New Proximal Methods for Variational Inequalities
(R Burachik with J Dutta)
Available convergence results for inexact versions of Generalized
Proximal Point Methods (GPPM) require strong assumptions on the original
problem such as paramonotonicity or use summable error criteria. Moreover,
each distance-like function usually requires its own convergence analysis.
This motivates the following natural questions:
Q1) Can new inexact versions
of GPPM be devised, in such a way that standard assumptions (such as
paramonotonicity) on the original problem are no longer necessary?
Q2) Can
the same convergence analysis be used for a wide family of distance-like
functions?
Q3) Can the resulting inexact iterations use the most recent
approach, of relative error analysis?
The aim of our project is to address the three questions posed above.
More precisely, we aim to:
A1) Define a common, inexact iteration using
relative error analysis for the subproblems of GPPM, which (i) can be posed
for a wide family of distance-like functions, and (ii) it includes the
existing ones as a particular case (thus addressing (Q2) and (Q3) above).
A2) Define a new family of distance-like functions (which includes the
Euclidean distance and the Second Order Kernels) such that (i) strong
assumptions on the problem are no longer required, and (ii) the same
convergence analysis can be used for all the members of the family (thus
addressing (Q1) and (Q3) above).
Researchers: R. S. Burachik (UniSA) and J. Dutta (Indian Institute of Technology, Kanpur)
Funding
IRSS