Mukund N. Thapa

Linear and Nonlinear Optimization

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NLP MODELS AND APPLICATIONS

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LP MODELS AND APPLICATIONS

We now turn to the study of nonlinear programming, also known as nonlinear optimization. In this chapter we discuss the differences between linear and nonlinear programming and state the form of the general nonlinear optimization model. We will see that in contrast to linear programming, it does make sense to talk about unconstrained nonlinear optimization problems.

INTERIOR-POINT METHODS

We begin this chapter with an example of a linear programming problem and then we go on to define linear programs in general. After that we discuss the main topic of this chapter: the classical models and applications of linear programming.

THE SIMPLEX ALGORITHM

This chapter discusses interior-point methods for linear programming as promised in Chap. 7. Their placement here is justified by the fact that they rely on the theory and methods of nonlinear optimization for which they were originally developed.

PROBLEMS WITH LINEAR CONSTRAINTS

There are several ways to solve linear programs, but even after its invention in 1947 and the emergence of many new rivals, George B. Dantzig’s Simplex Algorithm stands out as the foremost method of all.

THE SIMPLEX ALGORITHM CONTINUED

The optimization methods presented in this chapter are for solving the important class of nonlinear programs with linear constraints, that is, linear equations and/or linear inequalities. The algorithms covered here are based on ones designed for unconstrained optimization, but they are modified to take account of the constraints.

SOME COMPUTATIONAL CONSIDERATIONS

The discussion of the Simplex Algorithm for linear programming presented in the previous chapter included two crucial assumptions. The first was that a starting basic feasible solution was known and, moreover, that the system was in canonical form with respect to this feasible basis.

DUALITY AND THE DUAL SIMPLEX ALGORITHM

In this chapter we take up some additional techniques of a mostly practical nature: the handling of linear programs with explicitly bounded variables; the construction of a starting (feasible) basis; structured linear programs; the steepest-edge rule for column selection; the rare (but possible) exponential behavior of the Simplex Algorithm. Each of these is a large subject in its own right, so we shall limit our discussion to the most important fundamental concepts.

PROBLEMS WITH NONLINEAR CONSTRAINTS

In this chapter we present what is probably the most important theoretical aspect of linear programming: duality. This beautiful topic has more than theoretical charms. Indeed, it gives valuable insights into computational matters and economic interpretations such as the value of resources at an optimal solution.