The Expanded Lagrangian System for Constrained Optimization Problems

Smooth penalty functions can be combined with numerical continuation and bifurcation techniques to produce a class of robust and potentially fast algorithms for constrained optimization problems. The key to the development of these algorithms is an expanded Lagrangian system which is derived and analyzed in this work. This parametrized system of nonlinear equations contains the penalty path as a solution, provides a smooth path into the first-order necessary conditions, and sometimes yields a method for finding multiple optima. Furthermore, the inevitable ill-conditioning present in a sequential optimization algorithm is removed for three and essentially only three smooth penalty functions: the quadratic penalty function for equality constraints, the logarithmic barrier function (an interior method), and the quadratic loss function (an exterior method) for inequality constraints. Preliminary test results for several nontrivial test problems are included to demonstrate the robustness, efficiency and potential of the methodology.