This paper develops a parameter-free adaptive proximal bundle method with two important features: 1) adaptive choice of variable prox stepsizes that "closely fits" the instance under consideration; and 2) adaptive criterion for making the occurrence of serious steps easier.
Computational experiments show that our method performs substantially fewer consecutive null steps (i.e., a shorter cycle) while maintaining the number of serious steps under control. As a result, our method performs significantly less number of iterations than its counterparts
based on a constant prox stepsize choice and a non-adaptive cycle termination criterion. Moreover, our method is very robust relative to the user-provided initial stepsize.