ISSN 1064-5624, Doklady Mathematics, 2011, Vol. 83, No. 3, pp. 418–420. © Pleiades Publishing, Ltd., 2011. Original Russian Text © V.D. Noghin, O.V. Baskov, 2011, published in Doklady Akademii Nauk, 2011, Vol. 438, No. 4, pp. 456–459. 418 Many applied problems in economics and engi- neering can be formulated as a multicriteria choice problem with several numerical functions. A specific feature of multicriteria problems is that, starting a choice procedure, the decision maker (DM) cannot, as a rule, precisely express his or her interests and pref- erences, which underlie the choice made. Thus, beginning the search for a set (in a special case, a sin- gleton) of “best” elements, the DM does not have the exact definition of this concept. Frequently, these best elements are detected in the course of decision making based on available information about the DM’s prefer- ences. Numerous procedures and methods have been pro- posed for solving multicriteria problems depending on the type and character of information on the DM’s preferences [1, 2]. Frequently, these are heuristic pro- cedures that yield substantially different best solutions. According to the overwhelming majority of research- ers, best solutions have to be sought in the set of Pareto optimal (effective, tradeoff) alternatives. This circum- stance is expressed in the Edgeworth–Pareto princi- ple, which has relatively recently been axiomatically substantiated [3]. Thus, the problem of choosing a set of best alternatives can be reformulated as the problem of Pareto set reduction. Consider the model of multicriteria choice [3], which contains a set X of initial alternatives, a vector criterion y = f(x) = (f 1 (x), f 2 (x), …, f m (x)), and an asymmetric binary preference relation  X defined on X. Let Y = f(X), and let  Y be a binary relation on the set Y induced by the relation  X as follows: x 1  X x 2 ⇔ f(x 1 )  Y f (x 2 ) for all x 1 ∈ , x 2 ∈ ; , ∈ , where is the collection of equivalence classes gener- ated by the equivalence relation x 1 ~ x 2 ⇔ f (x 1 ) = f (x 2 ) x ˜ 1 x ˜ 2 x ˜ 1 x ˜ 2 X ˜ X ˜ on X. The sets of chosen (best) alternatives and vectors are denoted by C(X) and C(Y) = f (C(X)), respectively. These are the sets to be determined in the course of making a choice. The set of Pareto optimal alternatives with respect to the vector criterion f on X is denoted by P f (X). The Pareto set P f (X) can be reduced (i.e., certain Pareto-optimal elements can be eliminated) if we have additional information on the DM’s preferences. The most reliable and simple from a practical point of view is information on the DM’s readiness to make a cer- tain compromise. Frequently, this compromise means that the DM agrees to lose according to insignificant criteria while gaining according to significant criteria. Taking into account information on the DM’s preference relation, which the DM is usually initially totally unaware of, underlies an axiomatic approach to the reduction of the Pareto set. This approach has been developed for almost three decades [3]. First, it was established how one should use a single piece of information (“information quantum”) in the form of two collections of numbers, of which one indicates the admissible upper limits of losses for the group of insig- nificant criteria and the other shows the gain values for the significant criteria that are larger than or equal to those the DM would agree to receive when making the compromise. Later, possibilities of using various col- lections of such information (several information quanta) for Pareto set reduction were studied. In the geometric language, an information quan- tum about the DM’s preference relation means [3] that the relation u  Y 0 holds for some u ∈ N m , where N m is a set of m-dimensional vectors with at least one strictly positive and one strictly negative component. The specification of a collection of information quanta is equivalent to the fulfillment of the relations u i  Y 0, i = 1, 2, …, k. In this paper, we propose two universal algorithms (methods) based on taking into account an arbitrary finite collection of information quanta for Pareto set reduction. The first (geometric) algorithm was devel- oped by the first author. It involves the solution of the Pareto Set Reduction Based on an Arbitrary Finite Collection of Numerical Information on the Preference Relation V. D. Noghin and O. V. Baskov Presented by Academician S.K. Korovin January 21, 2011 Received February 17, 2011 DOI: 10.1134/S1064562411030288 St. Petersburg State University, Universitetskii pr. 28, Petrodvorets, St. Petersburg, 198504 Russia e-mail: noghin@gmail.com COMPUTER SCIENCE