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,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,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.
|Linear objective function vector f|
|Matrix for linear inequality constraints|
|Vector for linear inequality constraints|
|Matrix for linear equality constraints|
|Vector for linear equality constraints|
|lb||Vector of lower bounds|
|ub||Vector of upper bounds|
|Initial point for x, active set algorithm only|
|Options created with optimoptions|
Function Arguments contains general descriptions of arguments returned by linprog. This section provides function-specific details for exitflag, lambda, and output:
Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated.
Function converged to a solution x.
Number of iterations exceeded options.MaxIter.
No feasible point was found.
Problem is unbounded.
NaN value was encountered during execution of the algorithm.
Both primal and dual problems are infeasible.
Search direction became too small. No further progress could be made.
Structure containing the Lagrange multipliers at the solution x (separated by constraint type). Currently, the 'dual-simplex' algorithm returns  for lambda. The fields of the structure are:
Lower bounds lb
Upper bounds ub
Structure containing information about the optimization. The fields of the structure are:
Number of iterations
Optimization algorithm used
0 (interior-point algorithm only, included for backward compatibility)
Maximum of constraint functions
First-order optimality measure ( for 'dual-simplex' algorithm)
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 linprog algorithms use the following options:
Choose the optimization algorithm:
For information on choosing the algorithm, see Linear Programming Algorithms.
Display diagnostic information about the function to be minimized or solved. The choices are 'on' or the default 'off'.
Level of display.
Use Algorithm instead
Use the 'interior-point' 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.
Maximum number of iterations allowed, a positive integer. The default is:
Termination tolerance on the function value, a positive scalar. The default is:
TolFun measures dual feasibility tolerance.
The 'simplex' algorithm uses the following option:
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. See Active-Set and Simplex Algorithms for more information and an example.
The 'dual-simplex' algorithm uses the following options:
Maximum amount of time in seconds that the algorithm runs. The default is Inf.
Level of LP preprocessing prior to dual simplex algorithm iterations. Choices are 'none' or the default 'basic'.
Feasibility tolerance for constraints, a scalar from 1e-10 through 1e-3. TolCon measures primal feasibility tolerance. The default is 1e-4.
Find x that minimizes
f(x) = –5x1 – 4x2 –6x3,
x1 – x2 + x3 ≤
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).
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:
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.
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.
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.
Currently, 'dual-simplex' returns an empty Lagrange multiplier structure lambda and firstorderopt field in the output structure.
Coverage and Requirements
|For Large Problems|
A and Aeq should be sparse.
The interior-point method is based on LIPSOL (Linear Interior Point Solver, ), which is a variant of Mehrotra's predictor-corrector algorithm (), a primal-dual interior-point method. A number of preprocessing steps occur before the algorithm begins to iterate. See Interior-Point Linear Programming.
For a description of the 'dual-simplex' algorithm, see Dual-Simplex Algorithm.
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 . The algorithm finds an initial feasible solution by first solving another linear programming problem. For details, see Active-Set linprog Algorithm.
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.
 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.
 Mehrotra, S., "On the Implementation of a Primal-Dual Interior Point Method," SIAM Journal on Optimization, Vol. 2, pp. 575–601, 1992.
 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.