linprog - Solve linear programming problems

Equation

Finds the minimum of a problem specified by

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

Syntax

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

Description

linprog solves linear programming problems.

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

x = linprog(f,A,b,Aeq,beq) solves the problem above while additionally satisfying the equality constraints Aeq*x = beq. Set A = [] and b = [] if no inequalities exist.

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.

x = linprog(f,A,b,Aeq,beq,lb,ub,x0) sets the starting point to x0. This option is only available with the medium-scale algorithm (the LargeScale option is set to 'off' using optimset). The default large-scale algorithm and the simplex algorithm ignore any starting point.

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

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

Create the structure problem by exporting a problem from Optimization Tool, as described in Exporting to the MATLAB Workspace.

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

[x,fval,exitflag] = linprog(...) returns a value exitflag that describes the exit condition.

[x,fval,exitflag,output] = linprog(...) returns a structure output that contains information about the optimization.

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

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

Input Arguments

Function Arguments contains general descriptions of arguments passed into linprog. Options provides the function-specific details for the options values.

`problem`	`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
	`lb`	Vector of lower bounds
	`ub`	Vector of upper bounds
	`x0`	Initial point for `x`, active set algorithm only
	`solver`	`'linprog'`
	`options`	Options structure created with `optimset`

Output Arguments

Function Arguments contains general descriptions of arguments returned by linprog. This section provides function-specific details for exitflag, lambda, and output:

`exitflag`	Integer identifying the reason the algorithm terminated. The following lists the values of `exitflag` and the corresponding reasons the algorithm terminated.
	`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.
`lambda`	Structure containing the Lagrange multipliers at the solution `x` (separated by constraint type). The fields of the structure are:
	`lower`	Lower bounds `lb`
	`upper`	Upper bounds `ub`
	`ineqlin`	Linear inequalities
	`eqlin`	Linear equalities
`output`	Structure containing information about the optimization. The fields of the structure are:
	`iterations`	Number of iterations
	`algorithm`	Optimization algorithm used
	`cgiterations`	0 (large-scale algorithm only, included for backward compatibility)
	`constrviolation`	Maximum of constraint functions
	`message`	Exit message

Options

Optimization options used by linprog. Some options apply to all algorithms, and others are only relevant when using the large-scale algorithm. You can use optimset to set or change the values of these fields in the options structure, options. See Optimization Options for detailed information.

LargeScale

Use large-scale algorithm when set to 'on' (default). Use medium-scale algorithm when set to 'off'.

Medium-Scale and Large-Scale Algorithms

Both the medium-scale and large-scale algorithms use the following options:

`Diagnostics`	Display diagnostic information about the function to be minimized or solved. The choices are `'on'` or the default `'off'`.
`Display`	Level of display. `'off'` displays no output. `'iter'` displays output at each iteration. The `'iter'` option only works with the large-scale and medium-scale simplex algorithms. `'final'` (default) displays just the final output.
`MaxIter`	Maximum number of iterations allowed, a positive integer. The default is: `85` for the large-scale algorithm `10numberOfVariables` for the simplex algorithm `10max(numberOfVariables, numberOfInequalities + numberOfBounds)` for the medium-scale active-set algorithm
`TolFun`	Termination tolerance on the function value, a positive scalar. The default is: `1e-8` for the large-scale algorithm `1e-6` for the simplex algorithm The option is not used for the medium-scale active-set algorithm

Medium-Scale Algorithm Only

The medium-scale algorithm uses the following option:

Simplex

If 'on', 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'. See Medium-Scale linprog Simplex Algorithm for more information and an example.

Examples

Find x that minimizes

f(x) = –5x₁ – 4x₂ –6x₃,

subject to

x₁ – x₂ + x₃ ≤ 20
3x₁ + 2x₂ + 4x₃ ≤ 42
3x₁ + 2x₂ ≤ 30
0 ≤ x₁, 0 ≤ x₂, 0 ≤ x₃.

First, enter the coefficients

f = [-5; -4; -6]
A =  [1 -1  1
      3  2  4
      3  2  0];
b = [20; 42; 30];
lb = zeros(3,1);

Next, call a linear programming routine.

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

Entering x, lambda.ineqlin, and lambda.lower gets

x = 
     0.0000
    15.0000
     3.0000
lambda.ineqlin =
     0
     1.5000
     0.5000
lambda.lower =
     1.0000
     0
     0

Nonzero elements of the vectors in the fields of lambda indicate active constraints at the solution. In this case, the second and third inequality constraints (in lambda.ineqlin) and the first lower bound constraint (in lambda.lower) are active constraints (i.e., the solution is on their constraint boundaries).

Algorithm

Large-Scale Optimization

The large-scale 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 Large Scale Linear Programming.

Medium-Scale Optimization

linprog 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.

Alternatively, you can use the simplex algorithm, described in Medium-Scale linprog Simplex Algorithm, by entering

options = optimset('LargeScale', 'off', 'Simplex', 'on')

and passing options as an input argument to linprog. The simplex algorithm returns a vertex optimal solution.

Note You cannot supply an initial point x0 for linprog with either the large-scale method or the medium-scale method using the simplex algorithm. In either case, if you pass in x0 as an input argument, linprog ignores x0 and computes its own initial point for the algorithm.

Diagnostics

Large-Scale Optimization

The first stage of the algorithm might involve some preprocessing of the constraints (see Large Scale Linear Programming). Several possible conditions might occur that cause linprog to exit with an infeasibility message. In each case, the exitflag argument returned by linprog is set to a negative value to indicate failure.

If a row of all zeros is detected in Aeq but the corresponding element of beq is not zero, 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, 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, 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, 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, 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 may cause such a row to occur.

Once the preprocessing has finished, the iterative part of the algorithm begins until the stopping criteria are met. (See Large Scale Linear Programming for more information about residuals, the primal problem, the dual problem, and the related stopping criteria.) 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:

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 six messages indicating that the dual, the primal, or both appear to be infeasible. The messages differ according to how the infeasibility or unboundedness was measured.

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.

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

Medium-Scale Optimization

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.

Limitations

Medium-Scale Optimization

At this time, the only levels of display, using the Display option in options, are 'off' and 'final'; iterative output using 'iter' is not available.

Large-Scale Optimization

Large-Scale Problem Coverage and Requirements

For Large Problems
A and Aeq should be sparse.

References

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

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

[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.

linprog - Solve linear programming problems

Equation

Syntax

Description

Input Arguments

Output Arguments

Options

Medium-Scale and Large-Scale Algorithms

Medium-Scale Algorithm Only

Examples

Algorithm

Large-Scale Optimization

Medium-Scale Optimization

Diagnostics

Large-Scale Optimization

Medium-Scale Optimization

Limitations

Medium-Scale Optimization

Large-Scale Optimization

References

See Also

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