Optimization basics

Summary of the course

This is an introduction to the theory of optimization. We will study the optimality conditions of an optimization problem. The course is organized around five parts: Prerequisites, Unconstrained optimization, Optimization with linear constraints, Constrained optimization, Lagrangian duality.

Prerequisites

Basic numerical linear algebra, differential calculus and convex analysis.

Introduction to machine learning

Title : Linear and Quadratic Programming

Summary of the course : The course covers the basic aspects of linear (LP) and quadratic programming (QP), that is, optimization of linear or quadratic functions subject to linear equalities and inequalities. We address the duality theory of LP, the simplex method, complementarity and optimality conditions, interior-point methods, classical methods from optimization for QP. Examples from applications in data science, decision theory and financial mathematics will be analysed, such as portfolio optimization, scheduling, transportation problems, network design. Practical experiments will be performed with Matlab/Scilab.

Prerequisites : Basic linear algebra and convex geometry.

Keywords : LP, QP, duality, optimality conditions, simplex method, interior-point method, software experiments.

Stochastic processes

Title : Probability and stochastic processes

Summary of the course :

The aim of this course is to introduce some important classes of stochastic processes, both in discrete and continuous time, which are relevant in Biology, Operation Research, Physics and Social Sciences.

In the first part of the course, we will recall some basic notions in measure theory and their role in the foundations of modern probability theory. Then, after introducing random variables, expectations and conditional expectations, we will cover the basic theory of stochastic processes and their applications. The main topics here will be random walks and discrete and continuous time Markov chains. The course will also cover some important applications in mathematical biology and, if time permits, in mathematical finance.

Prerequisites :

Basic probability, linear algebra and measure theory.

Keywords :

Random variables, conditional expectations, stochastic processes, Markov chains in discrete and continuous time, applications in mathematical biology and in mathematical finance.

Practical Optimization

The course Practical Optimization will be done on 5 weeks and starts on March 04, 2024

Optimization is used everywhere in all scientific fields, in high technology, and in industry. In recent fields like artificial intelligence, and in particular in deep learning, optimization is at the heart of the algorithms. The goal of this course is to learn how to effectively solve an optimization problem. We will see how to model an optimization problem and to compute an optimal solution in some concrete situations. We will basically work on optimality conditions of optimization problems and study how to numerically solve them with the help of a powerful modeling language AMPL.