30552  ADVANCED ANALYSIS AND OPTIMIZATION  MODULE 2
Department of Decision Sciences
ALESSANDRO PIGATI
Lezioni della classe erogate in presenza
Suggested background knowledge
Mission & Content Summary
MISSION
CONTENT SUMMARY
 Convex analysis and optimization:
 Convex sets, convex functions, lower semicontinuity and closed epigraphs
 Duality and LegendreFenchel 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.
Intended Learning Outcomes (ILO)
KNOWLEDGE AND UNDERSTANDING
 Know fundamental notions in convex analysis: convex sets, convex functions, lower semicontinuity, LegendreFenchel 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.
APPLYING KNOWLEDGE AND UNDERSTANDING
 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.
Teaching methods
 Facetoface lectures
 Exercises (exercises, database, software etc.)
DETAILS
Online lectures have the same conceptual role as facetoface lectures. The actual blend of face toface lectures and online lectures will mainly depend on external constraints.
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 in 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 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.
Teaching materials
ATTENDING AND NOT ATTENDING STUDENTS
 M. HiriartUrruty, JeanBaptiste; Lemaréchal, Claude. Fundamentals of convex analysis.
Grundlehren Text Editions. SpringerVerlag, Berlin, 2001. x+259 pp. ISBN: 35404220569001
 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: 9780817643744; 0817643745