MATH.20:4.2 - Construct a comparator from available structure
Choose a construction whose relation to the unknown can be proved with available information.
| Requested result | Useful construction | Reason for the direction |
|---|---|---|
| Minimum of an objective | A feasible candidate gives an upper bound; minimizing over a larger feasible set gives a lower bound. | The minimum is no larger than any admitted value; added choices can only lower the infimum. |
| Maximum of an objective | A feasible candidate gives a lower bound; maximizing over a larger feasible set gives an upper bound. | The maximum is no smaller than any admitted value; added choices can only raise the supremum. |
| Unknown set | Build a subset of verified members or a superset containing all admitted possibilities. | Membership proofs establish the two inclusions in opposite directions. |
| Error in a result | Relate a computable discrepancy to that error through the governing equation or map. | The derived inequality, such as the inverse-map bound in :5.2, connects the measured quantity to the requested one. |
Other constructions can use symmetry, a conserved quantity, a norm inequality or a comparison theorem. Select the property that supplies the missing direction. A familiar shape or similar-looking value is a candidate for investigation until that relation is established.
A comparator can be a different kind of object from the result. In :5.1 a node potential produces a numerical lower bound on every path length. The derivation makes those resulting values comparable; it does not compare a potential directly with a path.