## Contributed Session Fri.2.H 2053

#### Friday, 13:15 - 14:45 h, Room: H 2053

**Cluster 9: Global optimization** [...]

### Advances in global optimization V

**Chair: Zulfiya Ravilevna Gabidullina**

**Friday, 13:15 - 13:40 h, Room: H 2053, Talk 1**

**Giancarlo Bigi**

Beyond canonical DC programs: The single reverse polar problem

**Coauthors: Antonio Frangioni, Qinghua Zhang**

**Abstract:**

We introduce the single reverse polar problem as a novel generalization of the canonical DC problem (CDC), and we extend to the former the outer approximation algorithms based on an approximated oracle, which have been previously proposed for the latter. In particular, we focus on the polyhedral case (PSRP), in which the problem amounts to a linear program with a single bilinear constraint which renders it nonconvex. Several important classes of nonconvex optimization problems (e.g., bilevel linear, integer and linear complementarity problems) can be easily formulated as a PSRP. In principle, this is true also for CDC, as most nonconvex programs have a DC representation, but the reformulation as a DC program is substantially more difficult to derive. Furthermore, the outer approximation algorithms for PSRP do away with some of the core assumptions required by the algorithms for the CDC case: These assumptions are not trivial to satisfy in practice and indeed cannot hold for some important classes of problems.

**Friday, 13:45 - 14:10 h, Room: H 2053, Talk 2**

**Simon Konzett**

Numerical enclosures of solution manifolds at near singular points

**Coauthor: Arnold Neumaier**

**Abstract:**

This work considers nonlinear real parameter-dependent equations for global optimization. By using interval analysis rigorous enclosures of solution paths dependent on a parameter are determined. Especially the method provides enclosures of (near) singular points of the solution manifold. The method uses an augmentation of the original problem to correct the singularities. Then a low-dimensional problem is deduced of the augmented problem which reflects locally the behaviour of the original problem. In particular the singular behaviour is reflected. The application of the method is illustrated with some numerical results to show the efficiency and robustness of the method.

**Friday, 14:15 - 14:40 h, Room: H 2053, Talk 3**

**Zulfiya Ravilevna Gabidullina**

Universal measure of the thickness of separator or pseudo-separator for sets of Euclidean space

**Abstract:**

At present, different approaches to linear separation of

the sets in a finite-dimensional Euclidean space are effectively used in medical diagnostics, discriminant analysis, pattern recognition etc. These approaches represent a particular interest from both theoretical and practical point of view.

In this contributed talk we define a separator and

pseudo-separator (the margin of unseparated points of sets) for

the sets of Euclidean space by help of generalized supporting

hyperplanes. We introduce new universal measure for estimation of the thickness of separator (when the sets are disjoint) as well as of pseudo-separator (when the sets are inseparable). The optimization problem maximizing the thickness of the separator and minimizing the thickness of the pseudo-separator is considered.