stopcriterium

Stopcriterium, or stop criterion, is a rule used to terminate an iterative algorithm when a predefined condition indicates sufficient accuracy or convergence. It is a fundamental concept in numerical analysis, optimization, and related fields, ensuring that computation stops once progress becomes negligible or a target level of precision is reached.

Common stop criteria fall into several categories. Absolute and relative tolerances measure when the error or residual is small enough, for example ||r_k|| ≤ tol or ||r_k|| ≤ tol·||b|| in a linear system. Gradient-based criteria use the norm of the gradient, such as ||∇f(x_k)|| ≤ tol, to signal near-optimality. Change-based criteria monitor improvement or stability, including ||x_{k+1} − x_k|| ≤ tol or |f(x_{k+1}) − f(x_k)| ≤ tol. Feasibility and duality gaps are used in constrained or primal–dual problems to declare convergence when these gaps fall below a threshold. A hard cap on iterations, or a maximum number of steps, is also a common stop criterion to prevent excessive computation.

In practice, stop criteria are chosen to balance accuracy with computational cost and may be scaled to problem magnitude. Degenerate or ill-conditioned problems may require relaunching with adjusted tolerances or alternative criteria. In machine learning, early stopping based on validation performance is a related technique to halt training before overfitting occurs.

See also: convergence, termination condition, stopping rule.