Time: 09h00 - 10h00, Tuesday December 20th, 2022

Location: Room 617, building A6, Institute of Mathematics (18 Hoang Quoc Viet, Cau Giay, Hanoi)

Abstract: The notion of error bound is a widely used concept in applied mathematics and thereby has received a lot of attention in the last years and decades. Indeed, it plays a key role in areas including variational analysis, mathematical programming, convergence properties of algorithms, sensitivity analysis, designing solution methods for non-convex quadratic problems, penalty functions, optimality conditions, weak sharp minima, stability and well-posedness of solutions, (sub)regularity and calmness of set-valued mappings, and subdifferential calculus. In this regard, Hoffman's estimation, as the starting point of the theory of error bounds, is very important and plays a considerable role in optimization. In this presentation, we will provide some sufficient criteria under which the function $f$, acting either between metrizable topological vector spaces or between metrizable subsets of some topological vector spaces, satisfies the error bound property at a point $bar{x}$. Then, we will discuss the Hoffman estimation and obtain some results for the estimate of the distance to the set of solutions to a system of linear equalities. Some applications of this presentation are illustrated by some examples. 

The talk is based on this paper:
M. Abassi, M. Théra. Strongly regular points of mappings. Fixed Point Theory Algorithms for Sciences and Engineering, 14 (2021), https://doi.org/10.1186/s13663-021-00699-z