Constrained optimization

Use this methodology to turn an optimization request into a defensible decision record.

Workflow

1. Frame the decision. Name the decision maker, feasible action window, variable domains (continuous, integer, categorical, binary), bounds, units, and whether each variable is controllable.
2. State objectives. Write objective direction, units, aggregation, baseline, and acceptable tradeoffs. Keep multiple objectives separate until a decision maker chooses weights, lexicographic priority, epsilon constraints, or a Pareto review.
3. Classify constraints. Mark each as hard (must hold), soft (preference with an explicit penalty or relaxation), or informational. Define equality/inequality tolerance, feasibility predicate, and how violations are measured. Never hide a hard violation in a penalty score.
4. Establish baselines. Validate a known feasible incumbent, a simple exact or bounded baseline where available, and a heuristic baseline when exact solving is impractical. State what each baseline proves and does not prove.
5. Choose the method. Match solver family to domains, smoothness, scale, and proof needs. An exact solver can establish optimality or a bound when its status and gap support it; a heuristic supplies a candidate and empirical evidence. Do not call timeout or unknown an infeasible result.
6. Run a fair comparison. Fix an objective/constraint implementation, feasibility checker, compute budget, stopping rule, and reporting schema. For stochastic methods, use equal budgets and declared seeds/repeats. Keep solver configuration separate from the formulation.
7. Independently validate. Recompute objective and every constraint from the returned decision using an independent implementation or checker. Report feasible-best, violation magnitude, bound/gap, status, runtime, and unresolved unknowns separately.
8. Decide and monitor. Select a feasible solution or Pareto set with the tradeoff rationale, sensitivity/robustness evidence, owner, and rollback/re-optimization trigger. Record “infeasible under these constraints” only when infeasibility is proven or independently certified.

Routing and exit

Load references/formulation-and-evidence.md for formulation, feasibility, solver status, bounds/gaps, and comparison rules. Load references/multiobjective-and-robustness.md for Pareto decisions and stochastic sensitivity. Use the formulation record to define the problem before solving, and the solution audit to verify a returned candidate.