Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. 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 rely on a new convex interpolation result: given a decreasing sequence of positively curved C^k smooth convex compact sets in the plane, we provide a level set interpolation of a C^k smooth convex function where k ≥ 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. (Joint work with E. Pauwels (Université Toulouse 1)

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