Optimality conditions for minimization over the (weakly or properly) efficient set. (English) Zbl 0796.90044

Bolintineanu, S.

J. Math. Anal. Appl. 173, No. 2, 523-541 (1993).

The paper considers the minimization problem of a nonlinear real-valued function over the (weak or properly) efficient set associated to a convex, multiple objective programming problem (MOP) in Banach spaces. Such problem was studied initially by Philip in 1972. The main difficulties in approaching the problem are created by the nonexplicit form of the efficient set, and the fact that, even in the linear case, the problem is not convex.
It is very interesting that the author gives optimality conditions for the problem via derivatives for set-valued functions using some results of Corley (1988). Moreover, it is valuable that these general optimality conditions are applied to the following two finite-dimensional cases: (i) all functions of the MOP are linear; (ii) all constraint functions of the MOP are convex and multi-objective function is strictly convex.

Reviewer: T.Tanaka (Hirosaki)

MSC:

90C29	Multi-objective and goal programming
90C48	Programming in abstract spaces

Keywords:

efficient set; convex, multiple objective programming; Banach spaces; optimality conditions; derivatives for set-valued functions