On efficient sets in vector maximum problems - A brief survey. (English) Zbl 0579.90090
Summary: The notion of an efficient (nondominated, noninferior; Pareto-optimal, functional efficient) set to a vector maximum problem (VMP) has been analyzed and developed in several directions during the last 30 years. Starting with the basic notion of efficiency given by Pareto (1896), formal descriptions of the efficient, properly efficient, locally (proper-) efficient and weak or strong efficient set have been developed. Based on these notions, various characteristics and properties of the efficient set have been studied. The structure of the efficient set and the existence of efficient solutions have also been analyzed based on the various properties of the feasible set X and the functions \(z_ k(x)\), \(k=1,...,K\), which constitute the vector-valued criterion z(x) of VMP (e.g., convexity of X, concavity of \(z_ k(x)\) for all k, differentiability). A portion of the literature is of a rather pure theoretical nature. Other portions try to develop theories to serve as the basis for developing methods for the determination of the efficient set or for developing interactive methods for determination of compromise solutions. Duality theories for more or less general cases have been developed and various aspects of stability of VMP have been investigated. A brief survey is given.
MSC:
| 90C31 | Sensitivity, stability, parametric optimization |
| 90B50 | Management decision making, including multiple objectives |
| 90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |
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