# METODI MATEMATICI PER L'OTTIMIZZAZIONE

Academic Year 2023/2024 - Teacher: Laura Rosa Maria SCRIMALI

## Expected Learning Outcomes

The course aims to present well-known optimization methods. The course provides students with the analytic tools to model and solve numerical 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 assess 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 a 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 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 provide 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.

## Course Structure

Teaching Organization

credit value 6 - 47 hours

total study 150 hours

103 hours of individual study
35 hours of frontal lecture

12 hours of exercises

No requirements.

## Detailed Course Content

The course will cover the following topics:

Fundamentals concepts (6 hours). 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.

Optimality conditions (15 hours). 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.

Methods (6 hours). First-order optimization methods: the steepest descent method, conjugate directions, and gradient-based methods. Second order optimization methods: Newton's method and modifications.

Nondifferentiable optimization (3 hours). Subgradients, subdifferential and methods.

Multiobjective optimization (5 hours). Pareto frontiers and solution methods.

Exercises (12 hours)

## Course Planning

SubjectsText References
1Decision models[4], teaching material
2Convex sets, convex functions[P], [1], [2], [3] teaching material
3Cone, tangent cone, normal cone [P], [1], [3], teaching material
4Optimality conditions for unconstrained optimization[P], [1], [2], teaching material
5Optimality conditions for constrained optimization[P], [1], [2], [3], teaching material
6Duality[P], [1], [2], [3] teaching material
7Solution methods[P], [4], teaching material

## Learning Assessment

### Learning Assessment Procedures

The final exam consists of an oral test during which the candidate is also requested to solve a numerical exercise. The final grade is established on the basis of the answers given by the candidate and the solving of the numerical example.

Final grades will be assigned taking into account the following criteria:

Rejected: Basic knowledge has not been acquired. The student is not able to solve simple exercises.

18-23: Basic knowledge has been acquired. The student solves simple exercises, has sufficient communication skills, and makes judgements.

24-27: All the knowledge has been acquired. The student solves all the proposed exercises making few errors and has good communication skills and making judgements.

28-30 cum laude: All the knowledge has been completely acquired. The student applies knowledge and has excellent communication skills, learning skills and making judgements.

Learning assessment may also be carried out online if the conditions should require it.

### Examples of frequently asked questions and / or exercises

Descent directions. duality, gradient method, Newton method, penalty method.