  Counterexamples to some old-standing optimization problems in the smooth
convex coercive setting are provided. We show that block-coordinate, steepest
descent with exact search or Bregman descent methods do not generally converge.
Other failures of various desirable features are established: directional
convergence of Cauchy's gradient curves, convergence of Newton's flow, finite
length of Tikhonov path, convergence of central paths, or smooth
Kurdyka-Lojasiewicz inequality. All examples are planar. These examples are
based on general smooth convex interpolation results. Given a decreasing
sequence of positively curved C k convex compact sets in the plane, we provide
a level set interpolation of a C k smooth convex function where k $\ge$ 2 is
arbitrary. If the intersection is reduced to one point our interpolant has
positive definite Hessian, otherwise it is positive definite out of the
solution set. Furthermore , given a sequence of decreasing polygons we provide
an interpolant agreeing with the vertices and whose gradients coincide with
prescribed normals.

Optimization and Control