SECOND-ORDER CONVEX ANALYSIS

The classical theorem of Alexandrov asserts that a finite convex function has a second-order Taylor expansion almost everywhere, even though its first partial derivatives may only exist almost everywhere. A theorem of Mignot provides a generic linearization of the subgradient mapping associated with such a function but leaves open the question of symmetry of the matrix that appears in this linearization. This paper clears up the gap between these results and goes on to a broader theory of second-order semi-derivatives of a convex function in relation to first-order semi-derivatives of its subgradient mapping. Connections with generalized derivatives based on approximations utilizing variational convergence are illuminated as well.

[1]  Willy Feller,et al.  Krümmungseigenschaften Konvexer Flächen , 1936 .

[2]  R. Wijsman.  Convergence of sequences of convex sets, cones and functions , 1964 .

[3]  F. Mignot.  Contrôle dans les inéquations variationelles elliptiques , 1976 .

[4]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[5]  R. Rockafellar,et al.  Maximal monotone relations and the second derivatives of nonsmooth functions , 1985 .

[6]  Jean-Baptiste Hiriart-Urruty,et al.  A new set-valued second-order derivative for convex functions , 1986 .

[7]  R. Rockafellar.  First- and second-order epi-differentiability in nonlinear programming , 1988 .

[8]  R. Rockafellar.  Proto-Differentiability of Set-Valued Mappings and its Applications in Optimization☆ , 1989 .

[9]  J. Hiriart-Urruty,et al.  Calculus rules on a new set—valued second order derivative for convex functions , 1989 .

[10]  R. Rockafellar.  Nonsmooth analysis and parametric optimization , 1990 .

[11]  R. T. Rockafellar,et al.  Generalized second derivatives of convex functions and saddle functions , 1990 .

[12]  B. Mordukhovich.  Sensitivity analysis in nonsmooth optimization , 1992 .

[13]  R. Rockafellar,et al.  A Calculus of EPI-Derivatives Applicable to Optimization , 1993, Canadian Journal of Mathematics.

[14]  B. Mordukhovich.  Lipschitzian stability of constraint systems and generalized equations , 1994 .

[15]  R. Tyrrell Rockafellar,et al.  Generalized Hessian Properties of Regularized Nonsmooth Functions , 1996, SIAM J. Optim..

[16]  J. F. Bonnans,et al.  Second Order Necessary and Sufficient Optimality Conditions under Abstract Constraints , 1996 .

[17]  R. T. Rockafellar,et al.  A Derivative-Coderivative Inclusion in Second-Order Nonsmooth Analysis , 1997 .

[18]  R. Tyrrell Rockafellar,et al.  Tilt Stability of a Local Minimum , 1998, SIAM J. Optim..

[19]  Adam B. Levy,et al.  Stability of Locally Optimal Solutions , 1999, SIAM J. Optim..