Geometric constraints in descent methods for shape optimisation

University of Graz, Institute of Mathematics, NAWI Graz, Universitätsplatz 3, 8010 Graz, Austria.
d@domob.eu

Received: 16 November 2015
Revised: 7 April 2017
Accepted: 18 April 2017

Abstract

Many shape-optimisation problems arising from practical applications feature geometric constraints. We are particularly interested in the situation that certain regions of a hold-all domain should always or never be part of the optimal shape. This can be used, for instance, to model design constraints or to include additional information into the optimisation process. In the context of descent methods, there are two fundamental ways to account for constraints by means of projections: one can project the descent direction before taking a step to stay feasible, or one can project the resulting point back to the feasible region after taking a step. The latter is usually called projected-gradient method and is more commonly used. For shape optimisation, both approaches create additional difficulties that are not present in the classical context of optimisation in vector spaces. In this paper, we analyse these issues. We are able to show that certain conditions ensure that both approaches behave in a very similar way, although one or the other can be better suited to special situations. Our theoretical results are confirmed by numerical experiments based on a Chan–Vese-like model for image segmentation.