Saddle point criteria and the exact minimax penalty function method in nonconvex programming

Abstract

A new characterization of the exact minimax penalty function method is presented. The exactness of the penalization for the exact minimax penalty function method is analyzed in the context of saddle point criteria of the Lagrange function in the nonconvex differentiable optimization problem with both inequality and equality constraints. Thus, new conditions for the exactness of the exact minimax penalty function method are established under assumption that the functions constituting considered constrained optimization problem are invex with respect to the same function η (exception with those equality constraints for which the associated Lagrange multipliers are negative - These functions should be assumed to be incave with respect to the same function η). The threshold of the penalty parameter is given such that, for all penalty parameters exceeding this treshold, the equivalence holds between a saddle point of the Lagrange function in the considered constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function.