Summary
The Schur complement relative to the linear mappingA of a functionf is denotedAf and defined as the image off underA. In this paper we give some estimates for the second-order differential ofAf whenf is either a partially quadratic convex function or aC 2 convex function with a nonsingular second-order differential. We then consider an arbitrary convex functionf and study the second-order differentiability ofAf in a more general sense.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, W. N. etDuffin, R. J.,Series and parallel addition of operators. J. Math. Anal. Appl.26 (1969), 576–594.
Anderson, W. N., Duffin, R. J. etTrapp, G. E.,Matrix operations induced by network connections. SIAM J. Control,13 (1975), 446–461.
Anderson, W. N. etTrapp, G. E.,Shorted operators II. SIAM. J. Appl. Math.28 (1973), 60–71.
Crouzeix, J. P.,A relationship between the second derivatives of a convex function and of its conjugate. Math. Programming13 (1977), 364–365.
Do, C. H.,Second-order nonsmooth analysis and sensitivity in optimization problems involving convex integral functionals. Thèse, Dept. of Mathematics, Univ. of Washington, Seattle, 1989.
Duffin, R. J. etMorley, T. D.,Inequalities induced by network connections. Pitman Research Notes in Mathematics66 (1982), 27–49.
Ekeland, I. etTeman, R.,Analyse convexe et problèmes variationnels. Dunod Gauthier-Villars, Paris, 1974.
Hiriart-Urruty, J. B.,Contributions à la programmation mathématique: cas déterministe et stochastique, Thèse, Univ. de Clermont-Ferrand II 1977.
Hiriart-Urruty, J. B.,A new set-valued second order derivative for convex functions. InMathematics for optimization, Mathematical Studies 129, North-Holland, Amsterdam, 1986.
Hiriart-Urruty, J. B. etSeeger, A.,Règles de calcul sur le sous-différentiel second d'une fonction convexe. C. R. Acad. Sc. Paris,304 (1987), 259–262.
Hiriart-Urruty, J. B. etSeeger, A.,Calculus rules on a new set-valued second order derivative for convex functions. Nonlinear Analysis: Theory, Meth. & Appl.13 (1989), 721–738.
Hiriart-Urruty, J. B. etSeeger, A.,The second-order subdifferential and the Dupin indicatrices of a nondifferentiable convex function. Proc. London Math. Soc.58 (1989), 351–365.
Kiselman, C.,How smooth is the shadow of a smooth convex body? J. London Math. Soc.33 (1986), 101–109.
Laurent, P. J.,Approximation et optimisation, Hermann, Paris, 1972.
Maury, S.,Formes quadratiques généralisées. C.R. Acad. Sci. Paris271, série A (1970), 1054–1057.
Mazure, M. L.,Analyse variationnelle des formes quadratiques convexes. Thèse, Univ. Paul Sabatier, Toulouse, 1986.
Mazure, M. L.,L'addition parallèle d'opérateurs interpretée comme inf-convolution de formes quadratiques convexes. Math. Modelling and Num. Anal.,20 (1986), 497–515.
Mazure, M. L.,La soustraction parallèle d'opérateurs interpretée comme de-convolution de formes quadratiques convexes. Optimization18 (1987), 465–484.
Mazure, M. L.,Shorted operators through convex analysis. InInternational Series of Numerical Mathematics, Vol. 854, Birkhäuser Verlag Basel, 1989, pp. 139–152.
Morley, T. D.,Parallel summation, Maxwell's principle and the infimum of projections. J. Math. Anal. Appl.70 (1979), 33–41.
Morley, T. D.,A Gauss — Markov theorem for infinite dimensional regression models with possibly singular covariance. SIAM J. Appl. Math37 (1979), 257–260.
Rockafellar, R. T.,Convex analysis. Princeton Univ. Press, Princeton, N.J, 1970.
Rockafellar, R. T.,Maximal monotone relations and the second derivatives of nonsmooth functions. Ann. Inst. H. Poincaré2 (1985), 167–184.
Rockafellar, R. T.,First and second-order epi-differentiability in nonlinear programming. Trans. Am. Math. Soc.307 (1988), 75–108.
Rockafellar, R. T.,Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives. Math. Op. Res. (à paraitre).
Rockafellar, R. T.,Generalized second derivatives of convex functions and saddle functions. Preprint, Dept. of Mathematics, Univ. of Washington, Seattle.
Seeger, A.,Analyse du second ordre de problèmes non diffёrentiables. Thèse, Univ. Paul Sabatier, Toulouse (1986).
Seeger, A.,Second derivatives of a convex function and of its Legendre—Fenchel transformate. Preprint, Dept. of Mathematics, Univ. of Washington, Seattle, 1989.
Seeger, A.,Limiting behaviour of the approximate second-order subdifferential of a convex function. Preprint, Dept. of Mathematics, Univ. of Washington, Seattle, 1989.
Slodkowski, Z.,Complex interpolation of normed and quasinormed spaces in several dimensions. I. Trans. Am. Math. Soc.308 (1988), 685–711.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Seeger, A. Complément de Schur et sous-différentiel du second ordre d'une fonction convexe. Aeq. Math. 42, 47–71 (1991). https://doi.org/10.1007/BF01818478
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01818478