This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# LinearProgramming

 LinearProgrammingfinds a vector x that minimizes the quantity subject to the constraints and x≥0. LinearProgrammingfinds a vector x that minimizes subject to x≥0 and linear constraints specified by the matrix m and the pairs . For each row of m, the corresponding constraint is if , or if , or if . LinearProgrammingminimizes subject to the constraints specified by m and b and . LinearProgrammingminimizes subject to the constraints specified by m and b and . LinearProgrammingminimizes subject to the constraints specified by m and b and . LinearProgrammingtakes the elements of x to be in the domain dom, either Reals or Integers. LinearProgrammingtakes to be in the domain .
• All entries in the vectors c and b and the matrix m must be real numbers.
• None is equivalent to specifying no bounds.
• LinearProgramming gives exact rational number or integer results if its input consists of exact rational numbers.
• LinearProgramming finds approximate numerical results if its input contains approximate numbers. The option Tolerance specifies the tolerance to be used for internal comparisons. The default is Tolerance, which does exact comparisons for exact numbers, and uses tolerance for approximate numbers.
Minimize , subject to constraint and implicit non-negative constraints:
Solve the problem with equality constraint and implicit non-negative constraints:
Solve the problem with equality constraint and implicit non-negative constraints:
Minimize , subject to constraint and implicit non-negative constraints:
 Out[1]=
Solve the problem with equality constraint and implicit non-negative constraints:
 Out[2]=
Solve the problem with equality constraint and implicit non-negative constraints:
 Out[3]=
 Scope   (6)
Minimize , subject to constraint and lower bounds , :
Minimize , subject to constraint and bounds , :
Minimize , subject to constraint and upper bounds , :
Minimize , subject to constraint and implicit non-negative constraints:
Minimize subject to bounds and only:
Solve the same kind of problem, but with both variables integers:
Solve the same problem, but with the first variable an integer:
Solve larger LPs, in this case 200,000 variables and 10,000 constraints:
Objective, constraints, and bounds can all be specified as SparseArray:
 Options   (2)
is faster than or , though it only works for machine-precision problems:
If an approximated solution is sufficient, a loose Tolerance option makes the solution process faster:
A linear programming problem can also be solved using Minimize:
NMinimize or FindMinimum can be used to solve inexact linear programming problems:
The integer programming algorithm is limited to the machine-number problems:
The method only works for machine numbers:
The method may return a solution in the middle of the optimal solution set:
The method always returns a solution at a corner of the optimal solution set:
In this case the optimal solution set is the set of all points on the line segment between and :
The method may not always be able to tell if a problem is infeasible or unbounded:
This expresses the Klee-Minty problem of dimension in LinearProgramming syntax:
Because scaling is applied internally, the simplex algorithm converges very quickly: