More attention has been devoted to linear programming in this text than to any other single topic. It is a very versatile technique that can be and has been applied to a wide variety of problems. Besides chapters devoted specifically to linear programming models and applications, we have also presented several variations of linear programming, integer and goal programming, and unique applications of linear programming for transportation and assignment problems. In all these cases, all the objective functions and constraints were linear; that is, they formed a line or plane in space. However, many realistic business problems have relationships that can be modeled only with nonlinear functions. When problems fit the general linear programming format but include nonlinear functions, they are referred to as nonlinear programming problems.
Nonlinear programming problems are given a separate name because they are solved in a different manner than are linear programming problems. In fact, their solution is considerably more complex than that of linear programming problems, and it is often difficult, if not impossible , to determine an optimal solution, even for a relatively small problem. In linear programming problems, solutions are found at the intersections of lines or planes, and though there may be a very large number of possible solution points, the number is finite, and a solution can eventually be found. However, in nonlinear programming there may be no intersection or corner points; instead, the solution space can be an undulating line or surface, which includes virtually an infinite number of points. For a realistic problem, the solution space may be like a mountain range, with many peaks and valleys, and the maximum or minimum solution point could be at the top of any peak or at the bottom of any valley. What is difficult in nonlinear programming is determining if the point at the top of a peak is just the highest point in the immediate area (called a local optimal , in calculus terms) or the highest point of all (called the global optimal ).
The solution techniques for nonlinear programming problems generally involve searching the solution surface for peaks or valleysthat is, high points or low points. The problem encountered by these methods is that they sometimes have trouble determining whether the high point they have identified is just a local optimal solution or the global optimal solution. Thus, finding a solution is often difficult and can involve very complex mathematics that are beyond the scope of this text.
In this chapter we present the basic structure of nonlinear programming problems and use Excel to solve simple models.