Homotopy methods for higher order shape optimization: A globalized shape-Newton method and Pareto-front tracing
This work addresses the convergence limitations of higher-order shape optimization methods for practitioners needing robust global convergence.
The authors combine an unregularized shape-Newton method with homotopy continuation to enable higher-order shape optimization from poor initial guesses, and extend the approach to multi-objective optimization for Pareto-front tracing. Numerical experiments demonstrate the method's effectiveness.
First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local convergence properties, necessitating a sufficiently close initial guess. In this work, we present an unregularized shape-Newton method and combine shape optimization with homotopy (or continuation) methods in order to allow for the use of higher order methods even if the initial design is far from a solution. The idea of homotopy methods is to continuously connect the problem of interest with a simpler problem and to follow the corresponding solution path by a predictor-corrector scheme. We use a shape-Newton method as a corrector and arbitrary order shape derivatives for the predictor. Moreover, we apply homotopy methods also to the case of multi-objective shape optimization to efficiently obtain well-distributed points on a Pareto front. Finally, our results are substantiated with a set of numerical experiments.