WC-01: Analysis of Non Linear Algorithms I
Stream: Analysis of Non Linear algorithms
Room: Fermat
Chair(s): Tobias Seidel

Duality and penalty function methods for a nonlinear programming problem
Andrei Halanay, Miruna Beldiman
We introduce a multiobjective nonlinear programming problem defined on a product of sigma-algebras. First, using a partitioning scheme we propose a dual model for our problem and obtain duality results under different generalized univexity hypothesis. Then, considering a pseudometric corresponding to this context we study two resolution methods using an exponential penalty function and then a logarithmic one. We prove the convergence for these methods, closing with some examples.

New Results on Superlinear Convergence of Classical Quasi-Newton Methods
Anton Rodomanov, Yurii Nesterov
We present a new theoretical analysis of local superlinear convergence of classical quasi-Newton methods from the convex Broyden class. As a result, we obtain a significant improvement in the currently known estimates of the convergence rates for these methods. In particular, we show that the corresponding rate of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method depends only on the product of the dimensionality of the problem and the logarithm of its condition number.

Calculating local and global minima with quADAPT
Tobias Seidel, Karl-Heinz Kuefer
A classical solution approach to semi-infinite optimization is based on discretizing the semi-infinite index set. To reduce the number of discretization points, they can be chosen adaptively. Recently a new adaptive discretization method (quADAPT) has been introduced, which guarantees a quadratic convergence of the iterates under mild regularity assumptions. In this talk we establish conditions under which a limit point is a local minimum. We show that no such result is possible in the case of global minima. Finally, we introduce modifications to the algorithm that make a statement possible.