A Shifted Primal-Dual Penalty-Barrier Method for Nonlinear Optimization
by Philip E. Gill, Vyacheslav Kungurtsev, Daniel P. Robinson
In nonlinearly constrained optimization, penalty methods provide an effective strategy for handling equality constraints, while barrier methods provide an effective approach for the treatment of inequality constraints. A new algorithm for nonlinear optimization is proposed based on minimizing a shifted primal-dual penalty-barrier function. Certain global convergence properties are established. In particular, it is shown that a limit point of the sequence of iterates may always be found that is either an infeasible stationary point or a complementary approximate Karush-Kuhn-Tucker point, i.e., it satisfies reasonable stopping criteria and is a Karush-Kuhn-Tucker point under a regularity condition that is the weakest constraint qualification associated with sequential optimality conditions. It is also shown that under suitable additional assumptions, the method is equivalent to a shifted variant of the primal-dual path-following method in the neighborhood of a solution. Numerical examples are provided that
illustrate the performance of the method compared to a widely-used conventional interior-point method.