Models for Systems of Linear Equations and Inequalities. Alternative Theorems.Models for Linear Programming Problems

Abstract

In the present chapter we shall consider concrete mechanical and physical systems equilibrium problems for which are equivalent to problems of solving systems of linear equations and inequalities and linear programming problems. Using these models, we will be able to give not only physical interpretations of the basic theoretical results but also treat several algorithms for obtaining the numerical solution as mathematical description of controlled processes of transition to the equilibrium state of the given physical system. The reader will find that the models to be examined in the present chapter are models for dual pairs of systems of linear equations and inequalities or of linear programming problems, and that the equilibrium state solves both the primal and dual problems. This is due to the fact that both primal and dual problems are equilibrium problems for the same physical system, the only difference being that the former concerns the question of finding the equilibrium state of the physical system and the latter a set of balancing forces applied to bodies of the system in the equilibrium state.