I found a problem I could not really understand with fmincon. When the optimal solution is x=0 for a function under a contraint that x>=0, fmincon returns 2.0000e-08. It does not equal 0, though very close to. I wondered why it happenend and how I can get the correct result which is 0 instead of 2.0000e-08.
The example below is simple, but it shows the problem.
-------------script1: set up function
function cor = corners(ivm);
cor=ivm
end
-------------script2: use fmincon
fun = @(ivm)corners(ivm);
problem = createOptimProblem('fmincon','x0',0,...
'objective',fun,'lb',0,'ub',[]);
[cx,fval,exitflag,output,lambda,grad,hessian] = fmincon(problem)
-------------the results I get:
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the default value of the optimality tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.
<stopping criteria details>
cx =
2.0000e-08
fval =
2.0000e-08
exitflag =
1
output =
struct with fields:
iterations: 7
funcCount: 17
constrviolation: 0
stepsize: 1.9739e-06
algorithm: 'interior-point'
firstorderopt: 2.0000e-08
cgiterations: 0
message: 'Local minimum found that satisfies the constraints.↵↵Optimization completed because the objective function is non-decreasing in ↵feasible directions, to within the default value of the optimality tolerance,↵and constraints are satisfied to within the default value of the constraint tolerance.↵↵Stopping criteria details:↵↵Optimization completed: The relative first-order optimality measure, 1.999988e-08,↵is less than options.OptimalityTolerance = 1.000000e-06, and the relative maximum constraint↵violation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.↵↵Optimization Metric Options↵relative first-order optimality = 2.00e-08 OptimalityTolerance = 1e-06 (default)↵relative max(constraint violation) = 0.00e+00 ConstraintTolerance = 1e-06 (default)'
lambda =
struct with fields:
eqlin: [0×1 double]
eqnonlin: [0×1 double]
ineqlin: [0×1 double]
lower: 1.0000
upper: 0
ineqnonlin: [0×1 double]
grad =
1
hessian =
6.4000e-05
-------------when I clicked "stopping criteria details", I get:
Optimization completed: The relative first-order optimality measure, 1.999988e-08,
is less than options.OptimalityTolerance = 1.000000e-06, and the relative maximum constraint
violation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.
Optimization Metric Options
relative first-order optimality = 2.00e-08 OptimalityTolerance = 1e-06 (default)
relative max(constraint violation) = 0.00e+00 ConstraintTolerance = 1e-06 (default)
-------------possible reasons I am thinking of:
- fmincon can't identify corner solutions properly
- something to do with one of the tolerances
- the gradient at x=0 is not 0.
It looks like the reasons are shown in the command window, but I couldn't really understand it. I would appreciate your help and detailed explanations. Thanks!
