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

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.