A Nonreducible System of Constraints of a Combinatorial Polyhedron in a Linear-Fractional Optimization Problem on Arrangements

Abstract

A system of linear constraints is investigated. The system describes the domain of feasible solutions of a linear optimization problem to which a linear-fractional optimization problem on arrangements is reduced. A system of nonreducible constraints of a polyhedrom is established for the linear-fractional optimization problem on arrangements.Introduction. Recently, a great many works are published [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] that are devoted to the investigation of problems of combinatorial optimization and problems of fractional-linear programming. But problems on Euclidean combinatorial sets with homographic objective functions are insufficiently investigated.During the investigation of problems of Euclidean combinatorial optimization, the need arises for the description of convex envelopes of definite combinatorial sets, disclosure and substantiation of properties of sets of feasible solutions of such problems, and development of methods of solution of these problems. For sets of permutations, arrangements, combinations, polypermutations, and others, a part of these problems is solved [4][5][6][7][8][9][10][11][12][13][14][15][16][17].In [11], a nonreducible system of linear constraints of a general polyhedron of permutations is established. In [15], a theorem on redundant constraints of the system of inequalities of a combinatorial polyhedron in a linear-fractional optimization problem on permutations is proved. In [16], a nonreducible system of constraints for a general polyhedron of arrangements is established. Based on the investigated properties of Euclidean combinatorial optimization problems, algorithms for solution of these problems are proposed (see, for example [10,17] ) .The existence of optimization problems with homographic objective functions on arrangements requires the development of efficient optimization methods and algorithms that make it possible to solve individual problems and classes of such problems. This necessitates the study of properties of such optimization problems.Statement of the problem. In this article, we investigate a system of linear constraints of a combinatorial polyhedron for a problem with a linear objective function to which the problem on arrangements with a homographic objective function is reduced.Presentation of the basic material of this investigation with a complete substantiation of the scientific results obtained. We introduce the necessary terminology and cite facts from [7]. A multiset G g g g = { }

Solution of Euclidean combinatorial optimization problems by the method of construction of a lexicographic equivalence

“…Then we use a directed exhaustive search for l-classes, i.e., the passage from a given class to the previous or next one in order (2). If, at every step, the passage to the previous class is realized, then we speak of exhaustive search in lexicographically decreasing order and, otherwise, in increasing order.…”

“…A great number of investigations in the theory of optimization is devoted to Euclidean combinatorial optimization (see, for example, [1][2][3][4][5][6][7][8]). One of their aspects was the study of properties of convex envelopes of Euclidean combinatorial sets [1][2][3], and another aspect is the development of methods and algorithms for solution of optimization problems stated for some combinatorial sets [1, [4][5][6][7][8][9].…”

“…One of their aspects was the study of properties of convex envelopes of Euclidean combinatorial sets [1][2][3], and another aspect is the development of methods and algorithms for solution of optimization problems stated for some combinatorial sets [1, [4][5][6][7][8][9]. However, a further investigation of Euclidean combinatorial optimization problems showed that they have a number of common properties.…”

Solving optimization problems with linear-fractional objective functions and additional constraints on arrangements

“…Some properties of Euclidean combinatory sets are analyzed in [1][2][3][4][5]: general sets of combinations, permutations, and arrangements. Methods and algorithms for the solution of conditional linear problems on permutations and arrangements and linear-fractional conditional problems on a set of permutations are described in [1, [6][7][8].…”

“…of the problem (2)-(4) and that the objective function (2) is maximized. If the pair (2) is to be minimized, we pass to the maximization problem with the coefficients of the numerator of the objective function changed to opposite ones.Let us relax (2)-(4): replace condition(3)with (1).…”