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Projected Hessians for Preconditioning in One-Step One-Shot Design Optimization

Please always quote using this URN:urn:nbn:de:0296-matheon-2117
  • One-shot optimization aims at attaining feasibility and optimality simultane- ously, especially on problems where even the linearized constraint equations cannot be resolved economically. Here we consider a scenario where forming and factoring the active Jacobian is out of the question, as is for example the case when the constraints represent some discretization of the Navier Stokes equation. Assuming that the 'user' provides us with a linearly converging solver that gradually restores feasibility after each change in the design variables, we derive a corresponding adjoint iteration and attach an optimization (sub)step. The key question addressed is how the approximate reduced gradient generated by the adjoint iteration should be preconditioned in order to achieve overall convergence at a reasonable speed. An eigenvalue analysis yields necessary conditions on the preconditioning matrix, which are typically not satised by the familiar reduced Hessian. Some other projection of the Lagrangian Hessian appears more promising and is found to work very satisfactorily on a nonlinear test problem. The analyzed approach is one-step in that the normal, dual and design variables are always updated simultaneously on the basis of one function evaluation and its adjoint. Multi-step variants are promising but remain to be investigated.

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Metadaten
Author:Andreas Griewank
URN:urn:nbn:de:0296-matheon-2117
Referee:Fredi Tröltzsch
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2005/01/25
Release Date:2005/01/24
Preprint Number:207
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