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.
Accepté le :
DOI : 10.1051/m2an/2017020
Mots clés : Shape optimisation, geometric constraints, gradient-descent method, projected gradient, shape projection, image segmentation
@article{M2AN_2017__51_6_2193_0,
author = {Kraft, Daniel},
title = {Geometric constraints in descent methods for shape optimisation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2193--2212},
publisher = {EDP-Sciences},
volume = {51},
number = {6},
year = {2017},
doi = {10.1051/m2an/2017020},
mrnumber = {3745169},
zbl = {1382.49046},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017020/}
}
TY - JOUR AU - Kraft, Daniel TI - Geometric constraints in descent methods for shape optimisation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 2193 EP - 2212 VL - 51 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017020/ DO - 10.1051/m2an/2017020 LA - en ID - M2AN_2017__51_6_2193_0 ER -
%0 Journal Article %A Kraft, Daniel %T Geometric constraints in descent methods for shape optimisation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 2193-2212 %V 51 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017020/ %R 10.1051/m2an/2017020 %G en %F M2AN_2017__51_6_2193_0
Kraft, Daniel. Geometric constraints in descent methods for shape optimisation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2193-2212. doi : 10.1051/m2an/2017020. http://www.numdam.org/articles/10.1051/m2an/2017020/
, A Framework for the Construction of Level Set Methods for Shape Optimization and Reconstruction. Interface Free Bound. 5 (2003) 301–329. | DOI | MR | Zbl
T.F. Chan and L.A. Vese, Image Segmentation Using Level Sets and the Piecewise-Constant Mumford-Shah Model. Technical Report 00-14, UCLA CAM (2000).
M.C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edn. Advances in Design and Control. SIAM (2011). | MR | Zbl
J.W. Eaton, D. Bateman, S. Hauberg and R. Wehbring, GNU Octave version 4.0.0 manual: a high-level interactive language for numerical computations (2015).
Yoshikazu Giga, Surface Evolution Equations: A Level Set Approach, Vol. 99 of Monographs in Mathematics. Birkhäuser (2006). | MR | Zbl
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Vol. 23 of Mathematical Modelling: Theory and Applications. Springer (2009). | MR | Zbl
M. Keuthen, Second Order Shape Optimization with Geometric Constraints. Ph.D. thesis, Technical University of Munich (2015).
and , Shape Optimization of a Breakwater. Inv. Probl. Sci. Eng. 24 (2016) 936–956. | DOI | MR | Zbl
and , Moreau-Yosida Regularization in Shape Optimisation with Geometric Constraints. Comput. Optim. Appl. 62 (2015) 181–216. | DOI | MR | Zbl
D. Kraft, A Hopf–Lax Formula for the Level-Set Equation and Applications to PDE-Constrained Shape Optimisation. In Proc. of the 19th International Conference on Methods and Models in Automation and Robotics, pp. 498–503. IEEE Xplore (2014).
D. Kraft, The level-set Package for GNU Octave. Octave Forge, 2014–2015. Available at http://octave.sourceforge.net/level-set/
D. Kraft, A Level-Set Framework for Shape Optimisation. Ph.D. thesis, University of Graz (2015). Available at https://www.domob.eu/research/phd/thesis.pdf
, A Hopf–Lax Formula for the Time Evolution of the Level-Set Equation and a New Approach to Shape Sensitivity Analysis. Interface Free Bound. 18 (2016) 317–353. | DOI | MR | Zbl
, Self-Consistent Gradient Flow for Shape Optimization. Optim. Meth. Softw. 32 (2017) 790–812. | DOI | MR | Zbl
J. Nocedal and S.J. Wright, Numerical Optimization, 2nd edn. Springer Series in Operation Research and Financial Engineering. Springer (2006). | MR | Zbl
and , Level Set Methods for Optimization Problems Involving Geometry and Constraints: I. Frequencies of a Two-Density Inhomogeneous Drum. J. Comput. Phys. 171 (2001) 272–288. | DOI | MR | Zbl
and , Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations. J. Comput. Phys. 79 (1988) 12–49. | DOI | MR | Zbl
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