30552 - ADVANCED ANALYSIS AND OPTIMIZATION - MODULE 2

For a fruitful and effective learning experience, it is recommended to be fluent with multivariable calculus (limits, partial derivatives, integrals) and basic linear algebra (vector spaces, linear maps, change of basis, diagonalization).

The purpose of this course is to present the foundations of convex analysis, convex optimization, and basic results on the local structure of embedded differentiable manifolds and the corresponding constrained optimization. This is fundamental in any program involving mathematical and computing sciences: the theory and applications encountered in this course will create a strong foundation to solve unconstrained and constrained optimization problems appearing in real-world models, in settings as diverse as operations research, economics and game theory, finance, physics, and so on.

• Convex analysis and convex optimization:
  • Convex sets, convex hull, and operations preserving convexity
  • Real-valued convex functions and subdifferential
  • Hahn-Banach separation theorem and l.s.c. convex functions
  • Duality and Legendre-Fenchel transform
  • LP duality and more general forms of convex duality
  • Minimax theorem of Von Neumann, with application to equilibria in zero-sum games with two players.
  • If time allows: convergence rate of gradient flows.

• Basics of differential geometry:
  • Inverse and implicit function theorem
  • Equivalent definitions of embedded differentiable manifolds
  • Tangent space and differential of smooth maps between manifolds
  • If time allows: hints towards Riemannian geometry and curvature.

• Non-convex optimization:
  • Constrained gradient descent
  • Constrained optimization: Lagrange multipliers.

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

• Know fundamental notions in convex analysis: convex sets, convex functions, subdifferential, lower semicontinuity, separation, Legendre-Fenchel transform, duality.
• Express basic notions and results about convex optimization.
• Understand the relevance of the fundamental definitions and theorems about manifolds: concrete examples, parametrization, tangent space, differential of maps.
• Know the fundamental properties of solutions to constrained optimization problems.

• Solve constrained and unconstrained optimization problems in simple settings.
• Operate with objects and tools from 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 choice between offering face-to-face lectures, online lectures, or both will depend on the university's policies.

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 into 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 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.
 

• Lecture notes (main reference, updated regularly on Blackboard)
   
• M. Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude. Fundamentals of convex analysis.
  Grundlehren Text Editions. Springer-Verlag, Berlin, 2001. ISBN: 3-540-42205-690-01
   
• Boyd, Stephen; Vanderberghe, Lieven. Convex Optimization.
  Cambridge University Press, 2004. ISBN: 9780521833783
  Available also online: 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. ISBN: 978-0-8176-4374-4; 0-8176-4374-5