Optimality and convexity conditions for piecewise smooth objective functions

In the recent book Optimization Stories edited by Martin Groetschel, Robert Mifflin and Claudia Sastizabal [4] describe how nonsmooth optimization arose through a rare collaboration between Soviet and Western scientists in the 1960s and 1970s. In the seemingly simple case of a convex objective function there exists everywhere a nonempty set of subgradients whose shortest element provides a direction of descent. While this set is also convex it may be quite difficult to compute and represent. On the other hand there are classical scenarios where it is quite easy to compute one of the subgradients, or in the more general Lipschitzian case, one generalized gradient. This lead to the widely accepted black-box paradigm, namely that nonsmooth optimization algorithms should be based on an evaluation oracle that provides the function value and just one generalized gradient at any point in the function domain. The resulting bundle methods were described and analyzed in great detail, for example in the classical bools of Jean Baptiste Hirriart Urruty and Claude Lemarechal [3] . However, there exist very simple examples, where due to the codependence of intermediate quantities, such a generalized gradient cannot be computed by the application of generalized differentiation rules. Moreover, even when generalized gradients are available everywhere, the resulting descent algorithms are typically quite slow and suffer from the lack of practical stopping criteria. Ideally, these should be based on computable and reasonably sharp optimality conditions, that are satisfied approximately in a small but open neighborhood of a local minimizer. It will be shown here that this desirable situation can be achieved for objectives in the so called abs-normal form, which also yields enough information to achieve, linear, superlinear, or even quadratic convergence, by successive piecewise linearization provided certain nondegeneracy conditions are satisfied. Any piecewise smooth function that is specified by an evaluation procedure involving smooth elemental functions and piecewise linear functions like min and max can be represented in abs-normal form [1]. This is in particular true for most popular nonsmooth test functions. By an extension of algorithmic, or