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  • Bài báo/Newspaper

  • Approximate Jacobian matrices for nonsmooth continuous maps and C¹-optimization

    • Authors: V. Jeyakumar (1998)

  • The notion of approximate Jacobian matrices is introduced for a continuous vector valued map. It is shown, for instance, that the Clarke generalized Jacobian is an approximate Jacobian for a locally Lipschitz map. The approach is based on the idea of convexificators of real valued functions. Mean value conditions for continuous vector-valued maps and Taylor’s expansions for continuously Gˆateaux differentiable functions (i.e., C1-functions) are presented in terms of ap proximate Jacobians and approximate Hessians, respectively. Second-order necessary and sufficient conditions for optimality and convexity of C1-functions are also given
  • Bài báo/Newspaper

  • Nonsmooth calculus, minimality, and monotonicity of convexificators

    • Authors: V. Jeyakumar (1999)

  • Noncompact convexificators, which provide upper convex and lower concave approximations for a continuous function, are defined. Various calculus rules, including extremality and mean-value properties, are presented. Regularity conditions are given for convexi ficators to be minimal. A characterization of quasiconvexity of a con tinuous function is obtained in terms of the quasimonotonicity of convexificators.