Foundations Mathematical Optimization

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

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.