Mathematical Optimization Foundations

Turn an allocation question into an explicit model and a defensible solution claim. Separate a good candidate, a feasible candidate, and a certified optimum.

When to Use

Trigger: constrained allocation, linear programming, convex optimization, integer programming, duality certificates, optimality gaps, or robust/stochastic optimization.

Examples: allocate a fixed capacity across products; verify an LP primal/dual witness; choose a formulation for uncertain demand.

Use decision theory to choose preferences or utilities, planning/search for action sequences, and theory of constraints to identify where improvement should focus. This skill owns the mathematical allocation after those choices. Ordinary prioritization without a quantitative constrained model need not activate it.

Quick Reference

Task	Resource
Choose LP, convex, or discrete formulation	formulation-and-methods.md
Verify a claimed optimum	certificates-and-status.md
Model uncertain coefficients	uncertainty-and-sensitivity.md

Workflow

1. Define decision variables, their domains and units, objective direction, resource constraints, and input provenance. Record which coefficients are estimates. Do not silently replace a disputed objective with a convenient proxy.
2. Use formulation-and-methods.md to distinguish continuous from indivisible choices and choose LP, convex QP/conic, MILP, or explicitly nonconvex methods. Check formulation fidelity before solver choice.
3. Identify the evidence needed for the claim: feasible point, global bound, certificate, or heuristic result. For duality, numerical tolerances, and termination status, use certificates-and-status.md.
4. When coefficients are uncertain, use uncertainty-and-sensitivity.md. Distinguish scenario performance from a probabilistic or worst-case guarantee.
5. Return the optimization contract with candidate, constraint residuals, bounds/gap, method and termination status, and sensitivity. If no feasible candidate was found, distinguish search failure from proven model infeasibility.

Exact LP Certificate Helper

Run python3 scripts/check_lp_certificate.py input.json, or pipe JSON to standard input using - (default). Python standard library only; it performs no search or optimization.

The helper accepts only the continuous canonical pair:

• Primal: maximize c^T x, subject to Ax <= b, x >= 0.
• Dual: minimize b^T y, subject to A^T y >= c, y >= 0.

Input has exactly A, b, c, x, y; A is a nonempty rectangular matrix with at least one column. Scalars are finite JSON numbers or decimal strings, including exponent notation. Decimal text is converted directly to exact fractions; no expressions or rational strings are evaluated. See certificates-and-status.md for schema and worked answer.

Output contains exact rational strings for objectives, gap, and slacks; violations are arrays of indexed constraint-name strings. optimal is true only when both supplied witnesses are feasible and their objective gap is exactly zero. Invalid input exits 2 with a JSON error on standard output; valid input exits 0 even when a witness fails.

A failed witness does not prove model infeasibility or unboundedness. This helper certifies the submitted rational continuous LP, not the fidelity of the model, a MILP optimum, a rounded approximation, or an external solver's floating-point result. Convert other LP forms explicitly and preserve the mapping back to original variables.

Completion Criteria

• Variables/domains, units, objective, and constraints match the stated problem.
• The global/local/heuristic claim has appropriate evidence and disclosed assumptions.
• Numerical feasibility, integrality, bounds, and stopping status are reported separately.
• Sensitivity includes important uncertain coefficients or a stated limitation.
• Any proposed external action stays within the user's existing authorization.

Fact-Checking

Use dated primary material for mathematical guarantees and vendor documentation for current solver semantics. Check solver status and conventions before interpreting its bounds; do not promote a heuristic or sample result into a global guarantee.

Navigation and Evidence

• data/script-contracts.json — parser limit documentation contract.
• data/sources.json — primary sources verified by 17 September 2026; no solver-version pin.
• scripts/check_lp_certificate.py — exact canonical LP witness checker.
• scripts/test_lp_certificate.py — hand-answer, precision, shape, and CLI regressions; run python3 scripts/test_lp_certificate.py.

The helper's tests establish its arithmetic and input behavior. They do not establish live routing or improved optimization decisions by an agent.