- ECTS Credits: 3
- Semester: 1
- Year: 2
- Campus: Ivan Franko National University of Lviv
- Language: English
- Code: DT0837
- Unit Coordinator: Giuseppe Pipoli
- ECTS Credits: 6
- Semester: 2
- Year: 1
- Campus: University of L'Aquila
- Aims:
Learning Objectives.
The course aims at providing the basics of the geometry of curves and surfaces, continuous and discrete.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence was verified at the time of the planning of the program.
Learning Outcomes
At the end of the course, the student should
1. know the fundamentals of geometry of curves and surfaces, as tangent and normal space, curvature in the smooth and discrete setting;
2. understand and be able to explain thesis and proofs in geometry of curves and surfaces;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics. - Content:
Geometry of curves: parametrization, Frenet frame, curvature and torsion.
Geometry of surfaces: parametrization, tangent and normal spaces, mean curvature and Gaussian curvature.Discrete curves and surfaces, discrete curvature.
- Pre-requisites:
Calculus in several variables, fundamental of Mathematical Analysis.
- Reading list:
Manfredo Do Carmo, Differential geometry of curves and surfaces.
Alexander I. Bobenko, Yuri B. Suris, Discrete Differential Geometry: Integrable Structure.
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: Brno University of Technology
- Unit Coordinator: Fabrizio Rossi, Andrea Manno
- ECTS Credits: 6
- Semester: 2
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Aims:
Learning Objectives
The course provides fundamental descriptive, predictive, and prescriptive analytics methodologies and software tools to examine raw data with the purpose of drawing data-driven decisionsLearning Outcomes
1. Descriptive and Predictive analytics
Knowledge of:
Statistical foundations of machine learning.
Main methods for supervised and unsupervised learning.
State-of-the-art techniques to extract information from data to orient decisions.
Practical ability to extract relevant information from raw data.2. Prescriptive analytics
Ability to recognize and model decision problems as mathematical optimization problems, with particular emphasis on Integer Linear Programming models.
Knowledge of fundamental algorithmic techniques for solving large scale Integer Linear Programming problems.
Ability to solve practical Integer Programming models with state-of-the-art solvers. - Content:
1. Descriptive and Predictive analytics (48 hours)
Introduction to analytics.
Statistical methods to support data analysis and learning.
Optimization models for data analysis and learning problems: regression models.
Data preprocessing, model validation, feature selection and extraction.
Clustering and Principal Component Analysis.
Outliers detection.
Performance metrics for learning models.Overview of models for supervised and unsupervised learning:
Logistic Regression, Decision Trees.
Support Vector Machines, Neural NetworksLearning from imbalanced data
Introduction to Time Series Analysis.2. Prescriptive analytics: Decision Optimization (48 hours)
Introduction to Mathematical Programming: linear, non-linear and integer linear programming models.
Integer programming models for fundamental combinatorial optimization problems:
Assignment Problem; Set Covering, Packing and Partitioning Problems; Minimum Spanning Tree Problem; Traveling Salesperson Problem (TSP). Formulations of logical conditions and fixed costs.
Uncapacitated Facility Location; Uncapacitated Lot Sizing; Discrete Alternatives; Disjunctive Formulations.Geometry of Integer Programming.
Optimality, Relaxation and Bounds.
Polyhedra: dimension, representations, valid inequalities, faces, vertices and facets. Good and Ideal formulations.LP based branch-and-bound algorithm:
Preprocessing, primal heuristics and branching strategies.
Cutting Planes algorithms. Automatic Reformulation: Gomory's Fractional Cutting Plane Algorithm.
Examples of strong valid inequalities: cover and clique inequalities.
Branch-and-cut algorithm.
Applications
Constrained Spanning Tree Problems; Constrained Shortest Path Problem; Clustering problems; Multicommodity Flows; Symmetric and Asymmetric Traveling Salesperson Problem; Vehicle Routing Problem; Steiner Tree Problem.Integration of predictive and prescriptive techniques in industrial applications.
- Pre-requisites:
Basic knowledge of: Statistics, Discrete Mathematics, Linear Programming,
Algorithms and Data Structures. Basic coding ability in Python. - Reading list:
Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani, An Introduction to Statistical Learning with applications in Python, Springer Texts in Statistics 2023.
Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong Mathematics for Machine Learning, Cambridge University Press 2020.
L.A. Wolsey, Integer Programming (Second Edition), Wiley. 2021.
