Proprietà degli operatori monotoni per problemi di ottimizzazione vettoriale

Abstract

In questa nota si assegnano condizioni per la monotonia della funzione di ottimo e si dimostrano alcune relazioni fra problemi reciproci, sia nel caso in cui si considera un problema di ottimizzazione scalare vincolata, che in quello in cui si introduce una funzione obiettivo vettoriale, in presenza di uno o piu vincoli. Si formulano infine alcune applicazioni economiche.

Abstract

First of all we consider a scalar problem withk inequality constraints max {f(x):g _i (x)≤b _i ,i=1, ...,k, x∈D} and analyze the conditions under which the optimal solution is not strictly dominated with respect to the constraint operatorG(x)=[g ₁(x),...,g _k (x)] in the sense of the vector programming. Furthermore we link optimal solutions for “reciprocal” problems such as$\phi (b)\underline{\underline {der}} \max \{ f(x):G(x) \leqslant b, x \in D\} and min \{ G(x):f(x) = \phi (b), x \in D\}$ and min {G(x):f(x)=ϕ(b), x∈D}. Since the minimization problem is a vector one we pass to study monotonicity properties for the optimum-value correspondence individuated by a vector optimization problem with one or more constraints in order to establish on one hand that the constraint is active at the solution point, on the other hand that the optimal solution is not strictly dominated with respect to the constraint operator.

These results allow us to give an interesting interpretation to some economic problems.