Appl Math Optim 49:123–144 (2004)
2004 Springer-Verlag New York Inc.
Optimality Conditions in Differentiable Vector Optimization
via Second-Order Tangent Sets
and Vicente Novo
Departamento de Econom´ıa e Historia Econ´omica,
Facultad de Econom´ıa y Empresa, Universidad de Salamanca,
Campus Miguel de Unamuno, s/n, 37007 Salamanca, Spain
Departamento de Matem´atica Aplicada, E.T.S.I. Industriales, UNED,
c/ Juan del Rosal 12, Apartado 60149, 28080 Madrid, Spain
Abstract. We provide second-order necessary and sufﬁcient conditions for a point
to be an efﬁcient element of a set with respect to a cone in a normed space, so that
there is only a small gap between necessary and sufﬁcient conditions. To this aim,
we use the common second-order tangent set and the asymptotic second-order cone
utilized by Penot. As an application we establish second-order necessary conditions
for a point to be a solution of a vector optimization problem with an arbitrary feasible
set and a twice Fr´echet differentiable objective function between two normed spaces.
We also establish second-order sufﬁcient conditions when the initial space is ﬁnite-
dimensional so that there is no gap with necessary conditions. Lagrange multiplier
rules are also given.
Key Words. Vector optimization, Second-order optimality conditions for efﬁ-
ciency, Second-order tangent set, Asymptotic second-order cone, Projective tangent
set, Lagrange multipliers, Strict efﬁciency.
AMS Classiﬁcation. 90C29, 90C46, 49K27.
This research was partially supported by Ministerio de Ciencia y Tecnolog´ıa (Spain), Project BFM2003-