Convex Analysis: Fall 2015

Course information

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
2018-02-09, Hallvard Norheim Bø