Chapter 15

A linear programming problem is characterized by the presence of linear constraints and linear objective function in decision variables. A linear programming problem can be viewed as

Optimize (maximize or minimize) c_jx_j

subject to
a_ijx_j ( £, =, ³ ) b_i; i = 1, 2, ....., m

x_j ³ 0; j = 1, 2, ....., n

There are, however, problems in real life situations where neither the objective function nor the constraints are linear functions in decision variables. For example, in a model for a steel-processing plant, a variable representing the temperature of a blast furnance can be described by a nonlinear function of variables indicating the amount and duration of heat energy applied. Each of these variables, in turn, is contained in other constraints as well as in the objective function. The term nonlinear programming usually refers to problems such as

Maximize c (x₁, x₂, ......., x_n)

subject to
a_i (x₁, x₂, ......., x_n) £ 0, for i = 1, 2, ...., m

where both c (x₁, x₂, ......., x_n) and a_i (x₁, x₂, ......., x_n) are real-valued, nonlinear functions of n real variables.