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Tolerances and Stopping Criteria

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.

Set tolerances and other criteria using optimoptions as explained in Set and Change Options.

    Tip   Generally set tolerances such as TolFun and TolX to be well above eps, and usually 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.

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 default tolerances. For example,

options = optimoptions('fmincon')

You can also find the default tolerances in the options section of the solver function reference page.

  • TolX 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 TolX, the iterations end. TolX is sometimes used as a relative bound, meaning iterations end when |(xi – xi+1)| < TolX*(1 + |xi|), or a similar relative measure.

  • TolFun is a lower bound on the change in the value of the objective function during a step. If |f(xi) – f(xi+1)| < TolFun, the iterations end. TolFun is sometimes used as a relative bound, meaning iterations end when |f(xi) – f(xi+1)| < TolFun*(1 + |f(xi)|), or a similar relative measure.

  • TolFun is also a bound on the first-order optimality measure. If the optimality measure is less than TolFun, the iterations end. TolFun can also be a relative bound. First-order optimality measure is defined in First-Order Optimality Measure.

  • TolCon is an upper bound on the magnitude of any constraint functions. If a solver returns a point x with c(x) > TolCon or |ceq(x)| > TolCon, the solver reports that the constraints are violated at x. TolCon can also be a relative bound.

      Note:   TolCon operates differently from other tolerances. If TolCon is not satisfied (i.e., if the magnitude of the constraint function exceeds TolCon), the solver attempts to continue, unless it is halted for another reason. A solver does not halt simply because TolCon is satisfied.

  • MaxIter is a bound on the number of solver iterations. MaxFunEvals is 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: TolPCG and MaxPCGIter. These relate to preconditioned conjugate gradient steps. For more information, see Preconditioned Conjugate Gradient Method.

There are several tolerances that apply only to the fmincon interior-point algorithm. For more information, see Interior-Point Algorithm.

There are several tolerances that apply only to intlinprog. See Some "Integer" Solutions Are Not Integers and Branch and Bound.

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