Necessary conditions for constrained optimization problems with semicontinuous and continuous data

We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions {\sl without} constraint qualification in terms of smooth subderivatives and normal cones. These results, in different versions, are set in reflexive and smooth Banach spaces.

[2]  Jonathan M. Borwein,et al.  Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity , 1996 .

[3]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[4]  J. Borwein,et al.  A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions , 1987 .

[5]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[6]  A. Kruger,et al.  Generalized differentials of nonsmooth functions, and necessary conditions for an extremum , 1985 .

[7]  J. M. Borwein,et al.  Proximal Analysis and Boundaries of Closed Sets in Banach Space. Part II: Applications , 1987, Canadian Journal of Mathematics.

[8]  R. DeVille,et al.  Smoothness and renormings in Banach spaces , 1993 .

[9]  R. Rockafellar.  Extensions of subgradient calculus with applications to optimization , 1985 .

[10]  A. D. Ioffe,et al.  Necessary Conditions in Nonsmooth Optimization , 1984, Math. Oper. Res..

[11]  R. DeVille,et al.  A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions , 1993 .

[12]  R. Tyrrell Rockafellar,et al.  Proximal Subgradients, Marginal Values, and Augmented Lagrangians in Nonconvex Optimization , 1981, Math. Oper. Res..

[13]  G. Pisier.  Martingales with values in uniformly convex spaces , 1975 .

[14]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[15]  B. Mordukhovich.  Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[16]  Jonathan M. Borwein,et al.  Proximal analysis in smooth spaces , 1996 .

[17]  A new characterization of Clarke's tangent cone and its applications to subgradient analysis and optimization , 1983 .

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

[19]  B. Mordukhovich,et al.  Nonsmooth sequential analysis in Asplund spaces , 1996 .

[20]  Philip D. Loewen.  Optimal Control Via Nonsmooth Analysis , 1993 .

[21]  Norms with locally Lipschitzian derivatives , 1983 .

[22]  A. Ioffe.  Proximal Analysis and Approximate Subdifferentials , 1990 .

[23]  F. Clarke.  Methods of dynamic and nonsmooth optimization , 1989 .

[24]  A. Ioffe.  Approximate subdifferentials and applications 3: the metric theory , 1989 .

[25]  D. Varberg.  Convex Functions , 1973 .

[26]  Jay S. Treiman,et al.  Clarke’s gradients and epsilon-subgradients in Banach spaces , 1986 .

[27]  Jonathan M. Borwein,et al.  Proximal analysis and boundaries of closed sets in Banach space, Part I: theory , 1986 .

[28]  J. Diestel.  Geometry of Banach Spaces: Selected Topics , 1975 .

[29]  J. Treiman,et al.  Lagrange Multipliers for Nonconvex Generalized Gradients with Equality, Inequality, and Set Constraints , 1999 .