Властивості Евклідових задач лексикографічної комбінаторної оптимізації на розміщеннях

Abstract

In the paper Euclidean problems of lexicographic combinatorial; optimization are discussed. These problems are to find lexicographically minimal (for minimization problems) or lexicographically maximal (for maximization problems) points among those that give the extremum of the objective function on a given Euclidean combinatorial set. The properties of linear and linear-fractional problems of lexicographic combinatorial optimization on a general set of arrangements are substantiated. The results obtained in the work are based on the previously known criteria of extremals of linear and linear-fractional functions on arrangements: any extremal is an element of certain set of polyarrangements (for linear problems the form of the extremal set is established explicitly, for linear-fractional problems the polyarrangement set is formed on the basis of some known extremal). In the paper we substantiate the form of points that are an lexicographic minimal and lexicographic maximal of linear function on the general set of arrangements. In particular if elements of multiset are in nondecreasing order, coefficients of objective function are in nonincreasing order and s is the least index such that corresponding coefficient of objective function is negative, tehn lexicographic minimal if formed as s – 1 first and k – s + 1 ( k is the dimension of space) last elements of multiset which are in nondecreasing order. For problems with linear-fractional function we obtain the method of forming solution of lexicographic combinatorial problem on arrangements, if any minimal (for minimization problems) or any maximal (for maximization problems) of objective function on given set of arrangements is known. In this case ordering of components of the extremal is carried out taking into account ordering for nonincreasing of coefficients of special linear function