PARETO-OPTIMALITY AND L-OPTIMAL FOR SOLVING SOME CLASSES OF OPTIMAL CONTROL PROBLEM

Abstract

Many practical problems of economic activities and a number of important issues of economic theory are connected to the choice of optimal solution. An adequate economic theory should reflect the process of continuous development of the economic system; therefore, it is necessary to consider the economic models in which all economic variables depend on time, and to have a mathematical tool that allows to find optimal values of these variables. The theory of optimal control is a mathematical tool for just such a purpose. The classical theory of optimal control considers models in which the behavior of the system is described by a set of differential equations while the functional is given to define the purpose of control and a variety of the limited control actions is set. An important tool for solving such problems is the Pontryagin principle of maximum. However, the use of the maximum principle leads to many computational problems. That is why the concept of Pareto optimality is used to solve certain classes of computational problems. The broader concept is a L-optimality, its definition was introduced in P.L. Yu (Cone. Cone convexity, cone extreme points, and non-dominated solutions in decision problems with multi-objectives // Optim. Theory Appl. 1974. Vol. 14. № 3). It shows that a plurality of L-optimal decisions can be wider or narrower than the set of Pareto-optimalities. The article highlights the classes of problems that are easy to solve using the Pareto- and L-optimality. The solution of advertising management problem is given for illustration purposes.

Keywords

• l-оптимальность
• pareto-optimality
• l-optimality
• convex closed set
• dimension matrix
• functional
• cone generators