Oberwolfach Reports

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Volume 14, Issue 4, 2017, pp. 3071–3120
DOI: 10.4171/OWR/2017/52

Published online: 2018-12-18

Copositivity and Complete Positivity

Abraham Berman[1], Immanuel M. Bomze[2], Mirjam Dür[3] and Naomi Shaked-Monderer[4]

(1) Technion - Israel Institute of Technology, Haifa, Israel
(2) Universität Wien, Austria
(3) Universität Augsburg, Germany
(4) The Max Stern Yezreel Valley College, Yezreel Valley, Israel

A real matrix $A$ is called copositive if $x^TAx \ge 0$ holds for all $x \in \mathbb R^n_+$. A matrix $A$ is called completely positive if it can be factorized as $A = BB^T$ , where $B$ is an entrywise nonnegative matrix. The concept of copositivity can be traced back to Theodore Motzkin in 1952, and that of complete positivity to Marshal Hall Jr. in 1958. The two classes are related, and both have received considerable attention in the linear algebra community and in the last two decades also in the mathematical optimization community. These matrix classes have important applications in various fields, in which they arise naturally, including mathematical modeling, optimization, dynamical systems and statistics. More applications constantly arise.

The workshop brought together people working in various disciplines related to copositivity and complete positivity, in order to discuss these concepts from different viewpoints and to join forces to better understand these difficult but fascinating classes of matrices.

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Berman Abraham, Bomze Immanuel, Dür Mirjam, Shaked-Monderer Naomi: Copositivity and Complete Positivity. Oberwolfach Rep. 14 (2017), 3071-3120. doi: 10.4171/OWR/2017/52