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A Shifted Primal-Dual Penalty-Barrier Method for Nonlinear Optimization

by Philip E. Gill, Vyacheslav Kungurtsev, Daniel P. Robinson

Abstract:

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.

UCSD-CCoM-19-03.pdf   March 2019