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

One-shot optimization aims at attaining feasibility and optimality simultaneously, 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.

[1]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[2]  T. Coleman,et al.  On the Local Convergence of a Quasi-Newton Method for the Nonlinear Programming Problem , 1984 .

[3]  F. Potra,et al.  Asymptotic mesh independence of Newton-Galerkin methods via a refined Mysovskii theorem , 1992 .

[4]  G. Charles Automatic differentiation and iterative processes , 1992 .

[5]  M. D. Salas,et al.  Aerodynamic design and optimization in one shot , 1992 .

[6]  Jorge Nocedal,et al.  A Reduced Hessian Method for Large-Scale Constrained Optimization , 1995, SIAM J. Optim..

[7]  Andreas Griewank,et al.  Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++ , 1996, TOMS.

[8]  Luís N. Vicente,et al.  An interface optimization and application for the numerical solution of optimal control problems , 1999, TOMS.

[9]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[10]  O. Pironneau,et al.  Applied Shape Optimization for Fluids , 2001 .

[11]  Volker Schulz,et al.  Simultaneous Pseudo-Timestepping for PDE-Model Based Optimization Problems , 2004, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[12]  George Biros,et al.  Inexactness Issues in the Lagrange-Newton-Krylov-Schur Method for PDE-constrained Optimization , 2003 .

[13]  Andreas Griewank,et al.  A mathematical view of automatic differentiation , 2003, Acta Numerica.

[14]  Andreas Griewank,et al.  Reduced Functions, Gradients and Hessians from Fixed-Point Iterations for State Equations , 2002, Numerical Algorithms.

[15]  A. Griewank,et al.  Time-lag in Derivative Convergence for Fixed Point Iterations , 2005 .