ors-methods

Proof & Algorithm Methodology (ors-methods)

When to trigger

• The model and claims exist (ors-theory-development) and now must be proved or guaranteed.
• You need to pick a proof strategy or design an algorithm with provable guarantees.
• A reviewer says "the proof of Theorem X has a gap" or "the rate is not established."

Match the machinery to the result

Operations Research is mathematically rigorous: the contribution lives or dies on

the soundness and strength of the analysis. Pick technique by methodology:

| Result you need | Typical machinery |

|-----------------|-------------------|

| Optimality / strong duality | LP/conic duality, KKT, polyhedral / total unimodularity, submodularity |

| Approximation guarantee | LP/SDP rounding, primal-dual, greedy + submodular bounds |

| Complexity / hardness | reductions (NP-hardness), oracle lower bounds |

| Convergence & rate | monotonicity/Lyapunov, fixed-point/contraction, first-order analysis |

| Steady-state / stability | Foster-Lyapunov, regenerative arguments, fluid/diffusion limits |

| Stochastic comparison / bounds | coupling, stochastic dominance, martingale/concentration inequalities |

| MDP / dynamic decisions | dynamic programming, value/policy iteration, ADP with error bounds |

| Heavy-traffic / asymptotics | functional CLT, weak convergence, state-space collapse |

Algorithm design with guarantees

• State what the algorithm guarantees: exact/optimal, an approximation factor, an

ε-stationary point, or a regret/convergence rate — and under which assumptions.

• Give complexity (time, iterations, oracle calls; per-iteration cost and total).
• Separate the method from its proof of correctness/convergence; a fast

heuristic without analysis is not an OR methodological contribution on its own.

Simulation methodology (when the analysis is empirical-stochastic)

• Specify the estimator and argue consistency; quantify error with valid

confidence intervals (batch means, regenerative, or replication-based).

• Use variance reduction (common random numbers, control variates) and justify it.
• For ranking-and-selection / simulation optimization, state the statistical

guarantee (e.g., probability of correct selection) and the budget rule.

Proof hygiene OR reviewers expect

• Every assumption used is invoked explicitly where the proof needs it.
• Long proofs go to an e-companion (which must not be longer than the manuscript);

the main text keeps the key idea and a proof sketch.

• Constants and rates are tracked, not hidden in "O(·)" when tightness is claimed.

Methodology pushback patterns and the OR fix

| Referee remark | Underlying defect | Fix that meets the OR bar |

|----------------|-------------------|----------------------------|

| "Proof of Theorem X has a gap" | an assumption invoked implicitly | name where each hypothesis is used; add a lemma to bridge the step |

| "The rate is asserted, not established" | rate read off numerical curves | prove it analytically (Lyapunov / contraction / first-order) with tracked constants |

| "Algorithm has no guarantee" | a fast heuristic without analysis | attach an approximation factor, ε-stationarity, or regret/convergence bound |

| "Bound may not be tight" | only an upper bound shown | exhibit a matching instance, or reframe explicitly as best-known |

| "Simulation conclusions unreliable" | point estimates, no error control | report CIs (batch-means/regenerative) and variance reduction with the rule |

| "Structural result not connected to the application" | theorem floats free of the decision | show the guarantee changes the operational policy it motivates |

Operations Research, as the INFORMS flagship, lives on **soundness and strength of

analysis**: a heuristic without a guarantee is an INFORMS Journal on Computing artifact,

not an OR methodological contribution. The machinery table above exists so each

claim is discharged by analysis a referee can verify line by line.

Worked machinery walk-through (illustrative)

Target result: an approximation algorithm for a stochastic-covering problem with a

claimed 1.5-factor guarantee (illustrative). Machinery selection from the table:

LP-rounding + primal-dual for the factor; concentration (martingale) to control

the stochastic constraint; an oracle lower bound to argue the factor cannot be

pushed below 1.5 without stronger assumptions. Proof hygiene: each of the three

assumptions (bounded second moment, independence across stages, integral demand) is

cited exactly where the argument needs it; the full rounding analysis goes to the

e-companion, the main text keeps the primal-dual sketch and the tight-instance

construction. This produces a theorem-grade result and a tightness statement — the

combination OR referees reward over a bare upper bound.

Execution bridge (StatsPAI / Stata MCP)

For the empirical / causal lane, estimate and audit rather than only specify. Full

map: [execution-with-mcp](../../../shared-resources/empirical-methods/execution-with-mcp.md). Operations Research is predominantly analytical / optimization / stochastic modeling; use the chain below only for its empirical/causal papers — modeling, optimization, and simulation are outside this causal-inference toolchain.

• detect_design → recommend → fit with as_handle=true → audit_result to

enumerate the checks the design owes.

• Panel / staggered DiD: callaway_santanna / sun_abraham + bacon_decomposition
• honest_did_from_result. IV: effective_f_test + anderson_rubin_ci. RDD:

rdrobust + mccrary_test.

• Experiments: randomization-based inference and romano_wolf for the many-outcome

family-wise correction reviewers expect.

Match the toolchain to the reviewer pool, and report the effect size the venue

wants. A run end-to-end (synthetic data, real returns) is in the

[JF execution walkthrough](../../../Journal-of-Finance-Skills/resources/worked-examples/02-execution-walkthrough.md).

Anti-patterns

• A "proof" that silently adds an assumption mid-argument.
• Claiming a rate from numerical curves rather than analysis.
• An algorithm with no guarantee presented as the central contribution.
• Simulation conclusions with no confidence intervals or variance control.

Output format

【Result → technique】each Thm/Prop mapped to its machinery
【Algorithm】guarantee (exact/approx/rate) + complexity
【Simulation】estimator, CI method, variance reduction (if used)
【Proof hygiene】assumptions invoked explicitly; e-companion plan
【Open gaps】[...]
【Next step】ors-data-analysis