Identification of Active Component Functions in Finite-Max Minimisation via a Smooth Reformulation

In this work, we consider a nonsmooth minimisation problem in which the objective function can be represented as the maximum of finitely many smooth “component functions”. First, we study a smooth min–max reformulation of the problem. Due to this smoothness, the model provides enhanced capability of exploiting the structure of the problem, when compared to methods that attempt to tackle the nonsmooth problem directly. Then, we present several approaches to identify the set of active component functions at a minimiser, all within finitely many iterations of a first order method for solving the smooth model. As is well known, the problem can be equivalently rewritten in terms of these component functions, but a key challenge is to identify this set a priori. Such an identification is clearly beneficial in an algorithmic sense, since we can discard those component functions which are not necessary to describe the solution, which in turn can facilitate faster convergence. Finally, numerical results comparing the accuracy of each of these approaches are presented, along with the effect they have on reducing the complexity of the original problem.