Numerical continuation and bifurcation techniques for parametric nonlinear programming

Abstract

Numerical methods are out,lined for the continuation, solution type determination, singularity detection, and bifurcation in parametric constrained optimization problems. The problem is first converted t o an 'active set' system of equations F ( z , CY) = 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 optin~ization. By exploiting the special structure in the parametric optimization problem, solution type classification and singularity detection require minimal expense beyond that involved in the continuation procedure itself. Due t o the special structure of these problems, singularity detection methods are more comprehensive than those for general nonlinear equations. These methods, which are applicable to large classes of design optimizat ion problems, are then combined to produce a global parametric analysis for the design of a simple truss. This example exhibits an extensive number of solution paths, each of the basic types of singularities, multiple optima, regions of sensitivity, and jump phenomena.