal Chain Problems
a. Chain Networks
b. Forward Solution I
c. Backward Solution I
d. Backward Solution II
e. Comparison of Forward and Backward Solutions
7-6 A Control Problem
a. Statement of the Problem
b. Backward Solutions
c. Forward Solutions
7-7 Numerical Considerations
7-8 A Principle of Optimality
7-9 Placement of Transmission-Line Towers
7-10 n State Variables: Discrete Processes
a. Problems and Difficulties
b. Series Approximations
c. Lagrange Multipliers
d. Region-Limiting Strategies and Iterated Dynamic Programming
7-11 Approximations in Function and Policy Space
a. A Control Problem
b. An Approximation in Function Space
c. An Approximation in Policy Space
d. Nonoriented Minimal Chain Problems
7-12 Continuous Decision Processes: Discrete Approximations with n State Variables
a. A General Control Problem
b. Recurrence Relations with Prespecified Time Increments
c. A Continuous Recurrence Relation
d. Recurrence Relations with Controlled Time Increments
7-13 Continuous Decision Problems: Calculus of Variations and Extensions
a. The Problem and Its Forward Recurrence Relation
b. Hamilton-Jacobi Equations
c. Costate Equations
d. Hamiltonian Functions
e. Necessary Conditions: A Maximum Principle
f. Necessary Conditions: Classical Calculus of Variations
7-14 Quadratic Minimum-Cost Function and Closed-Loop Control
a. A General Case
b. Steady-State Riccati Equations
7-15 A Stochastic Control Problem
7-16 Estimation of State Variables in the Presence of Noise
a. Modal Trajectory Estimation
b. Discrete Kalman-Bucy Filter
7-17 Conclusion
8 A Maximum Principle
8-1 Introduction
8-2 Preliminary Concepts
8-3 A Canonical Problem Form and Equivalent Problems
8-4 A Maximum Principle
8-5 The Constancy of H*
8-6 The General Transversality Condition
8-7 Time Optimal Control
a. Comments
b. A Second-Order System
c. Optimal Switch-Time Evaluation
8-8 Search Techniques for Solution of Boundary-Value Problems
a. Comments
b. Utilization of H in a Search Solution
c. A Newton-Raphson Algorithm for Linearization of Differential Equations and Solution of Two-Point Boundary-Value Problems
d. Iterative Solutions with Stabilization via Riccati Equations
e. A Riccati Transformation
8-9 Non-Normal Solutions
8-10 Singular Solutions
8-11 Equivalent Principles
a. An Equivalent Minimum Principle
b. Necessary Conditions for End-Point Functionals
8-12 Conclusion
A. Matrix Identities and Operations
B. Two-Sided Laplace Transform Theory
C. Correlation Functions and Power-Density Spectra
D. Inequalities and Abstract Spaces
Author Index
Subject Index
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