Exact penalty method for D-stationary point of nonlinear optimization

TL;DR

This paper introduces new stationary points for nonlinear optimization with constraint violations, and proposes an exact penalty SQP method that converges efficiently to these points, improving infeasibility detection.

Contribution

It defines D-, DL-, and DZ-stationary points using an exact penalty function and develops a new SQP method that converges to these points without reducing the penalty parameter to zero.

Findings

Method converges to a D-stationary point.

Rapid infeasibility detection without penalty parameter reduction.

Effective on infeasible and degenerate nonlinear problems.

Abstract

We consider the nonlinear optimization problem with least $\ell_1$-norm measure of constraint violations and introduce the concepts of the D-stationary point, the DL-stationary point and the DZ-stationary point with the help of exact penalty function. If the stationary point is feasible, they correspond to the Fritz-John stationary point, the KKT stationary point and the singular stationary point, respectively. In order to show the usefulness of the new stationary points, we propose a new exact penalty sequential quadratic programming (SQP) method with inner and outer iterations and analyze its global and local convergence. The proposed method admits convergence to a D-stationary point and rapid infeasibility detection without driving the penalty parameter to zero, which demonstrates the commentary given in [SIAM J. Optim., 20 (2010), 2281--2299] and can be thought to be a supplement of the theory of nonlinear optimization on rapid detection of infeasibility. Some illustrative examples and preliminary numerical results demonstrate that the proposed method is robust and efficient in solving infeasible nonlinear problems and a degenerate problem without LICQ in the literature.