In this course we will provide an introduction to Convex Analysis in finite and infinite-dimensional spaces. We will show also some applications to problems of nonlinear optimizations and control theory arising from physics and economics. At the end of the course, the student should be able to: - understand the deep link between this and the previous courses (in particular, Functional Analysis); - use the main tools of convex analysis to solve convex optimization problems; - formalize and analyze simple control system coming from physical and economics models, in the framework of optimal control theory; - be autonomous in the us of the textbook suggested for the course.
Table of contents
- Review of weak topology on Banach spaces: convex sets, Minkowski functional, linear continuous operators, weak topology, separation of convex sets.
- Convex functions: general properties, lower semicontinuous convex functions, convex conjugate, subdifferential in the sense of Convex Analysis. Introduction to Calculus of Variations.
- Generalizations of convexity: differential calculus in Hilbert and Banach spaces, proximal and limiting subdifferential, the density theorem, sum rule, chain rule, generalized gradient in Banach space.
- Introduction to control theory: multifunctions and trajectories of differential inclusions, viability,
equilibria, invariance, stabilization, reachability, Pontryagin Maximum Principle, necessary conditions
- Application to optimization problems arising from physical or economic models.
The course is divided in two part: 5 ECTS (Theory, 40 hours) and 1 ECTS (Exercises, 12 hours). Both part will be held as in class lectures.
During the course some cases of study will be assigned to groups of 4-5 student and will be discussed.
Ivar Ekeland and Roger Témam, Convex Analysis and Variational Problems, Ed. SIAM (1987)
F.H. Clarke, Y.S. Ledyaev, Ronald J. Stern, P.R. Wolenski, Nonsmooth Analysis and Control Theory, Ed. Springer-Verlag New York Inc. (1998)
Frank H. Clarke, Optimization and Nonsmooth Analysis, Ed. SIAM (1990)
|Ralph Tyrrell Rockafellar||Convex Analysis||Princeston University Press||1997||9780691015866|
|Ivar Ekeland and Roger Témam||Convex Analysis and Variational Problems||SIAM||1987||0-89871-450-8|
|F.H. Clarke, Y.S. Ledyaev, Ronald J. Stern, P.R. Wolenski||Nonsmooth Analysis and Control Theory||Springer-Verlag New York Inc.||1998||0387983368|
|Frank H. Clarke||Optimization and Nonsmooth Analysis||SIAM||1990||0-89871-256-4|
The exam is divided into a written and an oral test, the two tests must be passed in the same exam session. There are no difference between the assessment of attending or non-attending students.
There will be also two partial test, one at the mid of the semester (indicatively, end of November), and the other one at the end of the semester. The first part will concern the first part of the program (until the introduction to the Calculus of Variations included), and the second on the remaining part of the program. The students who will pass both the partial tests, can directly access to the oral part in the exam session of February.
After the oral part, the teacher will propose the final mark (on the Italian ranking system from 18 to 30).
Structure of the tests
The written test is concerns three exercise, and each of them will have the same contribute to the final mark. The first two exercises (one on the first part and the other on the second part of the program) will require the solution of specific problems. The third will be composed of questions on the whole of the program or on the material given to the students, asking for short open answers.
Each of the partial test will concern the relative part of the program, and will be made of three exercise. The first two will require the solution of specific problems, and the third e il terzo will be composed of questions on the relative part of the program or on the material given to the students, asking for short open answers. It will be mandatory for the student to solve the third exercise and choose one between the first and the second.
The oral part will test the whole of the program of the course.
Targets of the assessment procedure
- Knowledge and understanding: a part of the written and the oral tests will be devoted to verify the effective knowledge and understanding of the course's contents (mainly, the third exercise of the written test and the oral test).
- Applying knowledge and understanding: both during the written and the oral tests, the student will be required to solve problems based on the course's contents.
- Making judgements: during the tests, the student can be asked to solve problems requiring a contribution basing on the material of the course assigned for personal study.
- Communication skills: during the written and the oral tests, the solutions expressed in a clear, complete and short way will be preferred.
- Learning skills: part of the course's contents will be based on textbook or scientific articles left to the students for personal study.