#### Title of Poster / Presentation

The DC Algorithm & The Constrained Fermat-Torricelli Problem

#### Location

Portland State University

#### Start Date

4-5-2016 12:00 PM

#### End Date

4-5-2016 2:00 PM

#### Subjects

Mathematical optimization, Algorithms

#### Abstract

The theory of functions expressible as the Difference of Convex (DC) functions has led to the development of a rich field in applied mathematics known as DC Programming.We survey the work of Pham Dinh Tao and Le Thi Hoai An in order to understand the DC Algorithm (DCA) and its use in solving clustering problems. Further, we present several other methods that generalize the DCA for any norm. These powerful tools enable researchers to reformulate objective functions, not necessarily convex, into DC Programs.

The Fermat-Torricelli problem is visited in light of convex analysis and various norms. Pierre de Fermat proposed a problem in the 17th century that sparked interest in the location sciences: given three points in the plane, find a point such that the sum of its Euclidean distances to the three points is minimal. This problem was solved by Evangelista Torricelli, and is now referred to as the Fermat-Torricelli problem. We present the constrained version of the problem using the distance penalty method.

Faculty advisor: Nguyen Mau Nam

#### Persistent Identifier

http://archives.pdx.edu/ds/psu/17160

#### Included in

The DC Algorithm & The Constrained Fermat-Torricelli Problem

Portland State University

The theory of functions expressible as the Difference of Convex (DC) functions has led to the development of a rich field in applied mathematics known as DC Programming.We survey the work of Pham Dinh Tao and Le Thi Hoai An in order to understand the DC Algorithm (DCA) and its use in solving clustering problems. Further, we present several other methods that generalize the DCA for any norm. These powerful tools enable researchers to reformulate objective functions, not necessarily convex, into DC Programs.

The Fermat-Torricelli problem is visited in light of convex analysis and various norms. Pierre de Fermat proposed a problem in the 17th century that sparked interest in the location sciences: given three points in the plane, find a point such that the sum of its Euclidean distances to the three points is minimal. This problem was solved by Evangelista Torricelli, and is now referred to as the Fermat-Torricelli problem. We present the constrained version of the problem using the distance penalty method.

Faculty advisor: Nguyen Mau Nam

## Description

This research was partially supported by the USA National Science Foundation under grant DMS-1411817.

The poster was updated in 10/2016.