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Curiosities and counterexamples in smooth convex optimization

Author

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  • Bolte, Jérôme
  • Pauwels, Edouard
Abstract
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 Ck convex compact sets in the plane, we provide a level set interpolation of a Ck smooth convex function where k 2 is arbitrary. If the intersection is reduced to one point our interpolant has positive denite Hessian, otherwise it is positive denite out of the solution set. Further- more, given a sequence of decreasing polygons we provide an interpolant agreeing with the vertices and whose gradients coincide with prescribed normals.

Suggested Citation

  • Bolte, Jérôme & Pauwels, Edouard, 2020. "Curiosities and counterexamples in smooth convex optimization," TSE Working Papers 20-1080, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:124147
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    References listed on IDEAS

    as
    1. J. P. Crouzeix, 1980. "Conditions for Convexity of Quasiconvex Functions," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 120-125, February.
    2. Kannai, Yakar, 1977. "Concavifiability and constructions of concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 4(1), pages 1-56, March.
    3. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
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    Cited by:

    1. Jean-Pierre Crouzeix, 2022. "On Quasiconvex Functions Which are Convexifiable or Not," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 66-80, June.

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