Numerical Continuation and Singularity Detection Methods for Parametric Nonlinear Programming

Abstract

Numerical methods are developed for continuation, solution-type determination, and singularity detection in the parametric nonlinear programming problem. This problem is first converted to a closed, “active set” system of equations $\bar F ( z,\alpha ) = 0$, which includes a nonstandard normalization of the multipliers. A framework is then developed for combining various numerical continuation methods with a large number of null and range space methods from constrained optimization. By exploiting the special structure in the parametric optimization problem, solution-type classification and singularity detection are shown to require minimal additional expense beyond that involved in the continuation procedure itself. Due to the special structure of these problems, singularity detection methods are more comprehensive than those for general nonlinear equations. In this development, the Schur complement and related results play an important and unifying role. As an illustration, these methods are used to produce a “global” parametric analysis for a model problem from design optimization. This example exhibits an extensive number of solution paths, each of the basic types of singularities, multiple optima, regions of sensitivity, and jump phenomena.