CMP.5:4.3 - Obtain a result with the strength its next use requires

Select an obtaining procedure for the relaxed problem. It may return an optimum, a candidate with a bound, a dual bound, or a heuristic estimate. Retain that distinction when using the result.

For minimization, a feasible relaxed point with value P gives an upper bound on the relaxed minimum. It does not by itself give a lower bound on the original minimum. To support such a lower bound, obtain the relaxed optimum or another justified lower bound L, for example from a feasible dual construction. MATH.20 supplies the bound argument; the relevant solver or proof supplies its computed value.

With an original feasible candidate of value C, the useful comparison is L≤original optimum≤C. A relaxed candidate’s value P may lie on either side of the original optimum. Keep it separately when it guides recovery.

Check the computational bound direction when rounding, stopping tolerances or incomplete solving can change an exclusion or conclusion. A conservative weaker bound can be preferable to an expensive stronger one. C.11.DUA selects extra computation or assurance according to its possible effect.