Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 23 January 2018, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Keita Yokoyama

Title: Ekeland’s variational principle in reverse mathematics

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract: Ekeland’s variational principle is a key theorem used in various

areas of analysis such as continuous optimization, fixed point theory and

functional analysis. It guarantees the existence of pseudo minimal

solutions of optimization problems on complete metric spaces. Let f be

a positive real valued continuous (or lower semi-continuous) function

on a complete metric space (X,d). Then, a point x in X is said to be a

pseudo minimum if f(x)=f(y)+d(x,y) implies x=y. Now, Ekeland’s

variational principle says that for any point a in X, there exists a

pseudo minimum x such that f(x)<=f(a)-d(a,x). In reverse

mathematics, it is observed that many theorems for continuous

optimization problems are provable within the system of arithmetical

comprehension (ACA-0), and thus most such problems have arithmetical

solutions. However, this is not the case for pseudo minima. We will

see that Ekeland’s variational principle or its restriction to the

space of continuous functions C([0,1]) are both equivalent to

Pi-1-1-comprehenstion. This is a joint work with Paul Shafer and David

Fernandez-Duque.