An analysis of reduced Hessian methods for constrained optimization
Mathematical Programming: Series A and B
A comparison of optimization-based approaches for a model computational aerodynamics design problem
Journal of Computational Physics
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
| Hi-index | 0.00 |
The objective of this work is to solve a model one dimensional ductdesign problem using a particular optimization method. The design problem isformulated as an equality constrained optimization, called {\it all atonce} method, so that the analysis problem is not solved until theoptimal design is reached. Furthermore, the sparsity structure in theJacobian of the linearized constraints is exploited by decomposing thevariables into the design and flow parts. To achieve this, sequentialquadratic programming with BFGS update for the reduced Hessian of theLagrangian function is used with the {\it variable reductionmethod} which preserves the structure of the Jacobian in representingthe null space basis matrix. By updating the reduced Hessians of which thedimension is the number of design variables, the storage requirement for theHessians is reduced by a large amount. In addition, the flow part of theJacobian can be computed analytically.The algorithm with a line search globalization is described. A global andlocal analysis is provided with a modification of the paper by Byrd andNocedal [Mathematical Programming 49(1991) pp 285-323] in which theyanalyzed a similar algorithm with the {\it orthogonal factorizationmethod} which assumes the orthogonality of the null space basismatrix. Numerical results are obtained and compared favorably with resultsfrom the {\it black box method}, unconstrained optimization formulation.