  Gradient-based methods have been highly successful for solving a variety of
both unconstrained and constrained nonlinear optimization problems. In
real-world applications, such as optimal control or machine learning, the
necessary function and derivative information may be corrupted by noise,
however. Sun and Nocedal have recently proposed a remedy for smooth
unconstrained problems by means of a stabilization of the acceptance criterion
for computed iterates, which leads to convergence of the iterates of a
trust-region method to a region of criticality, Sun and Nocedal (2022).
  We extend their analysis to the successive linear programming algorithm, Byrd
et al. (2023a,2023b), for unconstrained optimization problems with objectives
that can be characterized as the composition of a polyhedral function with a
smooth function, where the latter and its gradient may be corrupted by noise.
This gives the flexibility to cover, for example, (sub)problems arising image
reconstruction or constrained optimization algorithms.
  We provide computational examples that illustrate the findings and point to
possible strategies for practical determination of the stabilization parameter
that balances the size of the critical region with a relaxation of the
acceptance criterion (or descent property) of the algorithm.
