METODI MATEMATICI PER L'OTTIMIZZAZIONE

The course aims at presenting well-known optimization methods. The course provides students with the analytic tools to model and solve numerically situations in which a single decision-maker has to find the best choice. The attention focuses on applications in economics, engineering, and computer science.

Learning outcomes:

Students should be able to classify optimization problems according to their mathematical properties.

Students should be able to perform a theoretical investigation of a given optimization problem in order to access its complexity.

Students should be able to write down first and second-order optimality conditions.

Students should be able to solve simple optimization problems without computer.

Students should be able to solve optimization problems in a computer environment.

Students should be able to analyze the obtained solutions.

The goals of the course are:

Knowledge and understanding: to acquire base knowledge that allows students to study optimization problems and apply opportune techniques to solve the decision-making problems. The students will be able to use algorithms for nonlinear programming problems.
Applying knowledge and understanding: to identify and model real-life decision-making problems. In addition, through real examples, the student will be able to find correct solutions for complex problems.
Making judgments: to choose and solve autonomously complex decision-making problems and to interpret the solutions.
Communication skills: to acquire base communication and reading skills using technical language.
Learning skills: to provides students with theoretical and practical methodologies and skills to deal with optimization problems, ranging from computer science to engineering; to acquire further knowledge on the problems related to applied mathematics.

For this course, there will be 2 hours of teaching per lecture twice a week. During the classroom lessons a graphics tablet will be used. The hand-written slides will be available. For each topic, exercises will be solved by the teacher or proposed to students.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

The course will cover the following topics:

One-dimensional optimization: convex and quasiconvex functions, first-order methods, local and global minima.

Existence of solutions: continuous and lower semicontinuous functions, coercive functions, Weierstrass theorem, unique and nonunique solutions.

Theory of optimality conditions: Fermat principle, the Hessian matrix, positive and negative semidefinite matrices, the Lagrange function and Lagrange multipliers, tangent cones, the Karush-Kuhn-Tucker conditions, regularity, complementarity constraints, stationary points.

First-order optimization methods: the steepest descent method, conjugate directions, gradient-based methods.

Second order optimization methods: Newton's method and modifications.

Convex optimization: optimality conditions, duality, subgradients and subdifferential.

Multiobjective optimization: Pareto frontiers and solution methods.

Teaching material will be given during the course.

Oral exam and resolution of a numerical example.

Learning assessment may also be carried out on line, should the conditions require it.