Exactness of penalization for exact minimax penalty function method in nonconvex programming

Abstract

The exact minimax penalty function method is used to solve a nonconvex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered constrained optimization problem are invex with respect to the same function η (with the exception of those equality constraints for which the associated Lagrange multipliers are negative—these functions should be assumed to be incave with respect to η). Thus, a threshold of the penalty parameter is given such that, for all penalty parameters exceeding this threshold, equivalence holds between the set of optimal solutions in the considered constrained optimization problem and the set of minimizer in its associated penalized problem with an exact minimax penalty function. It is shown that coercivity is not sufficient to prove the results.