Modified methods for nonlinear optimization and complementarity problems in the absence of strict complementarity

Abstract : For nonlinear constrained optimization and complementarity problems, we consider the case when the strict complementarity condition does not hold. In this situation, only a linear rate of convergence can be guaranteed for most classical algorithms. In this talk, we consider a Lagrange-Newton method and the modified Lagrangian method for nonlinear constrained optimization, and propose an approach that allows us to obtain modifications of these methods. The obtained modifications attain super-linear convergence even when the strict complementarity condition does not hold and subsume the case when this condition holds. Moreover, the proposed approach to modifying the methods can be applied to a variety of problems with some kind of degeneracy. We illustrate this by constructing a method for nonlinear complementarity problems in the absence of strict complementarity.