A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

Constrained nonsmooth problems of the calculus of variations

“…To derive optimality conditions for problem (3.7) we will use general optimality conditions for nonsmooth mathematical programming problems in infinite dimensional spaces in terms of quasidifferentials [34,35]. To this end, we will suppose that the equality constraints are polyhedrally codifferentiable, that is, they are codifferentiable and the sets dg j (u(α), u(β)) and dg j (u(α), u(β)) are polytopes (i.e.…”

“…This assumption is needed to ensure that certain cones generated by these sets are closed. It should be noted that this assumption can be replaced by a more restrictive constraint qualification (see [34,35] for more details). For the sake of shortness, we do not consider this alternative assumption and leave it to the interested reader.…”

“…With the use of optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials [35,Crlr. 4,Prp.…”

“…(ii) It should be noted that there are two Lagrange multipliers µ j and µ j corresponding to each equality constraint g j (u(α), u j (β)) = 0, which is a specific feature of optimality conditions for nonsmooth optimisation problems in terms of quasidifferentials. See [34,35] for more details.…”

“…With the use of the general result on the codifferentiability of an integral functional obtained in this paper and necessary optimality conditions for nonsmooth mathematical programming problems in Banach spaces in terms of quasidfferentials from [34,35] we derive optimality conditions for unconstrained nonsmooth problems of the calculus of variations, as well as problems with additional constraints at the boundary and isoperimetric constraints. Each of these optimality conditions is illustrated by a simple example, in which existing optimality conditions in terms of various subdifferentials are satisfied at a non-optimal point, while our optimality conditions are able to detect the non-optimality of this point.…”