This volume studies optimization of functionals and systems.
This volume studies optimization of functionals and systems. It connects analysis, geometry, control theory, and computation.
Part I. Foundations of Optimization
Chapter 1. Optimization Problems
1.1 Definitions and examples 1.2 Objective functions and constraints 1.3 Local and global optima 1.4 Existence of solutions 1.5 Examples
Chapter 2. Convexity
2.1 Convex sets 2.2 Convex functions 2.3 Properties 2.4 Jensen’s inequality 2.5 Applications
Chapter 3. Optimality Conditions
3.1 First-order conditions 3.2 Second-order conditions 3.3 Lagrange multipliers 3.4 KKT conditions 3.5 Examples
Part II. Calculus of Variations
Chapter 4. Functionals
4.1 Definitions 4.2 Variations 4.3 Minimization problems 4.4 Examples 4.5 Applications
Chapter 5. Euler–Lagrange Equations
5.1 Derivation 5.2 Boundary conditions 5.3 Examples 5.4 Applications 5.5 Extensions
Chapter 6. Direct Methods
6.1 Lower semicontinuity 6.2 Compactness 6.3 Existence theorems 6.4 Applications 6.5 Examples
Part III. Optimal Control
Chapter 7. Control Systems
7.1 State equations 7.2 Control variables 7.3 Constraints 7.4 Examples 7.5 Applications
Chapter 8. Pontryagin Maximum Principle
8.1 Statement 8.2 Adjoint equations 8.3 Necessary conditions 8.4 Applications 8.5 Examples
Chapter 9. Dynamic Programming
9.1 Bellman equation 9.2 Value functions 9.3 Optimal policies 9.4 Applications 9.5 Examples
Part IV. Convex Optimization
Chapter 10. Convex Optimization Problems
10.1 Problem formulation 10.2 Duality theory 10.3 Strong and weak duality 10.4 Applications 10.5 Examples
Chapter 11. Algorithms
11.1 Gradient methods 11.2 Subgradient methods 11.3 Interior-point methods 11.4 Applications 11.5 Examples
Chapter 12. Nonlinear Optimization
12.1 Nonconvex problems 12.2 Local optimization methods 12.3 Global optimization (overview) 12.4 Applications 12.5 Examples
Part V. Variational Methods in PDEs
Chapter 13. Variational Formulations
13.1 Weak formulations 13.2 Energy functionals 13.3 Euler–Lagrange PDEs 13.4 Applications 13.5 Examples
Chapter 14. Sobolev Spaces (Overview)
14.1 Definitions 14.2 Embedding theorems 14.3 Applications 14.4 Examples 14.5 Connections
Chapter 15. Minimization Problems
15.1 Existence results 15.2 Regularity 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Numerical Optimization
Chapter 16. Gradient-Based Methods
16.1 Gradient descent 16.2 Newton’s method 16.3 Quasi-Newton methods 16.4 Applications 16.5 Examples
Chapter 17. Constrained Optimization
17.1 Penalty methods 17.2 Barrier methods 17.3 Augmented Lagrangian 17.4 Applications 17.5 Examples
Chapter 18. Large-Scale Optimization
18.1 Sparse problems 18.2 Distributed optimization 18.3 Stochastic methods 18.4 Applications 18.5 Examples
Part VII. Applications
Chapter 19. Engineering Applications
19.1 Structural optimization 19.2 Control systems 19.3 Signal processing 19.4 Applications 19.5 Examples
Chapter 20. Economics and Decision Theory
20.1 Utility maximization 20.2 Game-theoretic optimization 20.3 Applications 20.4 Examples 20.5 Connections
Chapter 21. Machine Learning
21.1 Loss functions 21.2 Optimization algorithms 21.3 Regularization 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Non-smooth optimization 22.2 Optimal transport 22.3 Variational inequalities 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Nonconvex optimization challenges 23.2 Global optimality questions 23.3 Computational complexity 23.4 Stability issues 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of variational calculus 24.2 Key contributors 24.3 Evolution of optimization 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Optimality condition summary B. Algorithm reference C. Proof techniques checklist D. Numerical method tables E. Cross-reference to other MSC branches
This volume develops optimization and variational methods as central tools for analysis and applied mathematics. It emphasizes theory, computation, and real-world applications.