Proto-derivative Formulas for Basic Subgradient Mappings in Mathematical Programming
نویسندگان
چکیده
Subgradient mappings associated with various convex and nonconvex functions are a vehicle for stating optimality conditions, and their proto-differentiability plays a role therefore in the sensitivity analysis of solutions to problems of optimization. Examples of special interest are the subgradients of the max of finitely many C functions, and the subgradients of the indicator of a set defined by finitely many C constraints satisfying a basic constraint qualification. In both cases the function has a property called full amenability, so the general theory of existence and calculus of proto-derivatives of subgradient mappings associated with fully amenable functions is applicable. This paper works out the details for such examples. A formula of Auslender and Cominetti in the case of a max function is improved in particular.
منابع مشابه
Partial Extensions of Attouch's Theorem with Applications to Proto-derivatives of Subgradient Mappings
Attouch's Theorem, which gives on a reflexive Banach space the equivalence between the Mosco epi-convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients, has many important applications in convex optimization. In particular, generalized derivatives have been defined in terms of the epi-convergence or graph convergence of certain differe...
متن کاملVariational Conditions and the Proto-differentiation of Partial Subgradient Mappings
Subgradient mappings have many roles in variational analysis, such as the formulation of optimality conditions and, in the special case of normal cone mappings, the statement and analysis of variational inequalities and related expressions. Central to the study of perturbations of solutions to systems of conditions in which subgradient mappings appear are concepts of generalized differentiation...
متن کاملEqui-Calmness And Epi-Derivatives That Are Pointwise Limits
18. A. D. Iooe, \Variational analysis of a composite function: a formula for the lower second order epi-derivative," J. 843{865. 21. R. A. Poliquin, \An extension of Attouch's Theorem and its application to second-order epi-diierentiation of convexly composite functions," Trans. Amer. Math. Soc. 18 neighborhood of g ? F(x) and is twice epi-diierentiable with second-order epi-derivative given as...
متن کاملSOLVING FUZZY LINEAR PROGRAMMING PROBLEMS WITH LINEAR MEMBERSHIP FUNCTIONS-REVISITED
Recently, Gasimov and Yenilmez proposed an approach for solving two kinds of fuzzy linear programming (FLP) problems. Through the approach, each FLP problem is first defuzzified into an equivalent crisp problem which is non-linear and even non-convex. Then, the crisp problem is solved by the use of the modified subgradient method. In this paper we will have another look at the earlier defuzzifi...
متن کاملThe Differentiation of Hypoelliptic Diffusion Semigroups
Basic derivative formulas are presented for hypoelliptic heat semi-groups and harmonic functions extending earlier work in the elliptic case. Following the approach of 15], emphasis is placed on developing integration by parts formulas at the level of local martingales. Combined with the optional sampling theorem, this turns out to be an eecient way of dealing with boundary conditions, as well ...
متن کامل