Квадратичнi лексикографiчнi задачi оптимiзацiї i вiдображення Лагранжа

In different spheres of science and technology almost any appearing complicated problem of the optimal selection is multicriterion, since in course of searching for the best alternative we have to take into consideration a number of different requirements and some of them may even contradict each other. Meanwhile in practice linear and quadratic functions are the most frequently applied as criterion ones. They enable us to describe processes under investigation rather adequately and to apply known and studied algorithms for solving the problems. The first methods used for solving multicriterion problems of optimization were the methods based on the approach of reducing an input problem to a one-criterion one. However, the given procedure in most cases leads to dramatic changes of properties of an input problem and thus to unjustified replacing of a multicriterion model of the problem by a one-criterion model. We also should remember about calculation difficulties caused by the loss of properties of an input problem and about the impossibility of applying known algorithms in order to solve corresponding one-criterion problems. So developing methods for solving vector problems of optimization, in which original properties of criterion and limitation functions are not lost, remains relevant. In terms of one-criterion optimization a number of algorithms for extremum searching are elaborated using the dual approach. This issue is also of great importance to multicriterion optimization problems. The article deals with convex quadratic problems of lexicographic optimization within the set prescribed by a linear inequality system and with the issue of drawing up dual problems to them. Dual problems to an original one are constructed by means of Lagrange’s reflection where Lagrange multipliers are vector variables and the value set of each of them is a set of vectors of the space, the dimensionality of which is equal to a quantity of partial criteria and which is prescribed by some lexicographic order. Necessary and sufficient conditions of the existence and optimality of lexicographic solutions of the derived problem are established. We suggest an approach that enables the reducing of the solving of a lexicographic optimization problem to the sequence of inequality and equation systems in lexicographic order. The scalarization scheme analogue and applying of properties of Lagrange’s reflection constructed for a vector function and limitations underlie the approach. There is also promising possibility of constructing calculation algorithms for solving a lexicographic problem of quadratic optimization which are based on the dual approach.