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name: foundations-mathematical-optimization description: Formulates LP, MILP, CP-SAT models; diagnoses infeasibility and gaps. Use when a MILP is infeasible, shifts need scheduling under capacity, or choosing a solver. version: "1.0" last_validated: 2026-09-17

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, scheduling and assignment under capacity, solver choice, infeasible or unbounded models, duality certificates, optimality gaps, multi-objective tradeoffs, or robust/stochastic optimization.

Examples: allocate a fixed capacity across products; schedule GPU jobs under a GPU cap; find which constraints make a MILP infeasible; check an LLM-drafted model; verify an LP primal/dual witness.

Use decision theory to choose preferences or utilities, planning/search for action sequences with preconditions and effects (this skill owns resource-constrained scheduling), 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
Pick a solver or modeling layer; look up versions and features	solver-selection.md
Model is infeasible, unbounded, INF_OR_UNBD, or numerically unstable	infeasibility-diagnosis.md
MIP is slow or has a large gap; big-M, symmetry; several objectives	mip-formulation-and-multiobjective.md
Jobs, shifts, GPU or workflow scheduling; CP-SAT vs MIP vs greedy	scheduling-cp-sat.md
An LLM drafts the model from a text spec	llm-assisted-formulation.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, then pick the solver with solver-selection.md; scheduling goes to scheduling-cp-sat.md.
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 and diagnose with infeasibility-diagnosis.md.

Exact LP Certificate Helper

Run python3 scripts/check_lp_certificate.py input.json (or - for standard input) to check supplied primal/dual witnesses for the canonical pair max c^T x, Ax <= b, x >= 0 / min b^T y, A^T y >= c, y >= 0 in exact rational arithmetic. It performs no optimization. Schema, parser limits, output fields, exit codes, and a worked answer: certificates-and-status.md.

Exit code 0 does not mean the witness is valid: a failed witness exits 0 with optimal: false. Read optimal, or pass --strict to exit 1 unless it is true. Invalid input exits 2.

It certifies only the submitted continuous LP. It does not certify model fidelity, a MILP optimum, or a floating-point solver result, and a failed witness does not prove infeasibility or unboundedness.

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.
• 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 and vendor sources; no solver version is pinned. solver-selection.md says where to look versions up.
• 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.