This chapter introduces the concept of **nonlinear programming**.

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_{j}x_{j}

subject to

a_{ij}x_{j} ( ≤, =,≥) b_{i}; i = 1, 2,
....., m

x_{j } ≥ 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 (x_{1}, x_{2}, ......., x_{n})

subject to

a_{i} (x_{1}, x_{2}, ......., x_{n}) ≤ 0, for i = 1, 2, ...., m

where both c (x_{1}, x_{2}, ......., x_{n})
and a_{i} (x_{1}, x_{2}, ......., x_{n}) are real-valued, nonlinear functions of n real variables.