Course 2022-2023 a.y.

• Convex analysis and optimization:
  • Convex sets, convex functions, lower semicontinuity and closed epigraphs
  • Duality and Legendre-Fenchel transform
  • Subdifferential
  • Saddle points, the minimax theorem of Von Neumann, applications to game theory
  • Gradient flows

• Surfaces and level sets, submanifolds, tangent spaces, parametrizations.

• Inversion and implicit function theorem.

• Applications:
  • Critical points
  • Constrained optimization, Lagrange multipliers, saddle points.

• Brouwer fixed point theorem and applications to Nash equilibria and game theory.

• Know fundamental notions in convex analysis: convex sets, convex functions, lower semicontinuity, Legendre-Fenchel transform, subdifferential.
• Express basic notions and results about convex optimization.
• Understanding the relevance of the fundamental definitions and theorems about surfaces: submanifolds, tangent spaces, parametrizations, implicit function theorem.
• Know the fundamental properties of constrained optimization problems.
   

• Solve constrained and unconstrained optimization problems.
• Operate with objects and tools of convex analysis.
• Make use of the presented methodological tools in applied sciences such as computer science and physics.
   

• Face-to-face lectures
• Exercises (exercises, database, software etc.)

Online lectures have the same conceptual role as face-to-face lectures. The actual blend of face- to-face lectures and online lectures will mainly depend on external constraints.

Exercise sessions (again: both face-to face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.

Students will be evaluated on the basis of written exams, which can be taken in one of the two following ways. 

The exam can be split in two partial exams. Each partial may contain multiple-choice questions and open-answer questions; each partial weighs for one-half of the final mark. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating: 

• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects. 
• The ability to apply mathematical notions to the solution of more complex problems and exercises.

The exam can also be taken as a single general exam, which contains both multiple-choice questions and open-answer questions. The general exam covers the whole syllabus of the course and it can be taken in one of the four general sessions scheduled in the academic year. This option is mainly meant for students who have withdrawn from the two-partials procedure or could not follow it. Each type of question contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions and oral exams mainly aim at evaluating: 

• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects. 
• The ability to apply mathematical notions to the solution of more complex problems and exercises.

 
We will take care to obtain final grades whose distribution follows the grade distribution that is recommended by Università Bocconi. 
 

• M. Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude. Fundamentals of convex analysis.
  Grundlehren Text Editions. Springer-Verlag, Berlin, 2001. x+259 pp. ISBN: 3-540-42205-690-01 
   
• Boyd, Stephen; Vanderberghe, Lieven. Convex Optimization.
  Cambridge University Press, 2004. ISBN: 9780521833783
  https://web.stanford.edu/~boyd/cvxbook/
   
• Giaquinta, Mariano;  Modica, Giuseppe. Mathematical Analysis. An Introduction to Functions of Several Variables.
  Birkhäuser Boston, Inc., Boston, MA,  2009. xx+465 pp. ISBN: 978-0-8176-4374-4; 0-8176-4374-5