A globally convergent algorithm based on imbedding and parametric optimization

The continuation method, well-established for the solution of nonlinear equations is extended to restricted optimization problems. Only the locally active restrictions are considered along the homotopy path. It is assumed that there are only finitely many critical points, i. e. that there are only finitely many changes of the index set of active restrictions. The globally convergent algorithm which we present proceeds in three stages:

1. Within each stability region, the solution is computed by the classical continuation method.
2. On the boundary of a stability region, a critical point \(\bar t\) is determined.
3. A new active index set is determined when \(\bar t\) is passed.

For the class of convex problems, the hypotheses for the convergence of the algorithm may be secured. The algorithm is applied to several examples.