CMP.6:4 - Solution

Choose the comparison → construct the available changes → derive a useful update → control its extent → establish progress → return the result at its supported scope.

CMP.6:4.1 - State the candidate, criterion and wanted conclusion

Describe the current candidate x, its admissible set and the result wanted. A criterion J(x) can measure an objective, a residual or a potential used to establish progress. State which it is. A potential that decreases helps analyze the algorithm; its relation to the recipient’s requested answer still needs to be established.

Separate possible conclusions: one improved candidate, no improving change in a specified neighborhood, a point meeting a residual tolerance, a global optimum, or a sequence converging under stated conditions. Select the useful conclusion now rather than automatically pursuing the strongest one.

When the iteration serves a model or learner, retain its target through C.29.2 and the relevant modeling or learning method. Optimization error, error in the subject model and performance on later cases are different questions.

CMP.6:4.2 - Construct local changes and their obtainable consequences

Specify what may change at one step. Examples include flipping one binary choice, exchanging two assignments, updating a coordinate, adding a vector direction or replacing a violated part of a construction. MATH.10 supplies the comparison under admissible variation; the present task is to obtain a useful change by a finite computation.

Determine what information is available about a proposed change:

• a directly computed difference in J;
• a local approximation, such as a derivative or a small model of the objective;
• a response or estimate obtained from samples or feedback.

Choose a procedure that uses this information. A discrete local search can examine neighbor changes and select one with positive gain. A smooth minimization can use a negative gradient. A coordinate method can solve a smaller update problem while holding other coordinates fixed. A residual correction needs a relation showing how that correction changes the residual or another progress measure.

If obtaining the direction calls another hard problem, include its algorithm and cost or accept an approximate direction with an appropriate result condition. Writing an argmin expression identifies the desired update; it does not automatically supply an algorithm for obtaining it.

CMP.6:4.3 - Make the finite update admissible and useful

For discrete replacements, compute the whole finite difference when it is affordable. Apply only changes that retain the required constraints, or pair the change with a specified repair whose effect is included in the comparison.

For a direction d, construct x'=x+η*d with step size η. The local information must support that finite change. A known upper model can determine a suitable step; otherwise a trial-and-reduction rule can search for a step whose observed effect supports acceptance. Include every trial evaluation in the cost.

For example, suppose a differentiable minimizing objective satisfies

J(x+s)≤J(x)+grad J(x)·s+(L/2)*||s||²

on the relevant region, with known L>0. Choosing s=-grad J(x)/L gives

J(x+s)≤J(x)-||grad J(x)||²/(2L).

This derives a finite descent step from a bound on the local model’s error. When that bound is unavailable, an adaptive step rule needs its own termination or acceptance conditions. One unsuccessful trial can justify reducing or changing the step; it does not establish that the direction never helps.

For constrained problems, use an admissible parameterization, projection or other constraint-preserving update. Reestablish the progress account for that update. Simply clipping a coordinate can change the original unconstrained argument.

With noisy feedback, distinguish realized change from a conditional or expected improvement claim. Repeating a measurement or taking a larger sample is useful only when the additional information changes the step or its warranted use. A deterministic monotone-descent statement cannot be inferred from a noisy sign alone.

CMP.6:4.4 - Establish what repeated updates imply

Select a progress argument appropriate to the state space and update:

Use MATH.20 for the bound and MATH.21 for the convergence construction when needed. If each step improves but there is no useful stopping guarantee, return that limited procedure or change the method. Equal-value moves need their own cycle handling; strict-improvement reasoning does not cover them.

CMP.6:4.5 - Choose a stopping test that supports the receiving answer

Derive the test from the wanted conclusion. Exhausting a discrete neighborhood establishes local optimality for that neighborhood. A residual threshold answers a residual question; a condition or error bound is needed to convert it into distance from a solution. Small successive changes can result from a tiny step even while the candidate remains poor.

For a constrained optimum, the full gradient need not vanish. Use the absence of a feasible improving variation, a suitable projected update or the relevant constrained condition. Report that condition at the scope it establishes.

When time runs out, return the best admissible candidate or current approximation together with any supported bound. A tolerance or resource limit belongs to the use that needs the result.

CMP.6:4.6 - Revisit the source of a failed improvement

When the step fails, locate whether the direction, extent, feasibility repair, feedback or assumed relation to the objective changed. Revise that part. If local moves repeatedly stop at unsatisfactory candidates, enlarge or change the neighborhood, restart from another candidate, use CMP.4 to explore alternatives or obtain a bound through CMP.5.

Compare alternatives by the quality and cost relevant to the receiving use, using the existing characterization and improvement methods. Faster iteration is valuable only in relation to the result it obtains.