Generic minimizing behavior in semi-algebraic optimization

We present a theorem of Sard type for semi-algebraic set-valued mappings whose graphs have dimension no larger than that of their range space: the inverse of such a mapping admits a single-valued analytic localization around any pair in the graph, for a generic value parameter. This simple result yields a transparent and unified treatment of generic properties of semi-algebraic optimization problems: typical semi-algebraic problems have finitely many critical points, around each of which they admit a unique active manifold (analogue of an active set in nonlinear optimization); moreover, such critical points satisfy strict complementarity and second-order sufficient conditions for optimality are indeed necessary.

Transversality and alternating projections for nonconvex sets

We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear convergence. When the two sets are semi-algebraic and bounded, but not necessarily transversal, we nonetheless prove subsequence convergence.

Orbits of geometric descent

We prove that quasiconvex functions always admit descent trajectories bypassing all non-minimizing critical points.

Curves of Descent

Steepest descent is central in variational mathematics. We present a new transparent existence proof for curves of near-maximal slope --- an influential notion of steepest descent in a nonsmooth setting. We moreover show that for semi-algebraic functions --- prototypical nonpathological functions in nonsmooth optimization --- such curves are precisely the solutions of subgradient dynamical systems.

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 $n$.