Math Optimization

Contract

• Input: problem description and inputs defined by the skill body.
• Output: Markdown artifact with completed process steps.
• Side effects: none.
• Dependencies: none.
• Stop condition: all process steps executed; artifact saved with required sections.
• Risk: low.
• Boundary: produces reasoning artifact only; no system changes.

Mathematical Optimization

Turn a real problem into a mathematical program, solve it with an appropriate solver, and report what was found — including the gap to optimality when the optimum is not proven.

When to use

• The user wants to maximize/minimize something under constraints (cost, time, risk, distance, utility).
• A decision can be cast as variables + objective + constraints.
• Other skills (logistics, finance, ML) need an optimization backbone.

Process

1. Model

Define:

• Decision variables — what the agent controls.
• Objective — single scalar to extremize (or a Pareto front if multi-objective).
• Constraints — equalities, inequalities, integrality, bounds.
• Parameters — fixed inputs that distinguish instances.

State the canonical class: LP, QP, SOCP, SDP, convex, MIP, non-convex, combinatorial.

Completion criterion: variables, objective, constraints, parameters all explicit; canonical class named.

2. Diagnose hardness

Identify why the problem is hard (or easy):

• LP: poly-time (interior point / simplex).
• Convex: poly-time in practice.
• MIP: NP-hard in general; LP relaxation gives a bound.
• Non-convex / combinatorial: NP-hard; need heuristic or approximation with a guarantee.

Completion criterion: hardness class stated; relaxation or approximation strategy named if not convex.

3. Choose solver

Pick by class:

• LP/MIP: HiGHS, Gurobi, CPLEX, OR-Tools.
• Convex (QP/SOCP/SDP): CVXPY, SCS, MOSEK.
• Non-convex smooth: scipy.optimize, IPOPT, PyTorch (gradient-based).
• Combinatorial: greedy + local search, simulated annealing, tabu, ALNS, MILP relaxation.
• Multi-objective: weighted sum, ε-constraint, or NSGA-II.

Completion criterion: solver named with one reason it fits.

4. Solve

Run the solver; for non-trivial models:

• Warm-start when possible.
• Set tolerances and iteration caps explicitly.
• Log progress (objective, gap, time) at meaningful checkpoints.

Completion criterion: solver run with explicit tolerances and a logged trace.

5. Verify optimality / bound

• LP/Convex: verify dual feasibility or KKT conditions.
• MIP: report the optimality gap (e.g. 0.2% within time limit).
• Heuristic: report best-known value and a lower bound from a relaxation.
• Always run a sensitivity check — perturb a parameter by 1–5% and confirm the solution behaves sensibly.

Completion criterion: bound, gap, or sensitivity result reported.

6. Deliver

Markdown artifact with: model formulation, hardness, solver, trace summary, optimal (or best-known) solution, gap, sensitivity, and a sanity check against intuition.

Completion criterion: all six sections present; solution reproducible from the code shown.