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

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 inInput Arguments.

Create the structure problem by exporting a problem from Optimization Tool, as described inExporting 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.

Input Arguments

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

Output Arguments

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

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 Reference for detailed information.

Medium-Scale and Large-Scale Algorithms

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

Medium-Scale Algorithm Only

The medium-scale algorithms use the following option:

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);

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).

Algorithms

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 asingleton variable. In this case, the value of that component of x can be computed from Aeq andbeq. 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.

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:

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.

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