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Convex Optimization

M. Baes
Link zum Kurskatalog
Fall 2016
The aim of this course is to give to mathematicians and practitioners an overview of useful concepts and techniques in convex optimization. A particular attention is given to convex modeling and to algorithms for solving convex optimization problems. Some exercise sessions are devoted to an initiation to a convex optimization solver.
In summary, we will discuss one of the most challenging research areas of nonlinear optimization for which there are many interesting open questions both in theory and practice.
Here is a brief syllabus of the course. * Mathematical background (6 lectures) Introduction, convex sets, Semidefinite cone, separation theorems, Duality, Farkas Lemma, Optimality conditions, Lagrangian duality, Subgradients, conjugate functions, KKT conditions and applications.
*Applications, convex modeling (3 lectures) Conic Optimization and applications, Applications of Semidefinite Optimization Applications of Convex Optimization to Data Fitting and Statistical Estimation.
*Algorithms (5 lectures) Black-box methods, Self-concordant functions, Interior-point methods, Primal-dual interior-point methods.
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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), modeling issues, 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 separation theorems and their important consequences: the theory of Lagrange multipliers, the duality theory and some min-max theorems. On the algorithmic part, the course will study some simple first and second-order algorithms, as well as some efficient interior-point methods in the framework of self-concordant functions.

The slides of the course are available online, on the course website. An important reference book for the lecture is "S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004", available online for free.