CMP.6:1 - Problem frame

Use this when a desired result is hard to construct directly, but the current candidate reveals a computable improvement: a beneficial local replacement, a derivative, a violated condition or feedback about a proposed change. You need to turn that information into an update rule and determine what repeated updates can establish.

This is an algorithmic design problem. It occurs in discrete local search, iterative equation solving, optimization and learning procedures. The input need not be numerical: swapping two choices or changing one symbol can be the available operation. The difficulty is connecting an accessible local signal to a useful change of the whole candidate.

The gain is an executable update, an appropriate step or acceptance rule, and a stopping or progress account suited to the requested result. The reader needs to follow the candidate’s admissible changes and comparisons. Differentiation is a prerequisite only for the derivative-based branch; MATH.10 supplies the corresponding mathematical variation.

Use a direct construction when it already gives the required result affordably. An existing iterative method can also be used directly when its hypotheses fit. This pattern is needed when the update or its justification must be constructed or changed.