Abstract
We consider the task of design optimization where the constraint is a state equation that can only be solved by a typically rather slowly converging fixed point solver. This process can be augmented by a corresponding adjoint solver and based on the resulting approximate reduced derivatives also an optimization iteration which actually changes the design. To coordinate the three iterative processes, we use an exact penalty function of doubly augmented Lagrangian type. The main issue here is how to derive a design space preconditioner for the approximated reduced gradient which ensures a consistent reduction of the employed penalty function as well as significant design corrections. Some numerical experiments for an alternating approach where any combination and sequencing of steps are used to improve feasibility and optimality done on a variant of the Bratu problem are presented.
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Bonnons, J.F., Charles Gilbert, J., Lemaréchal, C., Sagastizábal, C.A.: Numerical Optimization Theoretical and Practical Aspects. Springer, Berlin/Heidelberg (2003)
Dennis, J.E. Jr., Schnabel, R.B.: Numerical Methods for unconstrained optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs (1983)
Dixon, L.C.W.: Exact penalty function methods in nonlinear programing. Report NOC, The Hatfield Polytechnic, n. 103 (1979)
Fontecilla, R., Steihaug, T., Tapia, R.A.: A convergence theory for a class of quasi-newton methods for constrained optimization. SIAM J. Numer. Anal. 24, 1133–1151 (1987)
Gherman, I., Schulz, V.: Preconditioning of one-shot pseudo-timestepping methods for shape optimization. Proc. Appl. Math. Mech. 5(1), 741–759 (2005)
Gilbert, J.Ch.: Automatic differentiation and iterative processes. Optim. Methods Softw. 1, 13–21 (1992)
Griewank, A.: Evaluating derivatives: principles and techniques of algorithmic differentiation. Society for Industrial and Applied Mathematics, Philadelphia, USA (2000)
Griewank, A., Kowarz, A., Utke, J., Vogel, O., Walther, A.: ADOL-C: A package for the automatic differentiation of algorithms written in C/C++. Version 1.9.0 (2004)
Griewank, A.: Projected Hessians for preconditioning in one-step one-shot design optimization. In: Large Scale Nonlinear Optimization, pp. 151–171. Springer, Berlin (2006)
Griewank, A., Faure, C.: Reduced functions, gradients and Hessians from fixed point iteration for state equations. Numer. Algorithms 30(2), 113–139 (2002)
Griewank, A., Gauger, N., Riehme, J.: Extension of fixed point PDE solvers for optimal design by single-step one-shot method. Eur. J. Comput. Mech. (REMN) 17, 87–102 (2008)
Griewank, A., Kressner, D.: Time-lag in derivative convergence for fixed point iterations. ARIMA, 87–102
Hamdi, A., Griewank, A.: Properties of an augmented Lagrangian for design optimization. Optim. Methods Softw. 99999:1 (2009)
Ito, K., Kunisch, K., Gherman, I., Schulz, V.: Approximate nullspace iterations for KKT systems in model based optimization
Jameson, A.: Optimum aerodynamic design using CFD and control theory. In: 12th AIAA Computational Fluid Dynamics Conference, AIAA paper 95-1729, San Diego, CA, 1995. American Institute of Aeronautics and Astronautics (1995)
Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids. Numerical Mathematics and Scientific Computation. Cladenen Press, Oxford (2001)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999)
Di Pillo, G., Grippo, L.: A continuously differentiable exact penalty function for nonlinear programming problems with inequality constraints. SIAM J. Control Optim. 23, 72–84 (1986)
Pillo, G.D.: Exact penalty methods. In: Spedicato, E. (ed.) Algorithms for Continuous Optimization: The State of the Art, pp. 209–253. Kluwer Academic, Amsterdam (1994)
Werner, J.: Numerische Mathematik 1: Lineare und nichtlineare Gleichungssysteme, Interpolation, numerische Integration. Vieweg, Braunschweig/Wiesbaden (1992)
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Hamdi, A., Griewank, A. Reduced quasi-Newton method for simultaneous design and optimization. Comput Optim Appl 49, 521–548 (2011). https://doi.org/10.1007/s10589-009-9306-x
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DOI: https://doi.org/10.1007/s10589-009-9306-x