The nonlinear optimization problem with possible infeasible constraints was studied early by Burke (J Math Anal Appl, 139:19–351, 1989) and was revisited by Dai and Zhang (CSIAM Trans Appl Math, 2:551–584, 2021; Math Program, 200:633–667, 2023) in a broad perspective. This paper considers nonlinear optimization with least \(\ell _1\) -norm measure of constraint violations and introduces 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 these specific stationary points, we propose an 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 Byrd et al (SIAM J Optim, 20:2281–2299, 2010) 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.
