Enhanced efficiency in multi-objective optimization (Q467454)

The authors introduce and study a generalized notion of proper efficiency in multi-objective optimization, denoted as \(\alpha\)-proper efficiency. After giving some preliminaries and definitions in Section 2, the main results are presented in Section 3. In particular, two characterizations of \(\alpha\)-proper efficiency are given: one in terms of the existence of an exact penalty function (Theorem 3.1), and one in terms of the stability of the associated parametric problem (Theorem 3.2). Moreover, a set of verifiable conditions for an efficient solution to be \(\alpha\)-properly efficient in polynomial multi-objective optimization is given (Theorem 3.3). Finally, some examples are presented to illustrate the results obtained. In Section 4, an example is given to demonstrate that \(\alpha\)-proper efficiency may not hold for problems with \(C^{\infty}\)-functions as objective functions.