Appl Math Optim 38:21–43 (1998)
1998 Springer-Verlag New York Inc.
Unicity Results for General Linear Semi-Inﬁnite Optimization
Problems Using a New Concept of Active Constraints
and M. I. Todorov
M¨orikestraße 6, D-60320 Frankfurt/Main 1, Germany
Bulgarian Academy of Sciences, Institute of Mathematics,
29 Ph. Macedonsky Street, 4002 Plovdiv, Bulgaria
Communicated by J. Stoer
Abstract. We consider parametric semi-inﬁnite optimization problems without
the usual asssumptions on the continuity of the involved mappings and on the com-
pactness of the index set counting the inequalities. We establish a characterization
of those optimization problems which have a unique or strongly unique solution
and which are stable under small pertubations. This result generalizes a well-known
theorem of N¨urnberger. The crucial roles in our investigations are a new concept of
active constraints, a generalized Slater’s condition, and a Kuhn–Tucker-type theo-
rem. Finally, we give some applications in vector optimization, for approximation
problems in normed spaces, and in the stability of the minimal value.
Key Words. Semi-inﬁnite linear optimization, Parametric optimization, Density
of the unicity set, Strong unicity.
AMS Classiﬁcation. 90C05, 90C34, 65K05, 49M39.
1. Problem Formulation and Some Preliminaries
Let P := F(T, R
) × F(T ) × R
, where N ∈ N, T is an arbitrary nonempty index
set, and F(T , R
) (resp. F(T )), is the set of mappings from T to R
(resp. to R). For
The second author was partially supported by the Deutsche Akademische Austauschdienst DAAD and
by the Bulgarian Ministry of Education and Science under Grant MM 44-701/97.