1. Speaker: Prof. Panos M. Pardalos, University of Florida, USA

http://www.ise.ufl.edu/pardalos/ & https://toxeus.org

Short CV

Title: Computational Approaches for Solving Systems of Nonlinear Equations

Abstract:

Finding one or more solutions to a system of nonlinear equations (SNE) is a
computationally hard problem with many applications in sciences and engineering.
First, we will briefly discuss classical approaches for addressing (SNE).
Then, we will discuss the various ways that a SNE can be transformed
into an optimization problem, and we will introduce techniques that can be utilized to search for solutions to the global optimization problem that arises when the most common reformulation is performed.
In addition, we will present computational results using different heuristics.

2. Speaker: Prof. Alexey Tret'yakov, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

https://www.researchgate.net/profile/Alexey_Tretyakov

Title: Degenerate Equality Constrained Optimization Problems and P-Regularity Theory

Abstract:

We consider necessary optimality conditions for optimization problems
with equality constraints given in the operator form as $F(x)=0$,
where $F$ is an operator between Banach spaces.
The paper addresses the case when the
Lagrange multiplier $\lambda_0$ associated with the objective function might be equal to zero.
If the equality constraints are not regular at some point $\xz$ in the sense that the Fr\'echet derivative of $F$ at $\xz$ is not onto,
then the point $\zz=(\xz, \lz_0, \lz)$ is a degenerate solution of the classical Lagrange system of optimality conditions ${\mathcal{L}}(x, \lambda_0, \lambda)=0$, where $\xz$ is a solution of the optimization problem and $(\lz_0, \lz)$ is a corresponding generalized Lagrange multiplier.
We derive new conditions that guarantee that $\zz$ is a locally unique solution
of the Lagrange system. We also introduce a modified Lagrange system
and prove that $\zz$ is its regular locally unique solution.
The modified Lagrange system introduced in the paper can be used as a basis for constructing numerical methods for solving degenerate optimization problems.
Our results are based on the construction of $p$--regularity and are illustrated by examples.

This is joint work with Olga Brezhneva, Yuri Evtushenko, and Vlasta Malkova.