Introduction to Linear Programming

Introduction to Linear Programming

Introduction to Linear Programming

Introduction to Linear Programming

Excerpt

Problems involving the determination of the minimum or maximum of a function are a part of classical mathematical analysis and have been denoted by the collective term extreme-value problems. The simplest extreme-value problem is the determination of the extremum (the minimum or maximum) of a function of a single unconstrained variable, and of the position where this extremum occurs. This problem is handled easily by solving the equation that results when the derivative of the function is set equal to zero. If we are dealing with a function of more than one variable, and the variables are free to assume any value, then each partial derivative of the function is set equal to zero, and we must then determine the solution of a system of equations. Situations also arise where the variables do not have complete freedom to assume any value but must satisfy a subsidiary system of equations. The problem then consists of finding an extreme value of the function, subject to the condition that the variables must satisfy the subsidiary system. This is a constrained extreme-value problem because the variables are constrained by the subsidiary system. Classical methods of analysis can be used to handle this problem, and they lead us eventually again to the determination of the solution of a system of equations.

The situation becomes more complex, however, if some or all of the variables of a constrained extreme-value problem are required to be nonnegative. Such problems arise, for example, if the subsidiary system consists of inequalities instead of equations. Classical methods are no longer applicable because they do not furnish us any assurance that the values of those variables that are constrained in sign will not turn out to be negative. The general constrained extreme-value problem with nonnegative variables is currently a very active field of research, but it is still too early, at the present time, to arrive at general conclusions. During the last decade, however, a great deal of attention has been devoted to the special class of constrained extreme-value problems with nonnegative variables where all relations are linear, and a method for handling . . .

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