Pareto-optimality conditions in discrete vector optimization problems

V. A. Emelichev ; O. A. Yanushkevich

Diskretnaya Matematika, Tome 9 (1997) no. 3, pp. 153-160

Résumé

For the vector optimization problem \begin{gather*} F = (f_1, f_2,\dots, f_n)\colon X\to\mathbf R^n,\qquad n\ge 2, \\ f_i(x)\to \min_X\qquad \forall\,i\in N_n=\{1, 2,\dots,n\}, \end{gather*} with a finite set of vector estimators $F(X)$ we give a wide class of efficiency (Pareto-optimality) criteria in terms of linear convolutions of transformed partial criteria. In particular, it is proved that an element $x^o\in X$ is efficient if and only if there exists a vector $(\lambda_1,\lambda_2,\dots,\lambda_n)$, $\lambda_i>0$, $i\in N_n$, such that $$ \sum_{i\in N_n}\lambda_i\alpha^{f_i(x^o)} \le\sum_{i \in N_n}\lambda_i\alpha^{f_i(x)}\qquad \forall\,x \in X, $$ where $\alpha=n^{1/\Delta}$, $\Delta=\min\{f_i(x)-f_i(x') >0\colon x, x' \in X,\ i \in N_n\}$. This research was supported by the Foundation for Basic Research of Republic Byelarus (grants F95–70 and MP96–35), and the DAAD and the International Soros Educational Program in Exact Sciences (grant ‘Soros Professor’ for the first of the authors).