Documentation

linprog

Solve linear programming problems

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Linear programming solver

Finds the minimum of a problem specified by

minxfTx such that {A⋅x≤b,Aeq⋅x=beq,lb≤x≤ub.

f, x, b, beq, lb, and ub are vectors, and A and Aeq are matrices.

Syntax

  • x = linprog(f,A,b)
    example
  • x = linprog(f,A,b,Aeq,beq)
    example
  • x = linprog(f,A,b,Aeq,beq,lb,ub)
    example
  • x = linprog(f,A,b,Aeq,beq,lb,ub,x0)
  • x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options)
    example
  • x = linprog(problem)
  • [x,fval] = linprog(___)
    example
  • [x,fval,exitflag,output] = linprog(___)
    example
  • [x,fval,exitflag,output,lambda] = linprog(___)
    example

Description

example

x = linprog(f,A,b) solves min f'*x such that A*x ≤ b.

example

x = linprog(f,A,b,Aeq,beq) includes equality constraints Aeq*x = beq. Set A = [] and b = [] if no inequalities exist.

example

x = linprog(f,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that the solution is always in the range lb ≤ x ≤ ub. Set Aeq = [] and beq = [] if no equalities exist.

    Note:   If the specified input bounds for a problem are inconsistent, the output x is x0 and the output fval is [].

x = linprog(f,A,b,Aeq,beq,lb,ub,x0) sets the starting point to x0.

    Note   linprog uses x0 for only the 'active-set' algorithm. For all other algorithms, linprog ignores x0.

example

x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options) minimizes with the optimization options specified by options. Use optimoptions to set these options.

x = linprog(problem) finds the minimum for problem, where problem is a structure described in Input Arguments.

Create the problem structure by exporting a problem from Optimization app, as described in Exporting Your Work. You can import a problem structure from an MPS file using mpsread.

example

[x,fval] = linprog(___), for any input arguments, returns the value of the objective function fun at the solution x: fval = f'*x.

example

[x,fval,exitflag,output] = linprog(___) additionally returns a value exitflag that describes the exit condition, and a structure output that contains information about the optimization process.

example

[x,fval,exitflag,output,lambda] = linprog(___) additionally returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.

Examples

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Linear Program, Linear Inequality Constraints

Solve a simple linear program defined by linear inequalities.

For this example, use these linear inequality constraints:

$$ x(1) + x(2) \le 2$$

$$x(1) + x(2)/4 \le 1$$

$$x(1) - x(2) \le 2$$

$$-x(1)/4 - x(2) \le 1$$

$$-x(1) - x(2) \le -1$$

$$-x(1) + x(2) \le 2.$$

A = [1 1
    1 1/4
    1 -1
    -1/4 -1
    -1 -1
    -1 1];

b = [2 1 2 1 -1 2];

Use the objective function $-x(1) - x(2)/3$.

f = [-1 -1/3];

Solve the linear program.

x = linprog(f,A,b)

Optimization terminated.

x =

    0.6667
    1.3333

Linear Program with Linear Inequalities and Equalities

Solve a simple linear program defined by linear inequalities and linear equalities.

For this example, use these linear inequality constraints:

$$ x(1) + x(2) \le 2$$

$$x(1) + x(2)/4 \le 1$$

$$x(1) - x(2) \le 2$$

$$-x(1)/4 - x(2) \le 1$$

$$-x(1) - x(2) \le -1$$

$$-x(1) + x(2) \le 2.$$

A = [1 1
    1 1/4
    1 -1
    -1/4 -1
    -1 -1
    -1 1];

b = [2 1 2 1 -1 2];

Use the linear equality constraint $x(1) + x(2)/4 = 1/2$.

Aeq = [1 1/4];
beq = 1/2;

Use the objective function $-x(1) - x(2)/3$.

f = [-1 -1/3];

Solve the linear program.

x = linprog(f,A,b,Aeq,beq)

Optimization terminated.

x =

    0.0000
    2.0000

Linear Program with All Constraint Types

Solve a simple linear program with linear inequalities, linear equalities, and bounds.

For this example, use these linear inequality constraints:

$$ x(1) + x(2) \le 2$$

$$x(1) + x(2)/4 \le 1$$

$$x(1) - x(2) \le 2$$

$$-x(1)/4 - x(2) \le 1$$

$$-x(1) - x(2) \le -1$$

$$-x(1) + x(2) \le 2.$$

A = [1 1
    1 1/4
    1 -1
    -1/4 -1
    -1 -1
    -1 1];

b = [2 1 2 1 -1 2];

Use the linear equality constraint $x(1) + x(2)/4 = 1/2$.

Aeq = [1 1/4];
beq = 1/2;

Set these bounds:

$$-1 \le x(1) \le 1.5$$

$$-0.5 \le x(2) \le 1.25 .$$

lb = [-1,-0.5];
ub = [1.5,1.25];

Use the objective function $-x(1) - x(2)/3$.

f = [-1 -1/3];

Solve the linear program.

x = linprog(f,A,b,Aeq,beq,lb,ub)

Optimization terminated.

x =

    0.1875
    1.2500

Linear Program Using the Dual-Simplex Algorithm

Solve a linear program using the 'dual-simplex' algorithm.

For this example, use these linear inequality constraints:

$$ x(1) + x(2) \le 2$$

$$x(1) + x(2)/4 \le 1$$

$$x(1) - x(2) \le 2$$

$$-x(1)/4 - x(2) \le 1$$

$$-x(1) - x(2) \le -1$$

$$-x(1) + x(2) \le 2.$$

A = [1 1
    1 1/4
    1 -1
    -1/4 -1
    -1 -1
    -1 1];

b = [2 1 2 1 -1 2];

Use the linear equality constraint $x(1) + x(2)/4 = 1/2$.

Aeq = [1 1/4];
beq = 1/2;

Set these bounds:

$$-1 \le x(1) \le 1.5$$

$$-0.5 \le x(2) \le 1.25 .$$

lb = [-1,-0.5];
ub = [1.5,1.25];

Use the objective function $-x(1) - x(2)/3$.

f = [-1 -1/3];

Set options to use the 'dual-simplex' algorithm.

options = optimoptions('linprog','Algorithm','dual-simplex');

Set the initial point to [], because the dual-simplex algorithm does not accept an initial point x0.

x0 = [];

Solve the linear program using the 'dual-simplex' algorithm.

x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options)

Optimal solution found.


x =

    0.1875
    1.2500

Return the Objective Function Value

Calculate the solution and objective function value for a simple linear program.

The inequality constraints are

$$ x(1) + x(2) \le 2$$

$$x(1) + x(2)/4 \le 1$$

$$x(1) - x(2) \le 2$$

$$-x(1)/4 - x(2) \le 1$$

$$-x(1) - x(2) \le -1$$

$$-x(1) + x(2) \le 2.$$

A = [1 1
    1 1/4
    1 -1
    -1/4 -1
    -1 -1
    -1 1];

b = [2 1 2 1 -1 2];

The objective function is $-x(1) - x(2)/3$.

f = [-1 -1/3];

Solve the problem and return the objective function value.

[x,fval] = linprog(f,A,b)

Optimization terminated.

x =

    0.6667
    1.3333


fval =

   -1.1111

Obtain More Output to Examine the Solution Process

Obtain the exit flag and output structure to better understand the solution process and quality.

For this example, use these linear inequality constraints:

$$ x(1) + x(2) \le 2$$

$$x(1) + x(2)/4 \le 1$$

$$x(1) - x(2) \le 2$$

$$-x(1)/4 - x(2) \le 1$$

$$-x(1) - x(2) \le -1$$

$$-x(1) + x(2) \le 2.$$

A = [1 1
    1 1/4
    1 -1
    -1/4 -1
    -1 -1
    -1 1];

b = [2 1 2 1 -1 2];

Use the linear equality constraint $x(1) + x(2)/4 = 1/2$.

Aeq = [1 1/4];
beq = 1/2;

Set these bounds:

$$-1 \le x(1) \le 1.5$$

$$-0.5 \le x(2) \le 1.25 .$$

lb = [-1,-0.5];
ub = [1.5,1.25];

Use the objective function $-x(1) - x(2)/3$.

f = [-1 -1/3];

Set options to use the 'dual-simplex' algorithm.

options = optimoptions('linprog','Algorithm','dual-simplex');

Set the initial point to [], because the dual-simplex algorithm does not accept an initial point x0.

x0 = [];

Solve the linear program and request the function value, exit flag, and output structure.

[x,fval,exitflag,output] = linprog(f,A,b,Aeq,beq,lb,ub,x0,options)

Optimal solution found.


x =

    0.1875
    1.2500


fval =

   -0.6042


exitflag =

     1


output = 

         iterations: 0
    constrviolation: 0
            message: 'Optimal solution found.'
          algorithm: 'dual-simplex'
      firstorderopt: 0

  • fval, the objective function value, is larger than Return the Objective Function Value, because there are more constraints.

  • exitflag = 1 indicates that the solution is reliable.

  • output.iterations = 0 indicates that linprog found the solution during presolve, and did not have to iterate at all.

Obtain Solution and Lagrange Multipliers

Solve a simple linear program and examine the solution and the Lagrange multipliers.

Use the objective function

$$f(x) = -5x_1-4x_2-6x_3.$$

f = [-5; -4; -6];

Use the linear inequality constraints

$$ x_1 - x_2 + x_3 \le 20$$

$$ 3x_1 + 2x_2 + 4x_3 \le 42$$

$$ 3x_1 + 2x_2 \le 30.$$

A =  [1 -1  1
      3  2  4
      3  2  0];
b = [20;42;30];

Constrain all variables to be positive:

$$x_1\ge 0$$

$$x_2\ge 0$$

$$x_3\ge 0.$$

lb = zeros(3,1);

Set Aeq and beq to [], indicating that there are no linear equality constraints.

Aeq = [];
beq = [];

Call linprog, obtaining the Lagrange multipliers.

[x,fval,exitflag,output,lambda] = linprog(f,A,b,Aeq,beq,lb);

Optimization terminated.

Examine the solution and Lagrange multipliers.

x,lambda.ineqlin,lambda.lower

x =

    0.0000
   15.0000
    3.0000


ans =

    0.0000
    1.5000
    0.5000


ans =

    1.0000
    0.0000
    0.0000

lambda.ineqlin is nonzero for the second and third components of x. This indicates that the second and third linear inequality constraints are satisfied with equalities:

$$ 3x_1 + 2x_2 + 4x_3 = 42$$

$$ 3x_1 + 2x_2 = 30.$$

Check that this is true:

A*x

ans =

  -12.0000
   42.0000
   30.0000

lambda.lower is nonzero for the first component of x. This indicates that x(1) is at its lower bound of 0.

Related Examples

Input Arguments

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f — Objective functionreal vector

Objective function, specified as a real vector. linprog attempts to find an x that minimizes the objective function

f'*x = f(1)*x(1) + ... + f(N)*x(N),

where N is the number of variables in the problem.

Example: f = [1,3,5,-6]

Data Types: double

A — Linear inequality constraintsreal matrix

Linear inequality constraints, specified as a real matrix. A is an M-by-N matrix, where M is the number of inequalities, and N is the number of variables (number of elements in x0). For large problems, pass A as a sparse matrix.

A encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and b is a column vector with M elements.

For example, to specify

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30,

give these constraints:

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the x-components add up to 1 or less, take A = ones(1,N) and b = 1

Data Types: double

b — Linear inequality constraintsreal vector

Linear inequality constraints, specified as a real vector. b is an M-element vector related to the A matrix. If you pass b as a row vector, solvers internally convert b to the column vector b(:). For large problems, pass b as a sparse vector.

b encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and A is a matrix of size M-by-N.

For example, to specify

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30,

give these constraints:

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the x-components sum to 1 or less, take A = ones(1,N) and b = 1

Data Types: double

Aeq — Linear equality constraintsreal matrix

Linear equality constraints, specified as a real matrix. Aeq is an Me-by-N matrix, where Me is the number of equalities, and N is the number of variables (number of elements in x0). For large problems, pass Aeq as a sparse matrix.

Aeq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and beq is a column vector with Me elements.

For example, to specify

x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20,

give these constraints:

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the x-components sum to 1, take Aeq = ones(1,N) and beq = 1

Data Types: double

beq — Linear equality constraintsreal vector

Linear equality constraints, specified as a real vector. beq is an Me-element vector related to the Aeq matrix. If you pass beq as a row vector, solvers internally convert beq to the column vector beq(:). For large problems, pass beq as a sparse vector.

beq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and Aeq is a matrix of size Meq-by-N.

For example, to specify

x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20,

give these constraints:

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the x-components sum to 1, take Aeq = ones(1,N) and beq = 1

Data Types: double

lb — Lower boundsreal vector | real array

Lower bounds, specified as a real vector or real array. If the number of elements in x0 is equal to that of lb, then lb specifies that

x(i) >= lb(i) for all i.

If numel(lb) < numel(x0), then lb specifies that

x(i) >= lb(i) for 1 <= i <= numel(lb).

In this case, solvers issue a warning.

Example: To specify that all x-components are positive, lb = zeros(size(x0))

Data Types: double

ub — Upper boundsreal vector | real array

Upper bounds, specified as a real vector or real array. If the number of elements in x0 is equal to that of ub, then ub specifies that

x(i) <= ub(i) for all i.

If numel(ub) < numel(x0), then ub specifies that

x(i) <= ub(i) for 1 <= i <= numel(ub).

In this case, solvers issue a warning.

Example: To specify that all x-components are less than one, ub = ones(size(x0))

Data Types: double

x0 — Initial pointreal vector | real array

Initial point, specified as a real vector or real array. Only the 'active-set' algorithm uses x0, all other solvers ignore its value.

Example: x0 = ones(size(f))

Data Types: double

options — Optimization optionsoutput of optimoptions | structure as optimset returns

Optimization options, specified as the output of optimoptions or a structure as optimset returns.

Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.

All Algorithms
Algorithm

Choose the optimization algorithm:

  • 'interior-point-legacy' (default)

  • 'interior-point'

  • 'dual-simplex'

  • 'active-set'

  • 'simplex'

For information on choosing the algorithm, see Linear Programming Algorithms.

Diagnostics

Display diagnostic information about the function to be minimized or solved. Choose 'off' (default) or 'on'.

Display

Level of display (see Iterative Display):

  • 'final' (default) displays just the final output.

  • 'off' or 'none' displays no output.

  • 'iter' displays output at each iteration. The 'iter' option is unavailable for the 'active-set' algorithm.

LargeScale

Use Algorithm instead

Use the 'interior-point-legacy' algorithm when set to 'on' (default). Use a medium-scale algorithm when set to 'off' (see Simplex in options). For more information, see Choosing the Algorithm.

MaxIter

Maximum number of iterations allowed, a positive integer. The default is:

  • 85 for the 'interior-point-legacy' algorithm

  • 200 for the 'interior-point' algorithm

  • 10*(numberOfEqualities + numberOfInequalities + numberOfVariables) for the 'dual-simplex' algorithm

  • 10*numberOfVariables for the 'simplex' algorithm

  • 10*max(numberOfVariables, numberOfInequalities + numberOfBounds) for the 'active-set' algorithm

See Tolerances and Stopping Criteria and Iterations and Function Counts.

TolFun

Termination tolerance on the function value, a positive scalar. The default is:

  • 1e-8 for the 'interior-point' algorithm

  • 1e-7 for the 'dual-simplex' algorithm

  • 1e-6 for the 'interior-point' and 'simplex' algorithms

  • The option is not used for the 'active-set' algorithm

TolFun measures dual feasibility tolerance.

interior-point Algorithm
Preprocess

Level of LP preprocessing prior to algorithm iterations. Specify 'basic' (default) or 'none'.

TolCon

Feasibility tolerance for constraints, a scalar from 1e-10 through 1e-3. TolCon measures primal feasibility tolerance. The default is 1e-6.

Dual-Simplex Algorithm
MaxTime

Maximum amount of time in seconds that the algorithm runs. The default is Inf.

Preprocess

Level of LP preprocessing prior to dual simplex algorithm iterations. Specify 'basic' (default) or 'none'.

TolCon

Feasibility tolerance for constraints, a scalar from 1e-10 through 1e-3. TolCon measures primal feasibility tolerance. The default is 1e-4.

Simplex Algorithm

Simplex

Use Algorithm instead

If 'on', and if LargeScale is 'off', linprog uses the simplex algorithm. The simplex algorithm uses a built-in starting point, ignoring the starting point x0 if supplied. The default is 'off', meaning linprog uses an active-set algorithm.

Example: options = optimoptions('linprog','Algorithm','interior-point','Display','iter')

problem — Problem structurestructure

Problem structure, specified as a structure with the following fields.

Field NameEntry

f

Linear objective function vector f

Aineq

Matrix for linear inequality constraints

bineq

Vector for linear inequality constraints

Aeq

Matrix for linear equality constraints

beq

Vector for linear equality constraints
lbVector of lower bounds
ubVector of upper bounds

x0

Initial point for x, active set algorithm only

solver

'linprog'

options

Options created with optimoptions

You must supply at least the solver field in the problem structure.

The simplest way to obtain a problem structure is to export the problem from the Optimization app.

Data Types: struct

Output Arguments

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x — Solutionreal vector | real array

Solution, returned as a real vector or real array. The size of x is the same as the size of f or x0.

fval — Objective function value at the solutionreal number

Objective function value at the solution, returned as a real number. Generally, fval = f'*x.

exitflag — Reason linprog stoppedinteger

Reason linprog stopped, returned as an integer.

1

Function converged to a solution x.

0

Number of iterations exceeded options.MaxIter.

-2

No feasible point was found.

-3

Problem is unbounded.

-4

NaN value was encountered during execution of the algorithm.

-5

Both primal and dual problems are infeasible.

-7

Search direction became too small. No further progress could be made.

output — Information about the optimization processstructure

Information about the optimization process, returned as a structure with these fields.

iterations

Number of iterations

algorithm

Optimization algorithm used

cgiterations

0 (interior-point algorithm only, included for backward compatibility)

message

Exit message

constrviolation

Maximum of constraint functions

firstorderopt

First-order optimality measure

lambda — Lagrange multipliers at the solutionstructure

Lagrange multipliers at the solution, returned as a structure with these fields.

lower

Lower bounds corresponding to lb

upper

Upper bounds corresponding to ub

ineqlin

Linear inequalities corresponding to A and b

eqlin

Linear equalities corresponding to Aeq and beq

Limitations

  • The 'active-set' algorithm has only 'off' and 'final' as Display options; you cannot choose the iterative display option, 'iter'.

More About

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Algorithms

Interior-Point-Legacy Algorithm

The 'interior-point-legacy' method is based on LIPSOL (Linear Interior Point Solver, [3]), which is a variant of Mehrotra's predictor-corrector algorithm [2], a primal-dual interior-point method. A number of preprocessing steps occur before the algorithm begins to iterate. See Interior-Point-Legacy Linear Programming.

The first stage of the algorithm might involve some preprocessing of the constraints (see Interior-Point-Legacy Linear Programming). Several conditions might cause linprog to exit with an infeasibility message. In each case, linprog returns a negative exitflag, indicating to indicate failure.

  • If a row of all zeros is detected in Aeq, but the corresponding element of beq is not zero, then the exit message is

    Exiting due to infeasibility: An all-zero row in the
constraint matrix does not have a zero in corresponding
right-hand-side entry.

  • If one of the elements of x is found not to be bounded below, then the exit message is

    Exiting due to infeasibility: Objective f'*x is unbounded below.

  • If one of the rows of Aeq has only one nonzero element, then the associated value in x is called a singleton variable. In this case, the value of that component of x can be computed from Aeq and beq. If the value computed violates another constraint, then the exit message is

    Exiting due to infeasibility: Singleton variables in
equality constraints are not feasible.

  • If the singleton variable can be solved for, but the solution violates the upper or lower bounds, then the exit message is

    Exiting due to infeasibility: Singleton variables in
the equality constraints are not within bounds.

    Note   The preprocessing steps are cumulative. For example, even if your constraint matrix does not have a row of all zeros to begin with, other preprocessing steps can cause such a row to occur.

When the preprocessing finishes, the iterative part of the algorithm begins until the stopping criteria are met. (For more information about residuals, the primal problem, the dual problem, and the related stopping criteria, see Interior-Point-Legacy Linear Programming.) If the residuals are growing instead of getting smaller, or the residuals are neither growing nor shrinking, one of the two following termination messages is displayed, respectively,

One or more of the residuals, duality gap, or total relative error 
has grown 100000 times greater than its minimum value so far:

or

One or more of the residuals, duality gap, or total relative error 
has stalled:

After one of these messages is displayed, it is followed by one of the following messages indicating that the dual, the primal, or both appear to be infeasible.

  • The dual appears to be infeasible (and the primal unbounded). (The primal residual < TolFun.)

  • The primal appears to be infeasible (and the dual unbounded). (The dual residual < TolFun.)

  • The dual appears to be infeasible (and the primal unbounded) since the dual residual > sqrt(TolFun). (The primal residual < 10*TolFun.)

  • The primal appears to be infeasible (and the dual unbounded) since the primal residual > sqrt(TolFun). (The dual residual < 10*TolFun.)

  • The dual appears to be infeasible and the primal unbounded since the primal objective < -1e+10 and the dual objective < 1e+6.

  • The primal appears to be infeasible and the dual unbounded since the dual objective > 1e+10 and the primal objective > -1e+6.

  • Both the primal and the dual appear to be infeasible.

For example, the primal (objective) can be unbounded and the primal residual, which is a measure of primal constraint satisfaction, can be small.

Interior-Point Algorithm

The 'interior-point' algorithm is similar to 'interior-point-legacy', but with a more efficient factorization routine, and with different preprocessing. See Interior-Point linprog Algorithm.

Dual-Simplex Algorithm

For a description, see Dual-Simplex Algorithm.

Active-Set and Simplex Algorithms

The 'active-set' algorithm uses a projection method as used in the quadprog algorithm. linprog is an active set method and is thus a variation of the well-known simplex method for linear programming [1]. The algorithm finds an initial feasible solution by first solving another linear programming problem. For details, see Active-Set linprog Algorithm.

For a description of the 'simplex' algorithm, see linprog Simplex Algorithm.

linprog gives a warning when the problem is infeasible.

Warning: The constraints are overly stringent;
there is no feasible solution.

In this case, linprog produces a result that minimizes the worst-case constraint violation.

When the equality constraints are inconsistent, linprog gives

Warning: The equality constraints are overly
stringent; there is no feasible solution.

Unbounded solutions result in the warning

Warning: The solution is unbounded and at infinity;
the constraints are not restrictive enough.

In this case, linprog returns a value of x that satisfies the constraints.

References

[1] Dantzig, G.B., A. Orden, and P. Wolfe. "Generalized Simplex Method for Minimizing a Linear Form Under Linear Inequality Restraints." Pacific Journal Math., Vol. 5, 1955, pp. 183–195.

[2] Mehrotra, S. "On the Implementation of a Primal-Dual Interior Point Method." SIAM Journal on Optimization, Vol. 2, 1992, pp. 575–601.

[3] Zhang, Y., "Solving Large-Scale Linear Programs by Interior-Point Methods Under the MATLAB Environment." Technical Report TR96-01, Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD, July 1995.

See Also

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Introduced before R2006a

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