in later chapters, though for nonlinear programs we will focus on the more tractable subclass of convex programs. A mathematical program with variables However, we will always describe the feasible region using constraint equations or inequalities. The notation for the constraints is introduced in the following text.
1.4.1 Linear Programs
We have already seen two examples of linear programs.
General Form of a Linear Program
There are
We will use matrix notation for linear programs whenever possible. Let
Here
Example 1.2 Consider the linear program
Converting the objective function and constraints to matrix form, we have
If we let
then this linear program can be written
It is important to distinguish between the structure of an optimization problem and the data that provides a specific numerical example. The structure of (1.4) is a minimization linear program with “
To write a mixture of “
and write, e.g.
1.4.2 Integer Programs
Many practical optimization problems involve making discrete quantitative choices, such as how many fire trucks of each type a fire department should purchase, or logical choices, such as whether or not each drug being developed by a pharmaceutical company should be chosen for a clinical trial. Both types of situations can be modeled by integer variables.
Consider again the linear program (1.1) with the alternative objective function (1.2). The variables represent the number of pallets of tents and food. If we restrict the variables to be integers, i.e. we can only load whole pallets, then the problem becomes an integer program and can be stated
Without the integer restriction, it is a linear program with optimal solution
