Vector-valued perturbed minimization principles
by
Catherine Finet
Université de Mons-Hainaut, Belgium

The lack of compactness in infinite dimensional Banach spaces results in functions not attaining their bounds and some perturbations are thus needed. The idea of perturbed optimization is simple : one starts with a real-valued function f which is, say, lower semi-continuous and bounded from below on a nice set and shows that there exist arbitrarily small perturbations g such that f+g attains a (strong) minimum on the set (every minimizing sequence converges). A most convenient approach to these problems is through smooth perturbed minimization principles (also called smooth variational principles). This approach can be chosen in spaces that admit non trivial smooth functions. A smooth perturbed minimization principle substitutes for the Hahn-Banach theorem in non convex analysis. Many authors have studied variational principles (see [1, 4, 5]). Instead of considering real-valued functions, we can look at vector-valued functions. For example, Göpfert and Tammer have studied extensions of Ekeland's variational principle to vector optimization [6, 11, 12]. In this context it is natural to look at some extension of the Deville-Godefroy-Zizler perturbed minimization principle [4]. This principle extends the Borwein-Preiss smooth perturbed minimization principle [1]. Our context will be the following : let X and Z be real Banach spaces. Suppose that Z is partially ordered by a closed convex pointed cone K such that the quasi-interior

q-int K * = {z * in Z * : z * (z) > 0, for allz in K \{0}}

of the dual cone K ^* is non empty. Let us mention that this is the case when K is a closed convex pointed cone of a separable Banach space Z. Let f be a function from X to Z. The vector minimization problem (P) under consideration is :

Starting with an arbitrary function f : X --> Z, there is no reason why the set Min f should be non empty. We want to find a ``nice" perturbation h of f as close as we want to f and such that Min h is not empty. In the scalar case, usual conditions on the function f are boundedness from below and lower semi-continuity. These notions have already been extended in the vector case [2, 8, 9, 10, 12].

A function f : X --> Z is said to be bounded from below if there exists z in Z such that for all x in X, z <= f(x).

A function f : X --> Z is said to be lower semi-continuous (lsc) if, for all z in Z, the set {x in X, f(x) <= z}, also denoted {f <= z}, is closed.

Let us remark that a notion of lsc function was also introduced by J.M. Borwein, J.P. Penot and M. Théra, see [2, 9, 10]. As an lsc mapping in the sense of [2] has a closed epigraph (Proposition 1.4 of [2]) it follows that such a mapping is also lsc in our sense. Let e ^* in q-int K ^* and e in K such that e ^* (e) = 1.

Our result is the following :

Theorem. Vector valued Deville-Godefroy-Zizler perturbed minimization principle.

Let (Y, ||·||_Y) be a complete convex cone of bounded, lsc functions from X to Z such that

1. for all g in Y, ||g||_\infty <= ||g||_Y,
2. for every g in Y and every u in X, ||T_u g||_Y = ||g||_Y, where T_ug(x) = g(u + x) for all x in X,
3. for every g in Y and every a > 0, the function h : X --> Z defined by h(x) = g(ax) belongs to Y,
4. There exists a bump function b : X --> IR such that b(0) > 0 and [b\tilde] = - be belongs to Y.

This result extends the one we got with R. Deville (see [3], where we supposed that the interior of K was non empty) and this answers a question of R. Paya.

We now give some applications of our result. As a Lipschitz continuous function g : X --> Z is lsc, we can take for Y the Banach space of all bounded Lipschitz continuous functions from X to Z equipped with the norm :

||g||Y = ||g||\infty + sup ì í î  ||g(x) - g(y)|| ||x-y|| , x =/= y ü ý þ .

Corollary. Let f : X --> Z be lsc, bounded from below. Then, for every \epsilon > 0, there exists g : X --> Z bounded Lipschitz continuous function such that

||g||Y < \epsilon ,  Min(f+g) =/= \emptyset.

Take now for Y the Banach space of all g : X --> Z that are bounded, Lipschitz continuous and Fréchet-differentiable (resp. Gateaux-differentiable) equipped with the norm : ||g||_Y = max( ||g||_\infty, ||g'||_\infty ). We then get a vector-valued version of the Borwein-Preiss smooth perturbed minimization principle.

Corollary. Let X be a Banach space that admits a Lipschitz continuous bump function which is Fréchet-differentiable (resp. Gateaux-differentiable). Then for every f : X --> Z lsc, bounded from below, and for every \epsilon > 0, there exists a function g : X --> Z which is Lipschitz continuous and Fréchet-differentiable (resp. Gateaux-differentiable) and such that ||g||_\infty < \epsilon, ||g'||_\infty < \epsilon, Min(f+g) =/= \emptyset.

References:

1. J. Borwein and D. Preiss, Smooth variational principle with applications to subdifferentiability of convex functions, Trans. Amer. Math. Soc. 303, 1987, 517-527 .

2. J. Borwein, J. Penot and M. Théra, Conjugate convex operators, J. Math. Anal. Appl. 102, 1984, 399-414 .

3. R. Deville and C. Finet, Vector-valued perturbed minimization principles, Canadian Math. Soc., Conference Proceedings (to appear).

4. R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman monographs and Surveys in Pure and Appl. Math. . Longman Scientific & Technical, 1993.

5. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47, 1974, 324-353 .

6. A. Göpfert and Chr. Tammer, A new maximal point theorem, Zeitschr. f. Analysis und Anwendungen , 1994.

7. J. Jahn, Mathematical vector optimization in partially ordered spaces , Lang Verlag Frankfurt, Bern, New York, 1986.

8. P. Loridan, Well-posedness in vector optimization, Cermsem 95.25 .

9. J. Penot and M. Théra, Applications sous-linéaires à valeurs dans un espace de fonctions continues, Annali di Matematica Pura ed Applicata, 136, 1984, 133-151 .

10. J. Penot and M. Théra, Semi-continuous mappings in general topology, Archiv der Math, 38, 1982, 158-166 .

11. Chr. Tammer, A generalization of Ekeland's variational principle, Optimization 25, 1992, 129-141 .

12. Chr. Tammer, A variational principle and applications for vectorial control approximation problems, Reports of the Inst. of Optimization and Stochastics . Martin-Luther Universität Halle - Wittenberg, 1996.

(T)

Date received: February 8, 2000