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Convex Optimization
Last Updated: 2026-02-05 15:10:09
Abstract
Convex optimization encompasses in a balanced manner theory (convex analysis, optimality conditions, duality theory) and algorithms for convex optimization. In particular the recent theory of semidefinite programming is discussed.
Content
Convexity plays a central role in the design and analysis of modern and highly successful algorithms for solving real-world optimization problems. The lecture (in English) on convex optimization will treat in a balanced manner theory (convex analysis, optimality conditions) and algorithms for convex optimization. Beginning with basic concepts and results about the structure of convex sets, continuity and differentiability of convex functions (including conjugate functions), the lecture will cover systems of inequalities, the minimum (or maximum) of a convex function over a convex set, Lagrange multipliers, duality theory and mini-may theorems. On the algorithmic part, we will cover efficient algorithms based on interior-point methods in the framework of self-concordant functions. In this way, we will obtain a simple algorithm for semi-definite optimization. Thus, we will be discussing one of the most challenging research areas of nonlinear optimization for which there are many interesting open questions both in theory and practice. The lecture will follow the textbook by S. Boyd, Convex Optimization, made available on the net. - Review of linear and convex quadratic programming. - Convexity of sets and functions. - Duality: weak and strong, complementary slackness. Certification of solutions. - Second-order cones and semidefinite programming, geometric programming. - Algorithms: penalty and barrier functions, ellipsoid method, outer approximations and cutting planes, interior point. - Applications: control systems analysis and design, signal processing, circuit design, classification and support vector machines, quantum mechanics, etc.
Resources
Literature
* A. Barvinok, A Course in Convexity. American Mathematical Society, 2003. * A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization - Analysis, Algorithms, and Engineering Applications, MPS-SIAM Series on Optimization, MPS-SIAM. * D. P. Bertsekas, A. Nedic and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, 2003. * D. Bertsimas and J. N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, 1997. * S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. * S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM, 1994. * E. de Klerk, Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications, Book Series: APPLIED OPTIMIZATION, Vol. 65. Kluwer Academic Publishers. * Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Book Series: APPLIED OPTIMIZATION, Vol. 87. Kluwer Academic Publishers, * R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985. * J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, MPS-SIAM Series on Optimization. * H. Wolkowicz, R. Saigal and L. Vandenberghe, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers.
General Information
- Language
- English
- Frequency
- Yearly recurring
Examination
- Type
- session examination
- Mode
- oral 20 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Convex Optimization |
|
2 h weekly |
| exercise |
Convex Optimization
im Wechsel mit 401-3902-00 U Diskrete Optimierung
|
|
1 h weekly |