PhD course by Marianna De Santis, Univ di Roma La Sapienza

Most real-world optimization problems in the areas of applied sciences, engineering and
economics involve multiple, often conflicting and nonlinear, goals. In the mathematical
model of these problems, under the necessity of reflecting discrete quantities, logical
relationships or decisions, integer and 0-1-variables need to be considered. We are
in the context of MultiObjective Mixed Integer Nonlinear Programming (MO-MINLP).

The design of efficient solution methods for MO-MINLP is a big challenge for people working in optimization, as these problems combine all the difficulties of both multiobjective problems and mixed integer nonlinear programming problems.

In this short course, after an introduction to the basic concepts and definitions of multiobjective optimization, we will give an overview on existing approaches for solving multiobjective mixed integer nonlinear programming problems.
We will present methods belonging to two main classes: decision space search algorithms (methods that work on the decision variables space) and criterion space search algorithms (methods that work in the space of the objectives).

The focus will be on exact algorithms, namely methods able to detect the complete nondominated set of a multiobjective mixed integer nonlinear programming problem.
We will discuss and show some theoretical results related to the exactness of the algorithms presented.