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. 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.
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
Source: brycewang-stanford/Awesome-Journal-Skills → Operations-Research-Skills/skills/ors-methods/SKILL.md
1---2name: ors-methods3description: Use when designing the proof technique, algorithm, or simulation protocol for an Operations Research (OR) manuscript — choosing the right machinery (duality, dynamic programming, probabilistic coupling, convergence analysis, simulation output analysis) to actually establish the claimed results. Establishes the results; it does not state the model (ors-theory-development) or run the experiments (ors-data-analysis).4---5
6
7# Proof & Algorithm Methodology (ors-methods)
8
9## When to trigger
10
11- The model and claims exist (`ors-theory-development`) and now must be *proved* or *guaranteed*.
12- You need to pick a proof strategy or design an algorithm with provable guarantees.
13- A reviewer says "the proof of Theorem X has a gap" or "the rate is not established."
14
15## Match the machinery to the result
16
17*Operations Research* is mathematically rigorous: the contribution lives or dies on
18the soundness and strength of the analysis. Pick technique by methodology:
19
20| Result you need | Typical machinery |
21|-----------------|-------------------|
22| Optimality / strong duality | LP/conic duality, KKT, polyhedral / total unimodularity, submodularity |
23| Approximation guarantee | LP/SDP rounding, primal-dual, greedy + submodular bounds |
24| Complexity / hardness | reductions (NP-hardness), oracle lower bounds |
25| Convergence & rate | monotonicity/Lyapunov, fixed-point/contraction, first-order analysis |
26| Steady-state / stability | Foster-Lyapunov, regenerative arguments, fluid/diffusion limits |
27| Stochastic comparison / bounds | coupling, stochastic dominance, martingale/concentration inequalities |
28| MDP / dynamic decisions | dynamic programming, value/policy iteration, ADP with error bounds |
29| Heavy-traffic / asymptotics | functional CLT, weak convergence, state-space collapse |
30
31## Algorithm design with guarantees
32
33- State **what the algorithm guarantees**: exact/optimal, an approximation factor, an
34 ε-stationary point, or a regret/convergence rate — and under which assumptions.
35- Give **complexity** (time, iterations, oracle calls; per-iteration cost and total).
36- Separate the **method** from its **proof of correctness/convergence**; a fast
37 heuristic without analysis is not an OR methodological contribution on its own.
38
39## Simulation methodology (when the analysis is empirical-stochastic)
40
41- Specify the estimator and argue **consistency**; quantify error with valid
42 confidence intervals (batch means, regenerative, or replication-based).
43- Use **variance reduction** (common random numbers, control variates) and justify it.
44- For ranking-and-selection / simulation optimization, state the statistical
45 guarantee (e.g., probability of correct selection) and the budget rule.
46
47## Proof hygiene OR reviewers expect
48
49- Every assumption used is invoked explicitly where the proof needs it.
50- Long proofs go to an **e-companion** (which must not be longer than the manuscript);
51 the main text keeps the key idea and a proof sketch.
52- Constants and rates are tracked, not hidden in "O(·)" when tightness is claimed.
53
54## Methodology pushback patterns and the OR fix
55
56| Referee remark | Underlying defect | Fix that meets the OR bar |
57|----------------|-------------------|----------------------------|
58| "Proof of Theorem X has a gap" | an assumption invoked implicitly | name where each hypothesis is used; add a lemma to bridge the step |
59| "The rate is asserted, not established" | rate read off numerical curves | prove it analytically (Lyapunov / contraction / first-order) with tracked constants |
60| "Algorithm has no guarantee" | a fast heuristic without analysis | attach an approximation factor, ε-stationarity, or regret/convergence bound |
61| "Bound may not be tight" | only an upper bound shown | exhibit a matching instance, or reframe explicitly as best-known |
62| "Simulation conclusions unreliable" | point estimates, no error control | report CIs (batch-means/regenerative) and variance reduction with the rule |
63| "Structural result not connected to the application" | theorem floats free of the decision | show the guarantee changes the operational policy it motivates |
64
65Operations Research, as the INFORMS flagship, lives on **soundness and strength of
66analysis**: a heuristic without a guarantee is an *INFORMS Journal on Computing* artifact,
67not an OR methodological contribution. The machinery table above exists so each
68claim is discharged by analysis a referee can verify line by line.
69
70## Worked machinery walk-through (illustrative)
71
72Target result: an approximation algorithm for a stochastic-covering problem with a
73claimed 1.5-factor guarantee (illustrative). Machinery selection from the table:
74**LP-rounding + primal-dual** for the factor; **concentration (martingale)** to control
75the stochastic constraint; **an oracle lower bound** to argue the factor cannot be
76pushed below 1.5 without stronger assumptions. Proof hygiene: each of the three
77assumptions (bounded second moment, independence across stages, integral demand) is
78cited exactly where the argument needs it; the full rounding analysis goes to the
79e-companion, the main text keeps the primal-dual sketch and the tight-instance
80construction. This produces a theorem-grade result *and* a tightness statement — the
81combination OR referees reward over a bare upper bound.
82
83## Execution bridge (StatsPAI / Stata MCP)
84
85For the **empirical / causal lane**, estimate and audit rather than only specify. Full
86map: [`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.
87
88- `detect_design` → `recommend` → fit with `as_handle=true` → `audit_result` to
89 enumerate the checks the design owes.
90- **Panel / staggered DiD:** `callaway_santanna` / `sun_abraham` + `bacon_decomposition`
91 + `honest_did_from_result`. **IV:** `effective_f_test` + `anderson_rubin_ci`. **RDD:**
92 `rdrobust` + `mccrary_test`.
93- **Experiments:** randomization-based inference and `romano_wolf` for the many-outcome
94 family-wise correction reviewers expect.
95
96Match the toolchain to the **reviewer pool**, and report the effect size the venue
97wants. A run end-to-end (synthetic data, real returns) is in the
98[JF execution walkthrough](../../../Journal-of-Finance-Skills/resources/worked-examples/02-execution-walkthrough.md).
99## Anti-patterns
100
101- A "proof" that silently adds an assumption mid-argument.
102- Claiming a rate from numerical curves rather than analysis.
103- An algorithm with no guarantee presented as the central contribution.
104- Simulation conclusions with no confidence intervals or variance control.
105
106## Output format
107
108```
109【Result → technique】each Thm/Prop mapped to its machinery
110【Algorithm】guarantee (exact/approx/rate) + complexity
111【Simulation】estimator, CI method, variance reduction (if used)
112【Proof hygiene】assumptions invoked explicitly; e-companion plan
113【Open gaps】[...]
114【Next step】ors-data-analysis
115```
116
117---
118
119**Source:** [`brycewang-stanford/Awesome-Journal-Skills`](https://github.com/brycewang-stanford/Awesome-Journal-Skills) → `Operations-Research-Skills/skills/ors-methods/SKILL.md`
thedixitjain