The Lagrangian methods, the penalty function methods, and the successive quadratic programming method have been the most efficient solution algorithms in solving constrained optimization problems. In the convex situation, the existence of a saddle point guarantees the success of the dual search via sequential minimization of the Lagrangian function. In a presence of nonconvexity, however, the conventional dual search methods often fail to locate the global optimal solution of the primal problem. Recent research results by Li (1995), Goh and Yang (1997), Yang and Li (2000), and Sun and Li (1998, 2000) represent an extension from the traditional linear Lagrangian theory to nonlinear Lagrangian theory in an advancement to achieve a guarantee of the identification of an optimal solution of the primal problem via dual search. This talk summarizes recent progress in new dual formulations for constrained nonlinear programming with clear motivation and full geometric interpretation in order to better our understanding of the fundamental properties in constrained nonlinear programming problems and in the newly developed nonlinear Lagrangian duality theory. Prominent features in both the theoretical achievements and computational implementation of the new dual formulations will be addressed in this talk.
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