Nonsmooth optimization using Taylor-like models: error bounds, convergence, and termination criteria

We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result is an explicit relationship between the step-size of any such algorithm and the slope of the function at a nearby point. Consequently, we (1) show that the step-sizes can be reliably used to terminate the algorithm, (2) prove that as long as the step-sizes tend to zero, every limit point of the iterates is stationary, and (3) show that conditions, akin to classical quadratic growth, imply that the step-sizes linearly bound the distance of the iterates to the solution set. The latter so-called error bound property is typically used to establish linear (or faster) convergence guarantees. Analogous results hold when the step-size is replaced by the square root of the decrease in the models value. We complete the paper with extensions to when the models are minimized only inexactly.

Polyhedrality,Complementarity and Regularity with Applications to Variational Inequalities over Polyhedral Sets

The regularity theory for variational inequalities over polyhedral sets developed in a series of papers by Robinson, Ralph and Dontchev-Rockafellar in the 90s has long become classics of variational analysis. But in the available proofs of almost all main results, fairly nontrivial as they are, techniques of variational analysis do not play a significant part. In the paper we develop a new approach that allows to obtain some generalizations of the the mentioned results without invoking anything beyond elementary geometry of convex polyhedra and some basic facts of the theory of metric regularity.

Clarke subgradients for directionally Lipschitzian stratifiable functions

Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that have appeared in the literature. Our techniques also yield a simplified proof that closed semialgebraic functions on $R^n$ have a limiting subdifferential graph of uniform local dimension $n$.