Convex Optimization
Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.
Why Read This Book
You will learn a clear, rigorous framework for recognizing and solving convex optimization problems and gain practical tools to apply them to real-world signal-processing tasks. The book balances theory, numerical methods, and worked examples so you can model DSP problems (filter design, beamforming, spectral estimation, sparse recovery) and solve them efficiently with modern solvers.
Who Will Benefit
Graduate students, researchers, and practicing engineers in signal processing, communications, radar, or audio who need to formulate and solve optimization-based algorithms and systems.
Level: Advanced — Prerequisites: Undergraduate linear algebra and multivariable calculus, basic probability and statistics, and familiarity with numerical computing (MATLAB or Python) — plus some DSP background to map examples to your domain.
Key Takeaways
- Formulate common DSP problems (filter design, beamforming, spectral estimation, waveform design, sparse recovery) as convex optimization problems
- Use convex duality and optimality conditions to derive insight, bounds, and efficient solver strategies for signal-processing tasks
- Apply and choose numerical algorithms (gradient methods, Newton and interior-point methods, specialized solvers) to solve medium- to large-scale convex programs
- Design and tune convex-based components: FIR/IIR filter approximations, linear receivers, adaptive filters (via convex relaxations), and sparse-signal recovery (L1, basis pursuit)
- Model problems using disciplined convex programming and deploy them with off-the-shelf tools (e.g., CVX/CVXPY, SeDuMi, SDPT3) for rapid prototyping
- Analyze solution sensitivity and trade-offs (regularization, constraints, and noise models) to make robust engineering choices in audio, speech, radar, and communications systems
Topics Covered
- 1. Introduction and Motivation: Where Convex Optimization Appears in Engineering
- 2. Convex Sets and Cones
- 3. Convex Functions and Inequalities
- 4. Convex Optimization Problems: Formulation and Examples
- 5. Duality: Lagrange Duality and KKT Conditions
- 6. Optimality Conditions, Sensitivity, and Bounds
- 7. Algorithms I: First-Order Methods and Subgradient Methods
- 8. Algorithms II: Newton's Method and Interior-Point Methods
- 9. Conic Formulations: Linear, Quadratic, and Semidefinite Programming
- 10. Numerical Considerations and Implementation Details
- 11. Applications in Signal Processing and Communications (filter design, beamforming, spectral estimation, sparse recovery)
- 12. Modeling Convex Problems and Using Solvers (disciplined convex programming)
- Appendices: Mathematical Background, Proofs, and Solver References
Languages, Platforms & Tools
How It Compares
More accessible and application-oriented than Ben-Tal & Nemirovski's rigorous mathematical treatment, and more specialized to convex problems than Nocedal & Wright's broad numerical optimization focus.
