Abstract

The differentiability of the metric projection PP onto a closed convex set KK in Rn\mathbf{R}^n is examined. The boundary ∂K\partial K can have singular points of orders k=−1,0,1…,n−1k=-1,0,1\ldots,n-1. Here k=−1k=-1 corresponds to the interior points of KK, k=0k=0 to regular points of the boundary (i.e., faces), k=1,…,n−2k=1,\ldots,n-2 to edges and k=n−1k=n-1 to vertices. It is assumed that for every kk the set of all singular points forms an n−k−1n-k-1 dimensional manifold Tk+1T_{k+1} (possibly empty) of class p≥2p\geq 2. Under a mild continuity assumption it is shown that then PP is of class p−1p-1 on an open set WW whose complement has null Lebesgue measure. The set WW is the union of the interiors of inverse images of Tk+1T_{k+1} under PP. Moreover, a formula for the Fréchet derivative DPP on each of these regions is given that relates DPP to the second fundamental form (i.e., the curvature) of the manifold Tk+1T_{k+1}. The results are illustrated (a) on the metric projection PP from the space Sym of symmetric matrices onto the convex cone Sym+^+ of positive semidefinite symmetric matrices and (b) on the metric projection from Sym onto the unit ball under the operator norm. We prove the indefinite differentiability of these projections on explicitly determined open sets with complements of measure 00 and give explicit formulas for the derivatives. In (a) the method of proof, based on the above general result, is different from the previous treatment of J. Malick and H. S. Sendov [Clarke Generalized Jacobian of the Projection onto the Cone of Positive Semidefinite Matrices, Set-Valued Analysis 14, (2006) 273–293] and applies to situations as described by C. Padovani and M. \v Silhavý [On the derivative of the stress-strain relation in a no-tension material (2015), in preparation] where the special methods of Malick and Sendov cited above cannot be used. The case (b) is new.

Author information

Contact details are reproduced from the original publication and may be historical.

Miroslav Šilhavý

Mathematical Institute, Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic

silhavy@math.cas.cz

Keywords

  • Metric projection
  • Fréchet derivative
  • normal cone

Mathematics Subject Classification

26B25; 26B05, 52A20

References

  1. P.-A. Absil, B. Vandereycken, S. Vandewalle: A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank, IMA J. Numer. Anal. 33 (2013) 481–514.https://doi.org/10.1093/imanum/drs006Google Scholar
  2. G. Alberti: On the structure of singular sets of convex functions, Calc. Var. Partial Differ. Equ. 2 (1994) 17–27.https://doi.org/10.1007/BF01234313Google Scholar
  3. G. Alberti, L. Ambrosio: A geometrical approach to monotone functions in Rn, Math. Z. 230 (1999) 259–316.https://doi.org/10.1007/PL00004691Google Scholar
  4. A. D. Alexandrov: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningr. Gos. Univ., Uch. Zap., Ser. Mat. Nauk 6 (1939) 3–35.Google Scholar
  5. L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000).https://doi.org/10.1093/oso/9780198502456.001.0001Google Scholar
  6. R. D. Anderson, V. L. Klee Jr.: Convex functions and upper semi-continuous collections, Duke Math. J. 19 (1952) 349–357.https://doi.org/10.1215/S0012-7094-52-01935-2Google Scholar
  7. G. Anzellotti: A class of convex non-coercive functionals and masonry-like materials, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2 (1985) 261–307.https://doi.org/10.1016/s0294-1449(16)30398-5Google Scholar
  8. M. G. Crandall, H. Ishii, P.-L. Lions: User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992) 1–67.https://doi.org/10.1090/S0273-0979-1992-00266-5Google Scholar
  9. H. Federer: Geometric Measure Theory, Springer, New York (1969).Google Scholar
  10. S. Fitzpatrick, R. R. Phelps: Differentiability of the metric projection in Hilbert space, Trans. Amer. Math. Soc. 270 (1982) 483–501.https://doi.org/10.1090/S0002-9947-1982-0645326-5Google Scholar
  11. M. Giaquinta, E. Giusti: Researches on the equilibrium of masonry structures, Arch. Ration. Mech. Anal. 88 (1985) 359–392.https://doi.org/10.1007/BF00250872Google Scholar
  12. U. Helmke, J. B. Moore: Optimization and Dynamical Systems, 2nd Ed., Springer, New York (1996).https://doi.org/10.1109/JPROC.1996.503147Google Scholar
  13. J.-B. Hiriart-Urruty: At what points is the projection mapping differentiable?, Amer. Math. Mon. 89 (1982) 456–458.https://doi.org/10.1080/00029890.1982.11995475Google Scholar
  14. R. B. Holmes: Smoothness of certain metric projections on Hilbert space, Trans. Amer. Math. Soc. 184 (1973) 87–100.https://doi.org/10.1090/S0002-9947-1973-0326252-2Google Scholar
  15. R. A. Horn, C. R. Johnson: Matrix Analysis, Cambridge University Press, Cambridge (1990).Google Scholar
  16. S. Kobyashi, K. Nomizu: Foundations of Differential Geometry, Vol. I, J. Wiley, New York (1963).Google Scholar
  17. S. Kobyashi, K. Nomizu: Foundations of Differential Geometry, Vol. II, J. Wiley, New York (1969).Google Scholar
  18. M. Lucchesi, C. Padovani, G. Pasquinelli, N. Zani: Masonry Constructions: Mechanical Models and Numerical Applications, Springer, Berlin (2008).https://doi.org/10.1007/978-3-540-79111-9Google Scholar
  19. J. Malick, H. S. Sendov: Clarke generalized Jacobian of the projection onto the cone of positive semidefinite matrices, Set-Valued Anal. 14 (2006) 273–293.https://doi.org/10.1007/s11228-005-0005-1Google Scholar
  20. F. Mignot: Contrôle dans les inéquations variationelles elliptiques, J. Funct. Anal. 22 (1976) 130–185.https://doi.org/10.1016/0022-1236(76)90017-3Google Scholar
  21. F. Morgan, M. Ritoré: Geometric measure theory and the proof of the double bubble conjecture, in: Global Theory of Minimal Surfaces (Berkeley, 2001), D. Hoffman (ed.), American Mathematical Society, Providence (2005) 1–18.Google Scholar
  22. D. Noll: Directional differentiability of the metric projection in Hilbert space, Pacific J. Math. 170 (1995) 567–591.https://doi.org/10.2140/pjm.1995.170.567Google Scholar
  23. C. Padovani, M. Šilhavý: On the derivative of the stress-strain relation in a notension material, Math. Mech. Solids, to appear.Google Scholar
  24. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  25. R. Temam: Problémes Mathématiques en Plasticité, Gauthier-Villars, Paris (1983).Google Scholar
  26. R. Temam, G. Strang: Duality and relaxation in the variational problems of plasticity, J. Méc. 19 (1980) 493–527.Google Scholar
  27. L. Zajíček: On the differentiation of convex functions in finite and infinite dimensional spaces, Czech. Math. J. 29 (1979) 340–348.https://doi.org/10.21136/CMJ.1979.101616Google Scholar
  28. E. H. Zarantonello: L’algebre des projecteurs coniques, in: Analyse Convexe et ses Applications, J.-P. Aubin (ed.), Lecture Notes in Econom. and Math. Systems 102, Springer, Berlin (1974) 232–243.https://doi.org/10.1007/978-3-662-00638-2_15Google Scholar

Suggested citation

M. Šilhavý. “Differentiability of the Metric Projection onto a Convex Set with Singular Boundary Points.” Journal of Convex Analysis 22 (2015), No. 4, 969–997. https://doi.org/10.68381/jca22052

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