Title: Continuous selections for inverse mappings in Banach spaces

Speaker: Radek Cibulka (University of West Bohemia)

Date and Time: October 28th, 2020, 17:00 AEDT (Register here for remote connection via Zoom)

Abstract: Influenced by a recent work by A. V. Arutyunov, A. F. Izmailov,  and S. E. Zhukovskiy,  we establish a general Ioffe-type criterion guaranteeing the existence of a continuous and calm selection for the inverse of a single-valued uniformly continuous mapping between Banach spaces with  a closed domain.  We show that the general statement yields elegant proofs  following  the same pattern as in the case of the usual openness with a linear rate by  considering mappings instead of points. As in the case of the Ioffe’s criterion for linear openness around the reference point, this allows us to avoid the iteration, that is, the construction of a sequence of continuous functions  the limit of which is the desired continuous selection for the inverse mapping, which is illustrated on the proof of the Bartle-Graves theorem.  Then we formulate sufficient conditions based on approximations given by positively homogeneous mappings and bunches of linear operators. The talk is based on a joint work with Marián Fabian.

Title:On diametrically maximal sets, maximal premonotone maps and promonote bifunctions

Speaker: Wilfredo Sosa (UCB)

Date and Time: October 21st, 2020, 17:00 AEDT (Register here for remote connection via Zoom)

Abstract: First, we study diametrically maximal sets in the Euclidean space (those which are not properly contained in a set with the same diameter), establishing their main properties. Then, we use these sets for exhibiting an explicit family of maximal premonotone operators. We also establish some relevant properties of maximal premonotone operators, like their local boundedness, and finally we introduce the notion premonotone bifunctions, presenting a canonical relation between premonotone operators and bifunctions, that extends the well known one, which holds in the monotone case.

Title: A Lyapunov perspective to projection algorithms

Speaker: Björn Rüffer (UoN)

Date and Time: October 14th, 2020, 17:00 AEDT (Register here for remote connection via Zoom)

Abstract: The operator theoretic point of view has been very successful in the study of iterative splitting methods under a unified framework. These algorithms include the Method of Alternating Projections as well as the Douglas-Rachford Algorithm, which is dual to the Alternating Direction Method of Multipliers, and they allow nice geometric interpretations. While convergence results for these algorithms have been known for decades when problems are convex, for non-convex problems progress on convergence results has significantly increased once arguments based on Lyapunov functions were used. In this talk we give an overview of the underlying techniques in Lyapunov’s direct method and look at convergence of iterative projection methods through this lens.