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LinearProgramming—Wolfram Language Documentation

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• See Also
  • NMinimize
  • Minimize
  • FindMinimum
• Related Guides
  • Matrices and Linear Algebra
  • Linear Systems
  • Graph Programming
• Tech Notes
  • Numerical Optimization
  • Constrained Optimization
  • Unconstrained Optimization
  • Implementation notes: Numerical and Related Functions

LinearProgramming[c,m,b]

finds a vector x that minimizes the quantity c.x subject to the constraints m.x≥b and x≥0.

LinearProgramming[c,m,{{b₁,s₁},{b₂,s₂},…}]

finds a vector x that minimizes c.x subject to x≥0 and linear constraints specified by the matrix m and the pairs {b_i,s_i}. For each row m_i of m, the corresponding constraint is m_i.x≥b_i if s_i==1, or m_i.x==b_i if s_i==0, or m_i.x≤b_i if s_i==-1.

LinearProgramming[c,m,b,l]

minimizes c.x subject to the constraints specified by m and b and x≥l.

LinearProgramming[c,m,b,{l₁,l₂,…}]

minimizes c.x subject to the constraints specified by m and b and x_i≥l_i.

LinearProgramming[c,m,b,{{l₁,u₁},{l₂,u₂},…}]

minimizes c.x subject to the constraints specified by m and b and l_i≤x_i≤u_i.

LinearProgramming[c,m,b,lu,dom]

takes the elements of x to be in the domain dom, either Reals or Integers.

LinearProgramming[c,m,b,lu,{dom₁,dom₂,…}]

takes x_i to be in the domain dom_i.

Minimize , subject to constraint and implicit non-negative constraints:

LinearProgramming has been superseded by LinearOptimization:

Solve the problem with equality constraint and implicit non-negative constraints:

Use LinearOptimization to solve the problem:

Solve the problem with equality constraint and implicit non-negative constraints:

Use LinearOptimization to solve the problem:

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:

"InteriorPoint" is faster than "Simplex" or "RevisedSimplex", 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 "InteriorPoint" method only works for machine numbers:

The "InteriorPoint" method may return a solution in the middle of the optimal solution set:

The "Simplex" 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 "InteriorPoint" 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:

Introduced in 1991 (2.0) | Updated in 2003 (5.0) ▪ 2007 (6.0)

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