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A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials
The paper is devoted to an analysis of a new constraint qualification and a
derivation of the strongest existing optimality conditions for nonsmooth
mathematical programming problems with equality and inequality constraints in
terms of Demyanov-Rubinov-Polyakova quasidifferentials under the minimal
possible assumptions. To this end, we obtain a novel description of convex
subcones of the contingent cone to a set defined by quasidifferentiable
equality and inequality constraints with the use of a new constraint
qualification. We utilize these description and constraint qualification to
derive the strongest existing optimality conditions for nonsmooth mathematical
programming problems in terms of quasidifferentials under less restrictive
assumptions than in previous studies. The main feature of the new constraint
qualification and related optimality conditions is the fact that they depend on
individual elements of quasidifferentials of the objective function and
constraints and are not invariant with respect to the choise of
quasidifferentials. To illustrate the theoretical results, we present two
simple examples in which optimality conditions in terms of various
subdifferentials (in fact, any outer semicontinuous/limiting subdifferential)
are satisfied at a nonoptimal point, while the optimality conditions obtained
in this paper do not hold true at this point, that is, optimality conditions in
terms of quasidifferentials, unlike the ones in terms of subdifferentials,
detect the nonoptimality of this point.Comment: In the second version a new section containing a comparative analysis
of constraint qualifications in terms of quasidifferentials was added and a
number of small typos and mistakes was corrected. In the third version
multiple typos and small mistakes were correcte