We present a method to approximate the solution mapping of parametric constrained optimization problems. The approximation, which is of the spectral stochastic finite element type, is represented as a linear combination of orthogonal polynomials. Its coefficients are determined by solving an appropriate finite-dimensional constrained optimization problem. We show that, under certain conditions, the latter problem is solvable because it is feasible for a sufficiently large degree of the polynomial approximation and has an objective function with bounded level sets. In addition, the solutions of the finite dimensional problems converge for an increasing degree of the polynomials considered, provided that the solutions exhibit a sufficiently large and uniform degree of smoothness. We demonstrate that our framework is applicable to one-dimensional parametric eigenvalue problems and that the resulting method is superior in both accuracy and speed to black-box approaches.
Preprint ANL/MCS-P1379-1006 Argonne National Laboratory, Mathematics and Computer Science Division, 9700 S Cass Avenue, Argonne IL, 60439, USA, October, 2006