Title: New Constraint Qualifications for Optimization Problems in Banach Spaces based on Asymptotic KKT Conditions

Speaker: Gerd Wachsmuth (BTU)

Date and Time: September 2nd, 2020, 17:00 AEST (Register here for remote connection via Zoom)

Abstract: Optimization theory in Banach spaces suffers from the lack of available constraint qualifications. Despite the fact that there exist only a very few constraint qualifications, they are, in addition, often violated even in simple applications. This is very much in contrast to finite-dimensional nonlinear programs, where a large number of constraint qualifications is known. Since these constraint qualifications are usually defined using the set of active inequality constraints, it is difficult to extend them to the infinite-dimensional setting. One exception is a recently introduced sequential constraint qualification based on asymptotic KKT conditions. This paper shows that this so-called asymptotic KKT regularity allows suitable extensions to the Banach space setting in order to obtain new constraint qualifications. The relation of these new constraint qualifications to existing ones is discussed in detail. Their usefulness is also shown by several examples as well as an algorithmic application to the class of augmented Lagrangian methods.

This is a joint work with Christian Kanzow (Würzburg) and Patrick Mehlitz (Cottbus).

Title: Two approaches for absolute value equation by using smoothing functions

Speaker: Jein-Shan Chen (NTNU)

Date and Time: August 26th, 2020, 17:00 AEST (Register here for remote connection via Zoom)

Abstract: In this talk, we present two approaches for solving absolute value equation.These two approaches are based on using some smoothing function. In particular, thereare several systematic ways of constructing smoothing functions. Numerical experimentswith comparisons are reported, which suggest what kinds of smoothing functions workwell along with the proposed approaches.

Title:  Projection Algorithms for Phase Retrieval with High Numerical Aperture

Speaker: Hieu Thao Nguyen (TU Delft)

Date and Time: August 19th, 2020, 17:00 AEST (Register here for remote connection via Zoom)

Abstract: We develop the mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework we first analyze the basic steps of solving this problem by projection algorithms and establish the closed forms of all the relevant prox-operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this work and the literature of phase retrieval mathematics which mostly deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that all available solution approaches for high-NA phase retrieval are outperformed by projection methods.