The number of iterations in an optimization depends on a solver's stopping criteria. These criteria include several tolerances you can set. Generally, a tolerance is a threshold which, if crossed, stops the iterations of a solver.
Generally set tolerances such as
StepTolerance to be well above
1e-14. Setting small tolerances does not always
result in accurate results. Instead, a solver can fail to recognize when it has
converged, and can continue futile iterations. A tolerance value smaller than
eps effectively disables that stopping condition. This tip
does not apply to
fzero, which uses a default value of
eps for the
You can find the default tolerances in the Optimization App. Some default tolerances differ for different algorithms, so set both the solver and the algorithm.
optimoptions displays tolerances.
options = optimoptions('fmincon'); [options.OptimalityTolerance,options.FunctionTolerance,options.StepTolerance]
ans = 1.0e-06 * 1.0000 1.0000 0.0001
You can also find the default tolerances in the options section of the solver function reference page.
StepTolerance is a lower bound on the size of a step,
meaning the norm of (xi – xi+1). If the solver attempts to take a step that is smaller than
StepTolerance, the iterations end.
StepTolerance is generally used as a
relative bound, meaning iterations end when |(xi – xi+1)| <
|xi|), or a similar relative measure.
For some algorithms,
FunctionTolerance is a lower bound on
the change in the value of the objective function during a step. For those
algorithms, if |f(xi) – f(xi+1)| <
FunctionTolerance, the iterations end.
FunctionTolerance is generally used as a
relative bound, meaning iterations end when |f(xi) – f(xi+1)| <
|f(xi)|), or a similar relative measure.
Unlike other solvers,
fminsearch stops when it
TolFun (the function tolerance) and
TolX (the step tolerance).
OptimalityTolerance is a tolerance
for the first-order optimality measure. If the optimality measure
is less than
OptimalityTolerance, the iterations
OptimalityTolerance can also be a relative
bound on the first-order optimality measure. First-order optimality
measure is defined in First-Order Optimality Measure.
ConstraintTolerance is an upper
bound on the magnitude of any constraint functions. If a solver returns
a point x with c(x) >
ConstraintTolerance or |ceq(x)| >
the solver reports that the constraints are violated at x.
also be a relative bound.
ConstraintTolerance operates differently
from other tolerances. If
not satisfied (i.e., if the magnitude of the constraint function exceeds
the solver attempts to continue, unless it is halted for another reason.
A solver does not halt simply because
MaxIterations is a bound on the
number of solver iterations.
a bound on the number of function evaluations. Iterations and function
evaluations are discussed in Iterations and Function Counts.
There are two other tolerances that apply to particular solvers:
These relate to preconditioned conjugate gradient steps. For more
information, see Preconditioned Conjugate Gradient Method.
There are several tolerances that apply only to the
algorithm. For more information, see Interior-Point