**Monday, 16:15 - 16:40 h, Room: H 2053**

**Achim Wechsung**

Improving relaxations of implicit functions

**Coauthor: Paul I. Barton**

**Abstract:**

A factorable function **f**:Y → **R**^{m}, *Y ⊂ ***R**^{n} can be represented as a DAG. While it is natural to construct interval extensions of factorable functions, the DAG representation has been shown to also enable the backward propagation of interval bounds on the function's range, i.e., to provide an enclosure of the intersection of *Y* with the function's pre-image. One application is to eliminate points in the domain where no solution of **f**(**y**)=**0** exists. This idea can be extended to the case of constructing convex relaxations of implicit functions. When *n>m*, it is possible to partition *Y* into *X ⊂ ***R**^{m} and *P ⊂ ***R**^{n-m}. Assuming that *X* and *P* are intervals and that there exists a unique **x**:P → X such that **f**(**x**(**p**),**p**)=**0**, it is then possible to construct relaxations of the implicit function **x** using the DAG representation of **f**, backward propagation and generalized McCormick relaxation theory. These relaxations can be used to initialize other methods that improve relaxations of implicit functions iteratively.

Talk 3 of the invited session Mon.3.H 2053

**"Algorithms and relaxations for nonconvex optimization Problems"** [...]

Cluster 9

**"Global optimization"** [...]