## News

• 23.11.: The exams will take place on Wednesday, December 9.
• 04.11:
• We have agreed that you continue to work with the exercises in Chapter 3.3, especially exercises 8 and 9. In particular, Theorem 3.3.5 is one of the central results in convex analysis, and therefore deserves special attention. Part (f) of exercise 9 (optimality conditions for primal-dual optimal solutions) is normally included in the formulation of the duality theorem.
• Having said this, we also need to move forward! Therefore, please also take a look at Chapter 4.2 (we tentatively skip Chapter 4.1).
• Finally, please send us your input to settle the date for the exam (i.e., the dates when you cannot participate in the exam.

Minutes from the first meeting:

• Tentatively we will try exercise presentations by you - for this we will need two volunteers/week. Andre has already volunteered for week #42, so only one additional volunteer is needed for this week. Please let us know if you would like to present an exercise for the coming weeks
• The exercises are of course voluntary. It is however helpful for us if you bring your solutions to the oral exam - this way we can use the exercises for the discussion

## Synopsis

The objective of this module course is to get acquainted with the techniques and theory of convex sets and functions, Lagrangian and Fenchel duality, and non-smooth optimization. The course will be run as a reading course, where we will meet regularly to discuss the material and the exercises.

## Course plan

Course week Topic Book section Exercises Hints Solutions
0 (prerequisite) Eucledian spaces 1.1
1 Symmetric matrices 1.2 1, 2, 7, 11, 12 1.2 1.2
Optimality conditions 2.1 1, 2, 3, 4, 7, 8 2.1
2 Theorems of the alternative 2.2 1, 3, 5, 6 2.2 2.2
Max-functions 2.3 1, 2, 5, 7 2.3 2.3
3 Subgradients and convex functions 3.1. 1, 2, 3, 4, 6, 9 3.1 3.1
The value function 3.2 1, 4, 6, 7, 11 3.2 3.2
4 The Fenchel conjugate 3.3 1 (select), 2, 3, 5 (b,d,e), 8, 9, 23 (only generalize Corollary 3.3.11) 3.3
5 Further work with the Fenchel conjugate 3.3 2, 3, 5 (b,d,e), 8, 9, 23 (only generalize Corollary 3.3.11) 3.3
Fenchel biconjugation 4.2 1, 3, 7, 17 (a,b) 4.2
6 Lagrangian duality 4.3 1, 3, 4, 9 4.3 4.3