  The primary focus of this paper is on designing an inexact first-order
algorithm for solving constrained nonlinear optimization problems. By
controlling the inexactness of the subproblem solution, we can significantly
reduce the computational cost needed for each iteration. A penalty parameter
updating strategy during the process of solving the subproblem enables the
algorithm to automatically detect infeasibility. Global convergence for both
feasible and infeasible cases are proved. Complexity analysis for the KKT
residual is also derived under mild assumptions. Numerical experiments exhibit
the ability of the proposed algorithm to rapidly find inexact optimal solution
through cheap computational cost.
