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Solve linear programming problems
Finds the minimum of a problem specified by
f, x, b, beq, lb, and ub are vectors, and A and Aeq are matrices.
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(...)
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 activeset algorithm. linprog ignores x0 with the interiorpoint 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.
Function Arguments contains general descriptions of arguments passed into linprog. Options provides the functionspecific 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 
Function Arguments contains general descriptions of arguments returned by linprog. This section provides functionspecific 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 (interiorpoint algorithm only, included for backward compatibility)  
message  Exit message  
constrviolation  Maximum of constraint functions  
firstorderopt  Firstorder optimality measure 
Optimization options used by linprog. Some options apply to all algorithms, and others are only relevant when using the interiorpoint algorithm. Use optimoptions to set or change options. See Optimization Options Reference for detailed information.
All linprog algorithms use the following options:
Algorithm  Choose the optimization algorithm:
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.

LargeScale Use Algorithm instead  Use largescale algorithm when set to 'on' (default). Use a mediumscale 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:

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

The simplex algorithm use the following option:
Simplex Use Algorithm instead  If 'on', and if LargeScale is 'off', linprog uses the simplex algorithm. The simplex algorithm uses a builtin starting point, ignoring the starting point x0 if supplied. The default is 'off', meaning linprog uses an activeset algorithm. See ActiveSet and Simplex Algorithms for more information and an example. 
Find x that minimizes
f(x) = –5x_{1} – 4x_{2} –6x_{3},
subject to
x_{1} – x_{2} + x_{3} ≤
20
3x_{1} +
2x_{2} + 4x_{3} ≤
42
3x_{1} +
2x_{2} ≤ 30
0
≤ x_{1}, 0 ≤ x_{2},
0 ≤ x_{3}.
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).
The first stage of the algorithm might involve some preprocessing of the constraints (see InteriorPoint 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 allzero row in the constraint matrix does not have a zero in corresponding righthandside 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 InteriorPoint 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.
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.
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.
Coverage and Requirements
For Large Problems 

A and Aeq should be sparse. 
[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 PrimalDual Interior Point Method," SIAM Journal on Optimization, Vol. 2, pp. 575–601, 1992.
[3] Zhang, Y., "Solving LargeScale Linear Programs by InteriorPoint Methods Under the MATLAB Environment," Technical Report TR9601, Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD, July 1995.