$Optimal Design and Control Math$

Science and Nature

Technology

Optimal Design and Control Math

Nonlinear and Dynamic Programming for Engineering, Machine Learning, and Inverse ProblemsBy Henrik Solberg

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$Optimal Design and Control Math$

Length14h 47m

About this audiobook

Stop blaming your solver. Start understanding why it fails. A solver that converges on the textbook problem and stalls on yours is the usual experience. This book explains why and what to do. You will see how conditioning wrecks steepest descent, how quasi-Newton updates recover curvature without a Hessian, and when a trust region beats a line search. You will compute derivatives by finite differences, complex-step, automatic differentiation and adjoints, then move into constrained problems through the Karush-Kuhn-Tucker conditions, duality, augmented Lagrangians, sequential quadratic programming and interior-point methods. Later chapters cover surrogate and Bayesian search, Pareto trade-offs, structural and topology optimisation, dynamic programming, the linear quadratic regulator and model predictive control, and close on regularised inverse problems and the stochastic gradient methods behind modern learning. Written for engineers, applied mathematicians and graduate students who build and tune solvers, this book connects numerical optimisation theory to the practical decisions that determine whether an algorithm converges, how fast, and to what. Each method is motivated by the failure mode it addresses, so you learn not just the update formula but the diagnostic reasoning that tells you which tool fits the problem in front of you. What you will learn: • Formulate optimisation problems with clear objectives, constraints and scaling • Diagnose conditioning and choose between line search, Newton and quasi-Newton updates • Apply conjugate gradient and trust-region methods to large and ill-behaved problems • Compute derivatives by finite differences, complex-step, automatic differentiation and adjoints • Solve constrained problems with KKT conditions, duality, SQP and interior-point methods • Use derivative-free, stochastic and surrogate methods, including Bayesian search • Optimise engineering designs, including structural and topology optimisation • Apply dynamic programming, LQR and model predictive control to optimal control • Regularise inverse problems and understand stochastic gradient methods for machine learning This book is for engineers, applied mathematicians, quantitative scientists and graduate students who need to build, tune and trust optimisation solvers in real applications. If you work at the boundary of numerical methods, engineering design, control and machine learning, the chapters give you the theory and the practical judgement to move from a method that almost works to one that reliably does.

Audiobook details

GenreScience and Nature, Technology

Length14 hrs 47 mins

Narrated byListen with 1,000+ voices

FormateBook with Audio

Publish dateSep 28, 2026

LanguageEnglish

1Optimal Design and Control Math

2Foreword

3Preface

4About This Book

5Chapter 1: Formulating Optimisation Problems

Show all chapters

61.1 Objectives, Design Variables and Constraints

71.2 Convex Sets, Convex Functions and Their Consequences

81.3 Local and Global Minima

91.4 First-Order Optimality Conditions

101.5 Second-Order Conditions and the Geometry of Stationary Points

111.6 Scaling, Units and Well-Posed Formulations

12Chapter 2: Line Search Methods

132.1 Descent Directions and the Generic Line Search Framework

142.2 Steepest Descent and Its Scaling Weakness

152.3 Armijo Backtracking and Sufficient Decrease

162.4 Wolfe and Strong Wolfe Conditions

172.5 Global Convergence of Line Search Methods

182.6 Practical Step-Length Strategies and Iteration Counts

19Chapter 3: Newton and Quasi-Newton Methods

203.1 Newton’s Method and the Newton Step

213.2 Hessian Modification and Modified Cholesky

223.3 Secant Conditions and the BFGS Update

233.4 DFP, Broyden Class and Update Properties

243.5 Limited-Memory BFGS for Large Problems

253.6 Superlinear Convergence and Practical Performance

26Chapter 4: Conjugate Gradient and Large Linear Systems

274.1 The Linear Conjugate Gradient Method

284.2 Preconditioning and Convergence Bounds

294.3 Nonlinear Conjugate Gradient Methods

304.4 Krylov Subspaces and Matrix-Free Implementation

31Chapter 5: Trust-Region Methods

325.1 The Trust-Region Subproblem

335.2 The Cauchy Point and Sufficient Reduction

345.3 Dogleg and Steihaug–CG Solutions

355.4 Radius Update Rules and Convergence

365.5 When Trust Regions Beat Line Searches

37Chapter 6: Computing Derivatives

386.1 Finite Differences: Truncation and Cancellation Error

396.2 Complex-Step Differentiation

406.3 Forward and Reverse Mode Automatic Differentiation

416.4 Adjoint Methods for Many-Variable Design

426.5 Choosing a Derivative Method

43Chapter 7: Nonlinear Least Squares and Data Fitting

447.1 The Least-Squares Problem and Residual Formulation

457.2 Gauss-Newton and Its Convergence

467.3 Levenberg–Marquardt Damping

477.4 Separable Problems and Robust Loss Functions

487.5 Parameter Confidence and Covariance

49Chapter 8: Linear and Quadratic Programming

508.1 Standard Forms and Geometry of Linear Programs

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