30552  ADVANCED ANALYSIS AND OPTIMIZATION  MODULE 2
Department of Decision Sciences
ALESSANDRO PIGATI
Suggested background knowledge
Mission & Content Summary
MISSION
CONTENT SUMMARY
 Convex analysis and convex optimization:
 Convex sets, convex hull, and operations preserving convexity
 Realvalued convex functions and subdifferential
 HahnBanach separation theorem and l.s.c. convex functions
 Duality and LegendreFenchel transform
 LP duality and more general forms of convex duality
 Minimax theorem of Von Neumann, with application to equilibria in zerosum 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.
 Nonconvex optimization:
 Constrained gradient descent
 Constrained optimization: Lagrange multipliers.
 Brouwer fixed point theorem and applications to topology and Nash equilibria in game theory.
Intended Learning Outcomes (ILO)
KNOWLEDGE AND UNDERSTANDING
 Know fundamental notions in convex analysis: convex sets, convex functions, subdifferential, lower semicontinuity, separation, LegendreFenchel 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.
APPLYING KNOWLEDGE AND UNDERSTANDING
 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.
Teaching methods
 Facetoface lectures
 Exercises (exercises, database, software etc.)
DETAILS
Online lectures have the same conceptual role as facetoface lectures. The actual choice between offering facetoface lectures, online lectures, or both will depend on the university's policies.
Exercise sessions (again: both faceto face and online) are dedicated to the application of the main theoretical results obtained during lectures to problems and exercises of various nature.
Assessment methods
Continuous assessment  Partial exams  General exam  


x  x 
ATTENDING AND NOT ATTENDING STUDENTS
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 multiplechoice questions and openanswer questions; each partial weighs for onehalf of the final mark. Multiplechoice 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 openanswer 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 multiplechoice questions and openanswer 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 twopartials procedure or could not follow it. Each type of question contributes in a specific way to the assessment of the students' acquired knowledge. Multiplechoice 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 openanswer 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.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
 Lecture notes (main reference, updated regularly on Blackboard)
 M. HiriartUrruty, JeanBaptiste; Lemaréchal, Claude. Fundamentals of convex analysis.
Grundlehren Text Editions. SpringerVerlag, Berlin, 2001. ISBN: 35404220569001
 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: 9780817643744; 0817643745