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Legacy Documentation

Mathematica 8 (2010)

This is documentation for Mathematica 8, which was
based on an earlier version of Wolfram Language.
View current documentation (Version 14.3)

LinearProgramming

LinearProgramming
finds a vector x that minimizes the quantity subject to the constraints and x≥0.
LinearProgramming
finds 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 .
LinearProgramming
minimizes subject to the constraints specified by m and b and .
LinearProgramming
minimizes subject to the constraints specified by m and b and .
LinearProgramming
minimizes subject to the constraints specified by m and b and .
LinearProgramming
takes the elements of x to be in the domain dom, either Reals or Integers .
LinearProgramming
takes 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 ->Automatic , which does exact comparisons for exact numbers, and uses tolerance for approximate numbers.
(1)
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]=
(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:
(1)
Objective, constraints, and bounds can all be specified as SparseArray :
(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:
(2)
A linear programming problem can also be solved using Minimize :
NMinimize or FindMinimum can be used to solve inexact linear programming problems:
(4)
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:
(1)
This expresses the Klee-Minty problem of dimension in LinearProgramming syntax:
Because scaling is applied internally, the simplex algorithm converges very quickly:
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