representation of the feasible integer solutions for."/>
Figure 1.4 Feasible integer solutions for (1.5).
Although the graphical method was fairly simple for this example, it is quite different than when solving linear programs. Note that:
There is not necessarily an optimal solution at the intersection of two constraints. In our example, lies only on the constraint line . In other examples, the optimal solution may not lie on any constraint.
The optimal solution is not necessarily obtained by rounding the linear program optimal solution. In fact, there is no limit to how far the optimal solution could be from the linear programming solution.
The integer program can be infeasible when the linear program is feasible.
These difficulties suggest that integer programs are harder to solve than linear programs, which we will see is true. Even solving a two‐variable integer program graphically can be tedious. However, we do not necessarily need to generate all integer feasible solutions. For example, if we start by generating the feasible point
The general form of an integer program is the same as a linear program with the added constraint that the variables are integers. If only some of the variables are restricted to be integer, it is called a mixed integer program. When variables represent logical choices, they are usually defined so that
General Form of an Integer Program
1.4.3 Nonlinear Programs
An optimization problem where the objective function and constraints may be nonlinear is called a nonlinear program. The variables are assumed to be continuous, as in a linear program. Thus, nonlinear programs are more general than linear programs, but do not include integer programs. While linear programs can be described in matrix notation, nonlinear programs are described in terms of functions.
General Form of a Nonlinear Program
We have not stated the nonnegativity constraints separately. However, a nonnegativity constraint can be included in the functional constraints if needed.
The ability to solve a nonlinear program depends on the type of objective function
Problems
For Exercises 1–6, solve the linear program graphically.
For Exercises 6–8, solve the integer program graphically.
1
2
3
4 a) Solve as stated. b) Change “min” to “max” and solve.
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6
7
8
9 For the linear program (1.1)Suppose the objective is to minimize the cost of aid given in 1.2. What is the optimal solution? Explain why minimizing cost is not a reasonable objective for this problem.Find an objective function for which , is optimal. Show graphically that this point is optimal.
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