Appl Math Optim 40:211–228 (1999)
1999 Springer-Verlag New York Inc.
Nonlinear Programming Problems Associated with
Closed Range Operators
and N. H. Pavel
Department of Mathematics, Ohio University,
Athens, OH 45701, USA
Department of Mathematics, University of Ia¸si,
Ro-6600 Ia¸si, Romania
Communicated by F. H. Clarke
Abstract. Necessary conditions for the optimality of a pair (y, u) with respect to
a locally Lipschitz cost functional L(y, u), subject to Ay + F (y) = Cu + B(u), are
given in terms of generalized gradients. Here A and C are densely deﬁned, closed,
linear operators on some Banach spaces, while F and B are (Fr´echet) differentiable
maps, which are suitably related to A and C. Various examples and potential appli-
cations to nonlinear programming models and nonlinear optimal control of partial
differential equations are also discussed.
Key Words. Optimality conditions, Closed range operators, Tangent cones,
Clarke’s generalized gradients, Wave operator.
AMS Classiﬁcation. Primary 49K27, 49K30, Secondary 47A62, 47H15.
This paper is mainly concerned with the following inﬁnite-dimensional nonlinear pro-
gramming problem with a special structure in a Banach space E:
(P) (Locally) Minimize L(y, u)
subject to Ay + F(y) = Cu + B(u),
where A: D(A) ⊂ E → E is an unbounded closed linear operator with dense domain
D(A) and closed range R(A), B: U → E is a (Fr´echet) differentiable map on an open