Seminario: Everything is possible: constructing convex sets with prescribed facial dimensions
19 de Agosto, 2025
El Instituto de Ingeniería Matemática y Computacional (IMC) invita al seminario que se dictará el próximo 27 de agosto.
Vera Roshchina, School of Mathematics and Statistics, University of New South Wales.
Miércoles 27 de agosto de 2025, 13:40 hrs. (Presencial en auditorio Edificio San Agustín)
ABSTRACT
Given any finite set of nonnegative integers, there exists a closed convex set whose facial dimension signature coincides with this set of integers, that is, the dimensions of its nonempty faces comprise exactly this set of integers. We show that such sets can be realized as solution sets of systems of finitely many convex quadratic inequalities, and hence are representable via second-order cone programming problems, and are, in particular, spectrahedral. It also follows that these sets are facially exposed, in contrast to earlier constructions. We obtain a lower bound on the minimum number of convex quadratic inequalities needed to represent a closed convex set with prescribed facial dimension signature, and show that our bound is tight for some special cases. We relate the question of finding efficient representations with indecomposability of integer sequences and other topics, and present a substantial number of open questions.
The talk is based on joint work with Levent Tunçel (University of Waterloo, Canada).
BIO
Vera Roshchina is an applied mathematician working on convex and nonsmooth problems that mostly come from optimization. She is currently an Associate Professor at the School of Mathematics and Statistics, UNSW Sydney. Before joining UNSW in 2018 she was an ARC DECRA Research Fellow at RMIT University and held postdoctoral positions at the The University of Melbourne, Federation University Australia and University of Évora (Portugal). In 2021 Vera was awarded the Christopher Heyde medal by the Australian Academy of Science. Vera got her undergraduate degree from St-Petersburg State University and her PhD from the City University of Hong Kong.
Vera Roshchina.
