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Operations Research Models and Methods
Models Section

Nonlinear Programming
Nonlinear Programming

The principal abstraction of the linear programming model is that all functions are linear. This leads to a number of powerful results that greatly facilitate our ability to find solutions. The first is that all local optima are global optima; the second is that if the optimal value of the objective function is finite, at least one of the extreme points in the set of feasible solutions will be an optimum.  Furthermore, starting at any extreme point in the feasible region, it is possible to reach an optimal extreme point in a finite number of steps by moving only to an adjacent extreme point in an improving direction.  The simplex method embodies these ideas and has proven to be extremely efficient.

Nevertheless, much of the world is nonlinear so it is natural to ask if it is possible to achieve the same efficiency with nonlinear models. In many contexts, the elements of a linear model are really approximations of more complex relationships. Economies of scale in manufacturing, for example, lead to decreasing costs, while biological systems commonly exhibit exponential growth.  In the design of a simple hatch cover, the shearing stress, bending stress and degree of defection are each polynomial functions of flange thickness and beam height. Similar relationships abound in engineering design, economics, and distribution systems, to name a few.

The appeal of nonlinear programming (NLP) is strong because of the modeling richness it affords. Unfortunately, NLP solvers have not yet achieved the same level of performance and reliability associated with LP solvers. For all but the most structured problems, the solution obtained from an NLP solver may not be globally optimal. This argues for caution. Before taking any action, the decision maker should have a full understanding of the nonlinearities governing the system under study.

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Operations Research Models and Methods
by Paul A. Jensen
Copyright 2004 - All rights reserved