Converged and practical solutions

In a strict mathematical sense, an algorithm is said to have converged if repeated iterations do not alter the coordinates of its solution significantly. A more practical view of convergence, as used in the nonlinear solvers of the Xpress suite, is to also consider the algorithm to have converged if repeated iterations have no significant effect on either the objective value or upon feasibility. This will be called extended convergence to distinguish it from the strict sense.

For some problems, a solver may visit points at which the local neighborhood is very complex, or even malformed due to numerical issues. In this situation, the best results may be obtained when convergence of some of the variables is forced. This leads to practical solutions, which are feasible and converged in most variables, but the remaining variables have had their convergence forced by the solver, for example by means of a trust region. Although these solutions are not locally optimal in a strict sense, they provide meaningful, useful results for difficult problems in practice.