Kozin I. Mathematical models of placing, packing and distributing with the condition of invariance in relation to the groups of transformations

Dissertation is devoted to the questions of substantiation and testing of new mathematical models and methods of search of approximate solutions in discrete optimization problems, in problems with the criterion of symmetry, methods of search of Pareto-optimal solutions of multi-criteria problems with the conditions of invariance in relation to the groups of transformations, research of fragmentary structures and evolutionary models for the search of optimal solutions, application of the developed methods. The concept of measure of symmetry is introduced, the set of problems of discrete optimization is formulated with the criterion of symmetry, computational complexity of these problems is set. For the search of approximate solutions in the problems of discrete optimization with hard-to-formalize criteria fragmentary approach is suggested which is perspective for the use in the decision support systems of "man-machine" class. Relationship of fragmentary model with the matroid theory is set. An universal evolutionary-fragmentary model is proposed for the search of approximate solutions in discrete optimization problems. Relationship of property of heredity in an evolutionary model with the axiomatic theory of convex sets in permutations space is set. The performance bound of evolutionary algorithm is generalized to the case of evolutionary-fragmentary model. For the substantiation of choice of solution in the multi-criteria optimization problems it is suggested to use principle of invariance of solution in relation to the certain groups of transformations. Application of such approach is given by possibility a priori to build equations of choice for an integral criterion in the problem of multi-criteria optimization. For the simplest groups of transformations it is succeeded to get the analytical solution of these equations. On the basis of principle of symmetry the pair domination methods of optimum choice by majority voting are generalized for the arbitrary relations. Keywords: decision making models, measure of symmetry, fragmentary models, matroids, evolutionary models, axiomatic theory of convexity, decision support systems, interactive systems of class "man-machine", multi-criteria optimization, Lie groups.