Generalized second-order directional derivatives and optimization with C

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C1,1 functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gateaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.

[1]  Vaithilingam Jeyakumar,et al.  On optimality conditions in nonsmooth inequality constrained minimization , 1987 .

[2]  A. Ben-Tal,et al.  A unified theory of first and second order conditions for extremum problems in topological vector spaces , 1982 .

[3]  F. Clarke.  Generalized gradients and applications , 1975 .

[4]  A. Ben-Tal,et al.  Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems , 1982, Math. Program..

[5]  J. Hiriart-Urruty,et al.  Generalized Hessian matrix and second-order optimality conditions for problems withC1,1 data , 1984 .

[6]  R. W. Chaney.  Second-Order Sufficiency Conditions for Nondifferentiable Programming Problems , 1982 .

[7]  Roberto Cominetti,et al.  A generalized second-order derivative in nonsmooth optimization , 1990 .

[8]  Vaithilingam Jeyakumar.  Composite Nonsmooth Programming with Gâteaux Differentiability , 1991, SIAM J. Optim..

[9]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

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

[11]  J. Hiriart-Urruty.  Generalized Differentiability / Duality and Optimization for Problems Dealing with Differences of Convex Functions , 1985 .

[12]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[13]  G. A. Garreau,et al.  Mathematical Programming and Control Theory , 1979, Mathematical Gazette.

[14]  Xiaoqi Yang,et al.  Convex composite multi-objective nonsmooth programming , 1993, Math. Program..

[15]  R. W. Chaney.  A general sufficiency theorem for nonsmooth nonlinear programming , 1983 .

[16]  Jonathan M. Borwein,et al.  The differentiability of real functions on normed linear space using generalized subgradients , 1987 .

[17]  Jean-Baptiste Hiriart-Urruty,et al.  Characterizations of the plenary hull of the generalized Jacobian matrix , 1982 .

[18]  A. D. Ioffe.  Second Order Conditions in Nonlinear Nonsmooth Problems of Semi-Infinite Programming , 1983 .

[19]  R. Holmes.  Geometric Functional Analysis and Its Applications , 1975 .

[20]  A. Ioffe,et al.  On some recent developments in the theory of second order optimality conditions , 1988 .

[21]  R. Tyrrell Rockafellar,et al.  Second-Order Optimality Conditions in Nonlinear Programming Obtained by Way of Epi-Derivatives , 1989, Math. Oper. Res..

[22]  A. Ioffe.  Nonsmooth analysis: differential calculus of nondifferentiable mappings , 1981 .