Convex Optimization Algorithms
This book, developed through class instruction at MIT over the last 15 years, provides an accessible, concise, and intuitive presentation of algorithms for solving convex optimization problems. It relies on rigorous mathematical analysis, but also aims at an intuitive exposition that makes use of visualization where possible. This is facilitated by the extensive use of analytical and algorithmic concepts of duality, which by nature lend themselves to geometrical interpretation. The book places particular emphasis on modern developments, and their widespread applications in fields such as large-scale resource allocation problems, signal processing, and machine learning.
Among its features, the book:
* Develops comprehensively the theory of descent and approximation methods, including gradient and subgradient projection methods, cutting plane and simplicial decomposition methods, and proximal methods
* Describes and analyzes augmented Lagrangian methods, and alternating direction methods of multipliers
* Develops the modern theory of coordinate descent methods, including distributed asynchronous convergence analysis
* Comprehensively covers incremental gradient, subgradient, proximal, and constraint projection methods
* Includes optimal algorithms based on extrapolation techniques, and associated rate of convergence analysis
* Describes a broad variety of applications of large-scale optimization and machine learning
* Contains many examples, illustrations, and exercises
* Is structured to be used conveniently either as a standalone text for a class on convex analysis and optimization, or as a theoretical supplement to either an applications/convex optimization models class or a nonlinear programming class
Why Read This Book
You will learn how to design, analyze, and implement state-of-the-art algorithms for solving convex optimization problems that regularly arise in signal processing, communications, and machine learning. The book gives you rigorous convergence results together with geometric intuition and algorithmic variants (first- and second-order, primal, dual, augmented Lagrangian/ADMM) that are directly applicable to large-scale DSP tasks such as spectral estimation, filter design, and sparse recovery.
Who Will Benefit
Practicing engineers and graduate students with mathematical maturity who need practical, provable optimization algorithms to solve large-scale problems in DSP, communications, and machine learning.
Level: Advanced — Prerequisites: Linear algebra (including eigenvalues/SVD), multivariable calculus, basic convex analysis/optimization concepts, and familiarity with probability and numerical computation; programming experience (MATLAB/Python) helps for implementation.
Key Takeaways
- Formulate signal-processing and communications tasks (filter design, spectral estimation, sparse recovery) as convex optimization problems and derive their duals.
- Apply and tune first-order methods (gradient, accelerated gradient, subgradient, proximal) for large-scale DSP applications.
- Implement and analyze second-order and interior-point methods where high accuracy is required, and understand trade-offs in computation and memory.
- Use augmented Lagrangian methods and ADMM to split and parallelize problems common in audio, radar, and communications processing.
- Analyze convergence, complexity, and numerical stability of algorithms so you can choose methods suited to real-time or resource-limited DSP systems.
Topics Covered
- 1. Introduction to Convex Optimization and Applications in Signal Processing
- 2. Convex Sets, Functions, and Optimality Conditions
- 3. Duality Theory and Geometric Interpretation
- 4. Gradient and Descent Methods; Line Search and Step-Size Rules
- 5. Subgradient and Nonsmooth Optimization; Proximal Operators
- 6. Accelerated First-Order Methods and Complexity Bounds
- 7. Coordinate, Incremental, and Stochastic Methods for Large-Scale Problems
- 8. Newton and Quasi-Newton Methods; Local Convergence Analysis
- 9. Interior-Point Methods and High-Accuracy Solvers
- 10. Augmented Lagrangian Methods, ADMM, and Decomposition Techniques
- 11. Applications: Filter Design, Sparse Recovery, Spectral Estimation, and Detection
- 12. Implementation Issues: Scaling, Preconditioning, and Numerical Stability
- 13. Advanced Topics: Composite Objectives, Constraints, and Recent Developments
Languages, Platforms & Tools
How It Compares
Compared to Boyd & Vandenberghe's Convex Optimization, Bertsekas focuses more on algorithmic development, rigorous convergence proofs, and duality-based geometric intuition; compared to Nesterov, it is broader in algorithmic coverage rather than focused exclusively on optimal first-order methods.
