Adrian Lewis Orthogonally invariant functions of symmetric matrices often inherit properties from their diagonal restrictions: von Neumann’s theorem on matrix norms is an early example. We discuss the example of “identifiability”, a common property of nonsmooth functions associated with the existence of a smooth manifold of approximate critical points. Identifiability (or its synonym, “partial smoothness”) is the … Read more
Steepest descent drives both theory and practice of nonsmooth optimization. We study slight relaxations of two influential notions of steepest descent curves — curves of maximal slope and solutions to evolution equations. In particular, we provide a simple proof showing that lower-semicontinuous functions that are locally Lipschitz continuous on their domains — functions playing a … Read more
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 … Read more
Around a solution of an optimization problem, an “identifiable” subset of the feasible region is one containing all nearby solutions after small perturbations to the problem. A quest for only the most essential ingredients of sensitivity analysis leads us to consider identifiable sets that are “minimal”. This new notion lays a broad and intuitive variational-analytic … Read more
We prove that uniform second order growth, tilt stability, and strong metric regularity of the subdifferential — three notions that have appeared in entirely different settings — are all essentially equivalent for any lower-semicontinuous, extended-real-valued function. Citation Cornell University, School of Operations Research and Information Engineering, 206 Rhodes Hall Cornell University Ithaca, NY 14853. May … Read more
We compare two recent variational-analytic approaches to second-order conditions and sensitivity analysis for nonsmooth optimization. We describe a broad setting where computing the generalized Hessian of Mordukhovich is easy. In this setting, the idea of tilt stability introduced by Poliquin and Rockafellar is equivalent to a classical smooth second-order condition. Article Download View Partial Smoothness,Tilt … Read more
Examples exist of extended-real-valued closed functions on $\R^n$ whose subdifferentials (in the standard, limiting sense) have large graphs. By contrast, if such a function is semi-algebraic, then its subdifferential graph must have everywhere constant local dimension $n$. This result is related to a celebrated theorem of Minty, and surprisingly may fail for the Clarke subdifferential. … Read more
We show that minimizers of convex functions subject to almost all linear perturbations are nondegenerate. An analogous result holds more generally, for lower-C^2 functions. Citation Cornell University, School of Operations Research and Information Engineering, 206 Rhodes Hall Cornell University Ithaca, NY 14853. May 2010. Article Download View Generic nondegeneracy in convex optimization
We prove that the subdifferential of any semi-algebraic extended-real-valued function on $\R^n$ has $n$-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems. Citation Cornell University, School of Operations Research and Information Engineering, 206 Rhodes Hall Cornell University Ithaca, NY 14853. April 2010. Article Download View Semi-algebraic functions have small subdifferentials
We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, “tame”). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region is “partly smooth”, ensuring finite identification of the manifold by many optimization algorithms. Furthermore, second-order optimality conditions … Read more