**Monday, 15:45 - 16:10 h, Room: MA 005**

**Dennis Michaels**

The convex hull of vectors of functions

**Coauthors: Martin Ballerstein, Robert Weismantel**

**Abstract:**

A challenging task in global optimization is to construct tight convex relaxations that provide reasonably globally valid bounds on a mixed-integer nonlinear program (MINLP). For a general MINLP, convex relaxations are usually obtained by replacing each non-linearity by convex under- and concave overestimators. The mathematical object studied to derive such estimators is given by the convex hull of the graph of the function over the relevant domain.

To derive improved relaxations, we consider a finite set of

given functions as a vector-valued function and study the

convex hull of its graph. We establish a link between such

a convex hull object and the convex hulls of the graphs of a certain family of real-valued functions. This link can be used to define improved relaxations. We especially focus on small sets of well-structured univariate functions. Numerical examples are presented demonstrating the impact of this concept.

Talk 2 of the invited session Mon.3.MA 005

**"Tight relaxations"** [...]

Cluster 14

**"Mixed-integer nonlinear programming"** [...]