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ABSTRACT. We consider quasistable muiticriteria problems of discrete optimization on systems of subsets (trajectory problems). We single out the class of problems for which new Pareto optima can appear, while other optima for the problems do not disappear when the coefficients of the objective functions are slightly perturbed (in the Chebyshev metric). Forthe case of linear criteria (MINSUM), we obtain a formula for calculating the quasistability radius of the problem.KEY WORDS: multicriteria optimization, trajectory problem, Pareto optimum.Hadamard [1] included the stability condition in the definition of a well-posed (i.e., reasonably formulated) problem. Since his work the problem of stability is regarded as a central mathematical problem.Usually the stability of an optimization problem (monocriteria or multicriteria) means that the solutions depend continuously on the parameters of the problem. The most general approaches to the investigation of stability of optimization problems are based on the study of multivalued mappings that determine the selector function (see, for example, [2][3][4][5]).Ordinary methods of calculus are not well adapted to problems of discrete optimization because of the complexity of discrete models. In fact, the behavior of these models under slight perturbations of initial data is often unpredictable [6,7]. On the other hand, the stability problem can be stated more naturally if we do not use the terminology of general topology for the set of isolated points. In particular, the optixnal mapping is upper (lower) semicontinuous in the sense of Hausdorff if and only if no new solutions of a problem appear (the initial solutions survive) under slight perturbations of parameters of the problem. The notion of stability [6.--12] (quasistability [6,7,11,12]) of discrete optimization problems arises naturally in this context.In this paper we study the quasistability of vector trajectory problems (that is, problems on a system of subsets of a finite set) with the individual criteria most often used in discrete optimization. We find the lower (attainable) bound of the quasistability radius, as well as a quasistability criterion of this problem. We also obtain some (easily verified) sufficient conditions of quasistability.Previously a similar research was carried out for vector problems of integral linear programming [12], and also for trajectory problems with linear criteria [11]. w DefinitionsConsider a system of subsets (E, T), where E = {el,..., e,~} and T is a collection of ~rajectories,

Recent results of investigations concerning the domain of definition of both the two-index (classical) transportation linear programming problem and the multi-index axial transportation problem are observed from a general point of view. At the end of the survey open problems and conjectures are formulated.

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