fmincon

Find the constrained minimum of a real function.

Syntax

x = fmincon(@func,x0)

x = fmincon(@func,x0,A,b)

x = fmincon(@func,x0,A,b,Aeq,beq)

x = fmincon(@func,x0,A,b,Aeq,beq,lb,ub)

x = fmincon(@func,x0,A,b,Aeq,beq,lb,ub,nonlcon)

x = fmincon(@func,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

[x,fval,info,output] = fmincon(...)

Inputs

func
The function to minimize. See the optimset option GradObj for details.
x0
An estimate of the location of the minimum.
A
A matrix used to compute A*x for inequality constraints.
Use [ ] if unneeded.
b
The upper bound of the inequality constraints A*x<=b.
Use [ ] if unneeded.
Aeq
A matrix used to compute Aeq*x for equality constraints.
Use [ ] if unneeded.
beq
The upper bound of the equality constraints Aeq*x=beq.
Use [ ] if unneeded.
lb
The design variable lower bounds.
Use [ ] if unbounded. Support for this option is limited. See Comments.
ub
The design variable upper bounds.
Use [ ] if unbounded. Support for this option is limited. See Comments.
nonlcon
The non-linear constraints function.
The function signature is as follows: function [c, ceq, cj, ceqj] = ConFunc(x), where c and ceq contain inequality and equality constraints, respectively, and cj and ceqj contain their Jacobians. The inequality constraints are assumed to have upper bounds of 0. See the optimset option GradConstr.
The function can return 1, 2, or 4 outputs. Any required output that is unused should be set to [ ]. Thus, if only c and cj are desired, ceq and ceqj should be returned as [ ].
options
A struct containing option settings.
See optimset for details.

Outputs

x
The location of the function minimum.
fval
The minimum of the function.
info
The convergence status flag.
info = 3:
Converged with a constraint violation within TolCon.
info = 1
Function value converged to within TolX or TolKKT.
info = 0
Reached maximum number of iterations or function calls.
info = -2
The function did not converge.
output
A struct containing iteration details. The members are as follows:
iterations
The number of iterations.
nfev
The number of function evaluations.
xiter
The candidate solution at each iteration.
fvaliter
The objective function value at each iteration.
coniter
The constraint values at each iteration. The columns contain the constraint function values in the following order: linear inequality contraints, linear equality constraints, nonlinear inequality contraints, nonlinear equality constraints.

Examples

Minimize the function ObjFunc, subject to the linear inequality constraint: x1 + 4*x2 > 27.

The constraint must be expressed with an upper bound: -x1 - 4*x2 < -27.
function obj = ObjFunc(x)
    obj = 2*(x(1)-3)^2 - 5*(x(1)-3)*(x(2)-2) + 4*(x(2)-2)^2 + 6;
end

init = [8, 6];		% initial estimate
A = [-1, -4];		% inequality contraint matrix
b = [-27];		% inequality contraint bound
lb = [-10, -10];	% lower variable bounds
ub = [10, 10];		% upper variable bounds
[x,fval] = fmincon(@ObjFunc,init,A,b,[],[],lb,ub)
x = [Matrix] 1 x 2
7.00000  5.00000
fval = 14
Modify the previous example to pass an extra parameter to the function using a function handle.
function obj = ObjFunc(x,offset)
    obj = 2*(x(1)-3)^2 - 5*(x(1)-3)*(x(2)-2) + 4*(x(2)-2)^2 + offset;
end
handle = @(x) ObjFunc(x,7);
[x,fval] = fmincon(handle,init,A,b,[],[],lb,ub)
x = [Matrix] 1 x 2
7.00000  5.00000
fval = 15

Comments

fmincon uses a Sequential Quadratic Programming algorithm and a line search method.

Options for convergence tolerance controls and analytical derivatives are specified with optimset.

If large lb and ub values are specified, then it is essential to use option TolX in optimset. The default TolX is likely too large, since it is applied relative to the interval size.

The unbounded lb and ub options are not fully supported due to their relationship to the TolX. The unbounded options are set to -1000 and 1000, respectively.

To pass additional parameters to a function argument, use an anonymous function.

See the optimization tutorial for an example with nonlinear constraints.

The optimset options and defaults are as follows:

  • MaxIter: 400
  • MaxFunEvals: 1,000,000
  • MaxFail: 20,000
  • TolX: 1.0e-7
  • TolCon: 0.5%
  • TolKKT: 1.0e-7
  • GradObj: 'off'
  • GradConstr: 'off'
  • Display: 'off'

When convergence occurs with an acceptable contraint violation, the next to last iteration might have a lower objective value than the final iteration, but the final iteration might have the lower constraint violation of the two. The final iteration is reported as the solution, x. The value of TolCon affects the size of potential difference between the choices.