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This book offers a comprehensive introduction to optimization with a focus on practical algorithms. The book approaches optimization from an engineering perspective, where the objective is to design a system that optimizes a set of metrics subject to constraints. Readers will learn about computational approaches for a range of challenges, including searching high-dimensional spaces, handling problems where there are multiple competing objectives, and accommodating uncertainty in the metrics. Figures, examples, and exercises convey the intuition behind the mathematical approaches. The text provides concrete implementations in the Julia programming language.

Topics covered include derivatives and their generalization to multiple dimensions; local descent and first- and second-order methods that inform local descent; stochastic methods, which introduce randomness into the optimization process; linear constrained optimization, when both the objective function and the constraints are linear; surrogate models, probabilistic surrogate models, and using probabilistic surrogate models to guide optimization; optimization under uncertainty; uncertainty propagation; expression optimization; and multidisciplinary design optimization. Appendixes offer an introduction to the Julia language, test functions for evaluating algorithm performance, and mathematical concepts used in the derivation and analysis of the optimization methods discussed in the text. The book can be used by advanced undergraduates and graduate students in mathematics, statistics, computer science, any engineering field, (including electrical engineering and aerospace engineering), and operations research, and as a reference for professionals.

Contents
Preface
Acknowledgments
1 Introduction
1.1 A History
1.2 Optimization Process
1.3 Basic Optimization Problem
1.4 Constraints
1.5 Critical Points
1.6 Conditions for Local Minima
1.7 Contour Plots
1.8 Overview
1.9 Summary
1.10 Exercises
2 Derivatives and Gradients
2.1 Derivatives
2.2 Derivatives in Multiple Dimensions
2.3 Numerical Differentiation
2.4 Automatic Differentiation
2.5 Summary
2.6 Exercises
3 Bracketing
3.1 Unimodality
3.2 Finding an Initial Bracket
3.3 Fibonacci Search
3.4 Golden Section Search
3.5 Quadratic Fit Search
3.6 Shubert-Piyavskii Method
3.7 Bisection Method
3.8 Summary
3.9 Exercises
4 Local Descent
4.1 Descent Direction Iteration
4.2 Line Search
4.3 Approximate Line Search
4.4 Trust Region Methods
4.5 Termination Conditions
4.6 Summary
4.7 Exercises
5 First-Order Methods
5.1 Gradient Descent
5.2 Conjugate Gradient
5.3 Momentum
5.4 Nesterov Momentum
5.5 Adagrad
5.6 RMSProp
5.7 Adadelta
5.8 Adam
5.9 Hypergradient Descent
5.10 Summary
5.11 Exercises
6 Second-Order Methods
6.1 Newton’s Method
6.2 Secant Method
6.3 Quasi-Newton Methods
6.4 Summary
6.5 Exercises
7 Direct Methods
7.1 Cyclic Coordinate Search
7.2 Powell’s Method
7.3 Hooke-Jeeves
7.4 Generalized Pattern Search
7.5 Nelder-Mead Simplex Method
7.6 Divided Rectangles
7.7 Summary
7.8 Exercises
8 Stochastic Methods
8.1 Noisy Descent
8.2 Mesh Adaptive Direct Search
8.3 Simulated Annealing
8.4 Cross-Entropy Method
8.5 Natural Evolution Strategies
8.6 Covariance Matrix Adaptation
8.7 Summary
8.8 Exercises
9 Population Methods
9.1 Initialization
9.2 Genetic Algorithms
9.3 Differential Evolution
9.4 Particle Swarm Optimization
9.5 Firefly Algorithm
9.6 Cuckoo Search
9.7 Hybrid Methods
9.8 Summary
9.9 Exercises
10 Constraints
10.1 Constrained Optimization
10.2 Constraint Types
10.3 Transformations to Remove Constraints
10.4 Lagrange Multipliers
10.5 Inequality Constraints
10.6 Duality
10.7 Penalty Methods
10.8 Augmented Lagrange Method
10.9 Interior Point Methods
10.10 Summary
10.11 Exercises
11 Linear Constrained Optimization
11.1 Problem Formulation
11.2 Simplex Algorithm
11.3 Dual Certificates
11.4 Summary
11.5 Exercises
12 Multiobjective Optimization
12.1 Pareto Optimality
12.2 Constraint Methods
12.3 Weight Methods
12.4 Multiobjective Population Methods
12.5 Preference Elicitation
12.6 Summary
12.7 Exercises
13 Sampling Plans
13.1 Full Factorial
13.2 Random Sampling
13.3 Uniform Projection Plans
13.4 Stratified Sampling
13.5 Space-Filling Metrics
13.6 Space-Filling Subsets
13.7 Quasi-Random Sequences
13.8 Summary
13.9 Exercises
14 Surrogate Models
14.1 Fitting Surrogate Models
14.2 Linear Models
14.3 Basis Functions
14.4 Fitting Noisy Objective Functions
14.5 Model Selection
14.6 Summary
14.7 Exercises
15 Probabilistic Surrogate Models
15.1 Gaussian Distribution
15.2 Gaussian Processes
15.3 Prediction
15.4 Gradient Measurements
15.5 Noisy Measurements
15.6 Fitting Gaussian Processes
15.7 Summary
15.8 Exercises
16 Surrogate Optimization
16.1 Prediction-Based Exploration
16.2 Error-Based Exploration
16.3 Lower Confidence Bound Exploration
16.4 Probability of Improvement Exploration
16.5 Expected Improvement Exploration
16.6 Safe Optimization
16.7 Summary
16.8 Exercises
17 Optimization under Uncertainty
17.1 Uncertainty
17.2 Set-Based Uncertainty
17.3 Probabilistic Uncertainty
17.4 Summary
17.5 Exercises
18 Uncertainty Propagation
18.1 Sampling Methods
18.2 Taylor Approximation
18.3 Polynomial Chaos
18.4 Bayesian Monte Carlo
18.5 Summary
18.6 Exercises
19 Discrete Optimization
19.1 Integer Programs
19.2 Rounding
19.3 Cutting Planes
19.4 Branch and Bound
19.5 Dynamic Programming
19.6 Ant Colony Optimization
19.7 Summary
19.8 Exercises
20 Expression Optimization
20.1 Grammars
20.2 Genetic Programming
20.3 Grammatical Evolution
20.4 Probabilistic Grammars
20.5 Probabilistic Prototype Trees
20.6 Summary
20.7 Exercises
21 Multidisciplinary Optimization
21.1 Disciplinary Analyses
21.2 Interdisciplinary Compatibility
21.3 Architectures
21.4 Multidisciplinary Design Feasible
21.5 Sequential Optimization
21.6 Individual Discipline Feasible
21.7 Collaborative Optimization
21.8 Simultaneous Analysis and Design
21.9 Summary
21.10 Exercises
A Julia
A.1 Types
A.2 Functions
A.3 Control Flow
A.4 Packages
B Test Functions
B.1 Ackley’s Function
B.2 Booth’s Function
B.3 Branin Function
B.4 Flower Function
B.5 Michalewicz Function
B.6 Rosenbrock’s Banana Function
B.7 Wheeler’s Ridge
B.8 Circle Function
C Mathematical Concepts
C.1 Asymptotic Notation
C.2 Taylor Expansion
C.3 Convexity
C.4 Norms
C.5 Matrix Calculus
C.6 Positive Definiteness
C.7 Gaussian Distribution
C.8 Gaussian Quadrature
D Solutions
Bibliography
Index
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