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# 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. linprog uses x0 only with the active-set algorithm. linprog ignores x0 with the interior-point and simplex algorithms.

x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options) minimizes with the optimization options specified in 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.

[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 created with optimoptions

## 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 (interior-point algorithm only, included for backward compatibility) message Exit message constrviolation Maximum of constraint functions firstorderopt First-order optimality measure

## Options

Optimization options used by linprog. Some options apply to all algorithms, and others are only relevant when using the interior-point algorithm. Use optimoptions to set or change options. See Optimization Options Reference for detailed information.

### All Algorithms

All linprog algorithms use the following options:

 Algorithm Choose the optimization algorithm:'interior-point' (default)'active-set''simplex'For information on choosing the algorithm, see Choosing the Algorithm. 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' or 'none' displays no output.'iter' displays output at each iteration. The 'iter' option only works with the interior-point and simplex algorithms.'final' (default) displays just the final output. LargeScaleUse Algorithm instead Use large-scale algorithm when set to 'on' (default). Use a medium-scale algorithm when set to 'off' (see Simplex in simplex Algorithm Only). For information on choosing the algorithm, see Choosing the Algorithm. MaxIter Maximum number of iterations allowed, a positive integer. The default is:85 for the interior-point algorithm10*numberOfVariables for the simplex algorithm10*max(numberOfVariables, numberOfInequalities + numberOfBounds) for the active-set algorithm TolFun Termination tolerance on the function value, a positive scalar. The default is:1e-8 for the interior-point algorithm1e-6 for the simplex algorithmThe option is not used for the medium-scale active-set algorithm

### simplex Algorithm Only

The simplex algorithm use the following option:

 SimplexUse 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. See Active-Set and Simplex Algorithms for more information and an example.

## Examples

Find x that minimizes

f(x) = –5x1 – 4x2 –6x3,

subject to

x1x2 + 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);`

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```

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).

## Diagnostics

### Interior-Point Algorithm

The first stage of the algorithm might involve some preprocessing of the constraints (see Interior-Point 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 Interior-Point 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:```

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 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.

### Active-Set and Simplex Algorithms

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

### Active-Set Algorithm

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.

### Interior-Point Algorithm

Coverage and Requirements

For Large Problems

A and Aeq should be sparse.

expand all

### Interior-Point Algorithm

The interior-point 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 Linear Programming.

### Active-Set and Simplex Algorithms

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 linprog Simplex Algorithm, by entering

`options = optimoptions('linprog','Algorithm','simplex')`

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

 Note   linprog ignores x0, and computes its own initial point for the interior-point and simplex algorithms. linprog uses x0 only with the active-set algorithm.

## 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, pp. 183–195, 1955.

[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.