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EndoWake: Modeling Wind by Linear Programming for Wind Farm Layout Optimization

Martina Fischetti, Matteo Fischetti

arXiv:2609.40157v1math.OCcs.CE

TL;DR

Wind-farm layout optimization must account for wake effects that complicate production evaluation and embedding in optimization. The paper introduces EndoWake, integrates its propagation into a MILP, and strengthens the root bound with band inequalities. On the 24 × 24 instance, the root gap falls from 21.8% to 10.0%.

  • Problem

    Wind-farm layout optimization seeks turbine placements that maximize energy, but wake effects complicate production evaluation and embedding the problem in an optimization framework.

    Source: §1 · p. 1

  • Method

    EndoWake makes cell-level wind speeds decision variables linked by linear upwind propagation, and the paper uses band inequalities and branch-and-check with upwind cuts.

    Source: §1 · p. 2, §9 · p. 26

  • Results

    On the 24 × 24 instance, band inequalities reduce the root gap from 21.8% to 10.0%.

    Source: §9 · p. 26

  • Takeaways & Limitations

    The paper presents a formulation whose calibrated flow field is part of the optimization model and whose relaxation can be strengthened with layout-power band inequalities.

    Source: §9 · p. 26

    The reported optima, bounds, and certificates concern the calibrated surrogate model, not the physical wake behavior it approximates.

    Source: §1 · p. 3

Abstract

from arXiv · show

The Wind Farm Layout Optimization problem consists of placing a given number of turbines within a given area so as to maximize energy production. One of its main challenges is the wake effect between turbines, which can strongly affect production and is a cumbersome function to evaluate and to embed in an optimization framework. In this paper we introduce a new endogenous wake model (EndoWake), based on the idea of treating the wind speed at every grid point as a decision variable, linked to its upwind neighbors through linear propagation constraints. This makes it possible to incorporate the wake equations directly into a Mixed-Integer Linear Programming (MILP) model. We then develop such a MILP model and, in particular, introduce a new class of valid "band" inequalities, showing that they substantially strengthen the root bound. Building on this, we design a branch-and-check scheme: the master problem retains only the site variables together with one hypograph variable per cell and scenario, integer solutions are evaluated by an exact linear-time oracle, and the optimality cuts are problem-specific "upwind cuts". Computational results are reported on a ladder of square grids under a four-sector axial wind rose, demonstrating the effectiveness of the proposed improvements.

1 Introduction

EndoWake makes wind speed at every grid cell a decision variable linked to upwind cells through linear propagation, bringing endogenous wake modeling into a MILP. The paper strengthens the weak relaxation with band inequalities and uses branch-and-check with upwind cuts to certify exact energy for the calibrated model.

1 Introduction

  • Propagation is written inside the optimization model as linear constraints, so the same mechanism supports the relaxation, valid inequalities, and exact layout evaluation.Source: §1 · p. 2
  • EndoWake places a wind-speed decision variable at each cell and scenario, with linear propagation linking it to upwind cells so wake aggregation and higher-order interference arise without a preset superposition rule.Its formulation grows linearly rather than quadratically with the number of cells.Source: §1 · p. 2
  • The ordinary linear relaxation can yield an uninformative dual bound, and two local repairs fail to strengthen it beyond the wake-free value.The paper reports the perspective reformulation's bound reaches only the wake-free value.Source: §1 · p. 2
  • Band inequalities significantly improve the root bound, cut nodes and solve time on instances that close, and lower the median certificate on every instance left open.They couple layout and power variables over bands ordered by the wind direction.Source: §1 · pp. 2–3
  • The branch-and-check master retains site and hypograph variables, while an exact oracle evaluates proposed layouts and returns optimality cuts supported only on upwind cells.On the largest instance, the scheme certifies exact energy in a fraction of the time the monolithic model takes to certify its interpolated surrogate.Source: §1 · p. 3

2 Related work

Prior mathematical-programming approaches generally price wakes using precomputed pairwise deficits and a fixed superposition rule, while engineering models allow wakes to depend on shaped inflow. This paper's distinction is to place linear flow propagation inside the optimization model and derive layout-flow inequalities and dual bounds.

2 Related work

  • Mathematical-programming WFLO models commonly use precomputed pairwise deficits under a fixed linear or root-sum-square superposition rule.Source: §2 · p. 3
  • Engineering wake models let a wake depend on inflow shaped by upwind machines, but the cited evaluators and gradient methods do not enter a mixed-integer model or return a dual bound.Examples include Zong and Porté-Agel (2020), Bastankhah et al. (2021), and Lanzilao and Meyers (2022).Source: §2 · p. 4
  • The paper's modeling novelty is to combine binary siting, linear local propagation, and acyclic flow in a mixed-integer optimization structure.The authors state that, to their knowledge, no valid inequalities coupling layout to flow had previously been proposed for WFLO.Source: §2 · p. 4
  • Zhang et al. (2014) used an exact evaluator and logic-based cuts for WFLO, but their approach re-solved the master each round and closed no instance within an hour.Source: §2 · p. 5

3 The problem, and the EndoWake wind model

The paper formulates discrete wind-farm layout as choosing separated turbine sites on a grid to maximize expected energy across wind scenarios. EndoWake carries cell-level wind speeds through upwind propagation, with turbine-induced reductions and calibrated spread and recovery parameters.

The problem

  • The discrete WFLO problem chooses candidate grid cells for a capped number of turbines, with minimum separation, to maximize expected energy over wind scenarios.Source: §3 · p. 5

Our new formulation: EndoWake

  • EndoWake replaces a precomputed interference matrix and fixed superposition rule with a wind-speed decision variable at each grid cell.Source: §3 · p. 5
  • For each wind scenario, cells are ordered into upwind-to-downwind planes, and each cell's speed is bounded using a weighted average of cells in the preceding plane plus recovery toward free stream.The five-offset stencil kernel is input data fitted before optimization; applying it plane by plane spreads wakes laterally.Source: §3 · p. 5
  • A turbine multiplies the speed passed to shadowed downstream cells by a factor below one, so wake merging, higher-order effects, and recovery follow from propagation.The calibration residual is below 8% of the reference field at the resolution used.Source: §3 · p. 6

4 The MILP model

This section presents the MILP’s wind-speed field formulation, linking cell speeds to upwind neighbors and turbine placement, and describes the acyclic propagation structure used to compute a maximal feasible field. It also identifies the structural hypothesis underpinning band and upwind cuts and calibrates EndoWake against an analytical wake model, with a 7.7% residual at the experimental grid resolution.

The optimization model

  • Treating flow outside the grid as undisturbed gives upwind-edge cells clean wind by construction, mildly favoring layouts that use those cells.A few non-allocatable cells can be added as a frame to move the allocatable region inside the flow domain; the experiments use no frame.Source: §4 · p. 7
  • The MILP jointly chooses turbine sites and scenario-specific cell speeds, maximizing expected production through flow and wake constraints and a power-curve block.The model also encodes minimum turbine separation and installation-count bounds.Source: §4 · p. 8
  • A turbine scales its local incoming speed by ρ, so a turbine already in a wake casts a proportionally weaker wake—an effect pairwise coefficients cannot express.The wake reduction is multiplicative on local speed and is switched off when no turbine is installed.Source: §4 · p. 8
  • The model has |A| binary variables and O(|S| |C|) continuous variables and rows, linear in grid size rather than the Θ(|C|^2) entries of an interference matrix.Source: §4 · p. 9

The maximal field, and its antitonicity

  • For each layout, a greatest feasible wind field exists and is computable in O(|C|) time by a forward sweep; any objective non-decreasing in wind speed is maximized at that field.Adding turbines can only lower the maximal field everywhere.Source: §4 · p. 9

The MILP model

  • Exact layout evaluation takes one grid sweep, using five multiply–adds and one comparison per cell and scenario, plus one power-curve lookup per installed cell.This is the oracle used in Section 7 and the evaluator behind the paper’s exact AEP figures.Source: §4 · p. 10
  • A single structural hypothesis supports the validity of both the band inequalities and the upwind cuts.The paper proves this hypothesis in the section because every cut in the paper rests on it.Source: §4 · p. 10

Calibration and instances

  • Weighted least squares fits the kernel to the Bastankhah–Porté-Agel model at CT = 8/9 over 3 D–12 D downstream, within a domain extending 20 D downstream and 5 D laterally.The amplitude is recovered in closed form at each evaluation because the field is linear in the source.Source: §4 · p. 10
  • At Δ = D, the calibrated wake field has a 7.7% RMS residual relative to the reference on the operational window.The residual is measured as RMS error divided by the RMS of the reference.Source: §4 · p. 10
  • The fit drives β to zero as an outcome, not a modeling choice; reported results concern the calibrated model rather than wake-model fidelity.The theory is stated for the full five-point stencil.Source: §4 · p. 10

5 A weak relaxation

This section examines why the linear relaxation remains weak, reaching the trivial rated-power bound by ignoring wake interactions and valuing fractional turbines at rated output. It shows that natural per-cell repairs do not strengthen the bound, because the needed improvement couples layout and power variables in aggregate.

5 A weak relaxation

  • Under Proposition 2’s condition, the relaxation’s optimal value is exactly NmaxPmax, the trivial rated-power bound.The condition uses the minimum of the power-ratio, packing, and 1/2 terms; a uniform fractional installation attains the bound.Source: §5 · p. 11
  • With qmax = 9, the bound is vacuous below one turbine per nine cells, the operating regime for industrial instances and solver-open grids from n = 12.The four smallest ladder grids exceed the threshold; on three, packing constraints bind before cardinality and the root falls short of the trivial value.Source: §5 · p. 11
  • The relaxation discards wake interactions and the power curve, pricing each fractional turbine at rated output; the big-M constant ρ is not the cause.For the Section 8 turbine, the model’s and exact power-curve wake-free maxima are smaller than NmaxPmax by 8.2% and 9%, respectively.Source: §5 · p. 11

5.1 Two local repairs, and why they do not work

  • The ideal per-cell power convexification lowers the bound to the wake-free value on all four ladder instances, then leaves band-inequality root bounds unchanged within solver noise.The worst reported change on top of the band inequalities is 5 · 10^-10 kW.Source: §5.1 · p. 12
  • The lifted constraint (6) dominates (3) and is componentwise maximal when Nmin = 0; for As = 11.4 m/s, the resulting density threshold is 0.274.The threshold passage also gives vr = 10.34 m/s and ρ = 0.293.Source: §5.1 · pp. 12–13
  • Lifting acts on the wrong variables: lowering one cell’s wind lets the relaxation retreat to a slightly lower uniform field.The objective can still price a turbine at rated output even when its fractional site variable is small.Source: §5.1 · p. 13
  • The relaxation’s bound remains vacuous after adding single-source inequalities, which constrain the wind of one downwind cell at a time.Their validity follows from antitonicity, but the relaxation absorbs them in the same way as the local repairs.Source: §5.1 · p. 13
  • Strengthening the bound requires coupling layout to power variables in aggregate, as the band inequalities in Section 6 do.This contrasts with repairs that act on one cell’s wind at a time.Source: §5.1 · p. 13

6 Band inequalities

Band inequalities strengthen the layout model by bounding production over strips of cells rather than one cell at a time. A dynamic program supplies valid production bounds, whose upper concave envelope yields linear inequalities.

6 Band inequalities

  • Band inequalities couple the turbine count and production in a strip, bounding its summed power with facets of an upper concave envelope.The envelope is built from upper bounds on the power achievable with each number of turbines in the band.Source: §6 · pp. 14–15
  • Bands narrower than the minimum separation order their turbines along the wind flow, allowing the strip layout to be treated as a chain.The width ensures that no two cells on the same propagation plane can both contain turbines.Source: §6 · p. 14
  • A dynamic program computes the capacity K_R and power bound Ψ_R(k), using the floor deficit from each turbine’s immediate predecessor as its transition reward.The floor deficit depends on the offset between cells, so a program for a width and scenario serves bands at different positions.Source: §6 · p. 15
  • Validity uses safe relaxations: enforce separation only between consecutive turbines, charge only the predecessor’s floor deficit, and rely on non-decreasing power.These choices make the chain program’s returned value an upper bound on feasible band production.Source: §6 · p. 15
  • The envelope must be an upper hull, and its power representation must be at least as large as the model’s representation.The paper uses the SOS2 interpolant for the monolithic model and the true power curve for the branch-and-check master.Source: §6 · p. 16

7 Branch-and-check with upwind cuts

The section combines a master over turbine locations and power hypographs with an exact oracle and problem-specific upwind cuts. This branch-and-check scheme carries certification onto the exact energy scale.

7.1 The two-stage reading, and an oracle that costs less than an LP

  • For fixed layout and scenario, the subproblem is solved exactly in O(|C|) time by a forward wind-field sweep followed by cellwise power evaluation.The paper notes that this exact oracle is cheaper than the linear program it replaces.Source: §7.1 · p. 17

7.2 The master

  • The master retains site variables, packing constraints, band inequalities, and one hypograph variable per cell and scenario, while dropping the wind-flow and power blocks.The band inequalities give a root bound on the exact energy scale, which the cuts then drive down.Source: §7.2 · pp. 17–18

7.3 The oracle, and posting the incumbent at its exact value

  • At each integer layout, the oracle computes exact cell powers and posts the layout as an incumbent at its exact value, rather than trusting optimistic master hypographs.The exact incumbent lets pruning operate on the exact energy scale, up to the flagging tolerance.Source: §7.3 · p. 18

7.4 The upwind cut

  • Upwind cuts are strengthened no-good optimality cuts: they restrict support to relevant upwind cells and use an exact oracle value instead of a big-M constant.Their recourse is antitone in the layout, and the disaggregation is per cell and scenario.Source: §7.4 · p. 19
  • Across final layouts, the upwind support averaged 2.8 cells, versus 12.9 cells in X \ {c}.Restricting support keeps a cut active on every layout containing the relevant upwind cells and target cell.Source: §7.4 · p. 19

8 Computational experiments

The computational experiments on a synthetic square-grid ladder evaluate EndoWake’s exact methods under four axial wind sectors, focusing on unstrengthened-model scalability and the effects of band inequalities and branch-and-check. They examine root-bound gains, search costs, exact-energy certificates within the one-hour budget, and the effects of band pooling, cut support, and exact incumbent posting.

8 Computational experiments

  • The campaigns use a square-grid ladder with Nmax = n + 2 turbines and four equiprobable axial sectors; existing benchmarks are not directly comparable because they use different wake models and lack certified optima for this formulation.Heuristics at industrial size and external validation against engineering wake simulators are outside the paper’s scope.Source: §8 · p. 20
  • Monolithic solver gaps vary by as much as 131.41 percentage points across seeds, while branch-and-check closes instances with every seed and open-instance spreads are 0.4 to 1.3 percentage points.In band-placement runs, certificate spread reaches 17 points on 20 × 20.Source: §8 · p. 21
  • The unstrengthened EndoWake MILP closes through 10 × 10, but from 12 × 12 hits the 3600-s limit and reaches a 77.75% gap at 24 × 24.The model grows linearly with the number of cells and builds in a fraction of a second; the reported difficulty is attributed to its weak relaxation.Source: Table 1 · p. 21, §8 · pp. 21–22

8.1 The band inequalities, at the root and inside the search

  • Across four examined instances, one partition per width (77% of the BI family) reproduces the full-family root bound exactly; width L = 2 alone closes its gap (L = 1 on the smallest).A single band is ineffective because fractional turbines spread across neighboring bands; the useful offsets are multiples of band width.Source: §8.1 · p. 22
  • Static BI constraints improve dual bounds by 2.7% to 4.6% on 16 × 16, 20 × 20, and 24 × 24, but 16 × 16 explores 9 793 nodes versus 63 048 for the plain model.The tightened static model yields fewer explored nodes, but good incumbents stop coming.Source: §8.1 · p. 22
  • On the largest grid, pooled inequalities yield a 54.0% median certificate, versus 77.3% with static constraints and 68.2% with none; on 12 × 12, static constraints win, 11.8% to 13.1%.Across the same three grids and five random seeds, pooling costs 1.1 to 3.0 percentage points of certificate on instances that do not close and wins on none of the 25 instance-seed pairs.Source: §8.1 · p. 22
  • 21.8%–25.1% root gaps fall to 9.7%–11.4% on the five instances left open by the reference solve when band inequalities are added.On 24 × 24, the gap falls from 21.8% to 10.0%; on the five open instances, the tightened root bound exceeds Gurobi’s one-hour untightened bound by 5.0% on 16 × 16 and at least 2.8% on the others, at a 4.1 s generation cost.Source: Table 2 · p. 23, §8.1 · p. 23
  • On instances that close either way (up to 10×10), band inequalities reduce the largest solve from 453 348 nodes to 152 822 and cut wall-clock time by 2.9×, including separation.On all five instances that do not close, the median certificate on the exact energy scale falls by 3% to 31% of its value.Source: §8.1 · p. 24

8.2 Branch-and-check

  • Branch-and-check closes every instance up to 10 × 10, solving the largest in 77 s versus 3472 s for the monolithic model; on open larger instances, its gaps are 7.98%–9.84% versus 13.11%–54.02%.The monolithic certificate on 10 × 10 is 2.18% on the exact energy scale, while branch-and-check’s is zero to the digit; no run from 12 × 12 onward closes within an hour.Source: Table 4 · p. 25, §8.2 · p. 25
  • Removing band inequalities from the branch-and-check master closes nothing above 8 × 8 and raises the median gap by 5.7 to 10.7 points on every one of 30 open instance–seed pairs.Using full support U = X \ {c} also closes nothing above 8 × 8; the 10 × 10 instance remains at 7.9% after an hour with 5.3× as many cuts.Source: §8.2 · p. 25
  • Exact incumbent posting proves the 10 × 10 instance in 77 s versus 96 s for the same certificate and improves incumbents on open instances by up to 0.85% on 16 × 16.Without posting, pruning uses the master’s optimistic hypographs rather than the oracle’s computed value, so the search cannot recognize or exploit its incumbent.Source: §8.2 · p. 25

9 Conclusions

EndoWake represents wind speeds as decision variables propagated by linear constraints, and the paper analyzes the resulting relaxation and strengthens it with band inequalities. The conclusions report stronger root bounds and an exact-energy branch-and-check scheme, while noting that exact solves remain below industrial-farm scale and that some settings remain open.

9 Conclusions

  • EndoWake makes the wind field a vector of decision variables propagated by linear constraints, rather than precomputing an interference matrix.The multi-scenario formulation is block-angular, and its second stage is evaluated exactly in O(|C|) time per scenario, supporting branch-and-check with an oracle and upwind cuts.Source: §9 · p. 26
  • Under a condition comfortably met by practical operating densities, the root bound equals turbine count times rated power, exceeding even wake-free production regardless of wind-sector count.The authors characterize this bound exactly and find that maximally lifted wake constraints and the power-block perspective reformulation do not improve it beyond the wake-free value.Source: §9 · p. 26
  • At the three densest grids, packing rows bind first, so tightening minimum separation does not help; elsewhere, branching closes the gap slowly as wakes must be discovered.The authors report that natural local repairs fail to strengthen the bound, including a maximally lifted wake constraint and a power-block perspective reformulation.Source: §9 · p. 26
  • On 24 × 24, band inequalities reduce the root gap from 21.8% to 10.0%, and they improve bounds on instances left open by the reference solve.The inequalities aggregate cells ordered by wind direction and are imposed at every band width.Source: §9 · p. 26
  • The branch-and-check scheme certifies exact energy, whereas the monolithic model certifies an energy scale affected by its SOS2 interpolation bias.On the largest instance, the monolithic model with band inequalities takes 15 times as long to close its own model.Source: §9 · p. 26
  • Exact solves remain an order of magnitude below industrial-farm size; oblique-to-grid sectors and irregular sites remain open, with a matheuristic proposed as a scale-up bridge.The authors also identify lazy cut-pool management and incumbent posting as engineering refinements.Source: §9 · p. 26
  • The band-inequality recipe can extend beyond fixed full-length strips to cell sets with limited lateral extent, whose chain property supports dynamic programming.The authors identify contaminant transport, indoor ventilation, and emission control with sensor or filter placement as possible future applications of the broader grid-state structure.Source: §9 · pp. 26–27

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

The authors used Claude Opus 5 and Claude Fable 5 to assist with experimental code, adversarial review, and manuscript editing, and ChatGPT 5.6 Sol for adversarial audits. They independently reviewed and edited the content, checked every proof, and accepted responsibility for the published article.

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

  • Claude Opus 5 and Claude Fable 5 assisted with experimental code, mathematical-statement review, and manuscript language and structure.The authors also obtained independent adversarial audits from ChatGPT 5.6 Sol.Source: Declaration of generative AI and… · p. 27
  • The authors reviewed and edited the content and independently checked every proof.They state that they take full responsibility for the published article.Source: Declaration of generative AI and… · p. 27

Supplementary material

The supplement provides pseudocode and a finite-termination proof for the branch-and-check scheme, plus computational tables for each solver seed, as part of the replication package.

Supplementary material

  • The supplement includes band dynamic-program pseudocode and the branch-and-check scheme with a proof of finite termination at the exact optimum in exact arithmetic.Proposition S1 states what remains under solver tolerances.Source: Supplementary material · p. 31
  • Computational-campaign tables cover each of the five solver seeds.The supplement is part of the replication package.Source: Supplementary material · p. 31

S1 The band dynamic program

The band dynamic program computes band capacities and tables used to construct band inequalities, with its propagation order adapted to the wind direction. Its power representation uses a piecewise-linear majorant of the true curve.

S1 The band dynamic program

  • The power representation bP uses eight breakpoints, with six segments up to rated speed and a seventh flat rated stretch.The breakpoints include the cut-in speed and rated speed.Source: S1 The band dynamic program · p. 31
  • Between cut-in and rated speed, the exact power curve is cubic, so its chords majorize it and support the bound bP ≥ P.The rated stretch is flat.Source: S1 The band dynamic program · p. 31
  • The band dynamic program returns each band’s capacity and a table from which the band inequalities are built.Its inputs include a band narrower than the minimum turbine separation, so no propagation plane can carry two turbines.Source: S1 The band dynamic program · p. 31
  • For a north wind, grid rows are propagation planes and the flow order is row order; other directions interchange row and column roles.The algorithm’s remaining structure is unchanged across directions.Source: S1 The band dynamic program · p. 31

S2 The branch-and-check scheme as a whole

The branch-and-check scheme evaluates integer layouts with a one-sweep-per-scenario oracle and adds problem-specific upwind cuts. Its supplement establishes finite termination and specifies how solver tolerances affect the reported bound.

S2 The branch-and-check scheme as a whole

  • For each integer candidate, one sweep per scenario returns the wind field and exact layout energy, and flagged candidates receive lazy upwind cuts.The scheme uses a full-support cut as a fallback.Source: S2 The branch-and-check scheme as a… · p. 32
  • Local search refined oracle-scored layouts, and the solver adopted 308 of the 422 layouts posted across 45 runs.The remaining posted layouts did not improve on the incumbent held by the solver.Source: S2 The branch-and-check scheme as a… · p. 32
  • The cut support is based on allocatable cells whose wake can reach the target cell; in floating point, the code’s support can be smaller than this full ancestor set.The measured code support was contained in the ancestor set with zero exceptions across 7,284 scenario-cell pairs.Source: S2 The branch-and-check scheme as a… · pp. 32–33
  • The authors checked the support lemma on 270 feasible layouts and 14,408 checks, reporting a largest difference of 0 m/s for the ancestor-set comparison.The code-set comparison’s reported constant difference was 1.1 · 10^-9 kW.Source: S2 The branch-and-check scheme as a… · p. 33
  • With zero flagging and solver tolerances, the scheme terminates finitely at the exact layout optimum; with default tolerances, its bound is valid up to solver feasibility and optimality tolerances.At termination, including a time limit, the optimality gap is bounded by the final bound minus the best exact layout value.Source: S2 The branch-and-check scheme as a… · p. 33
  • Flagging tolerance affects whether the solver flags a candidate and whether a posted layout improves the incumbent, but not the certificate.The cited largest-instance bound on candidate objective excess is 0.026 kW.Source: S2 The branch-and-check scheme as a… · p. 33

S3 The campaign tables, seed by seed

The campaign tables report runs across five Gurobi seeds, selecting a reproducible median-certificate run for each instance. They cover monolithic scalability, band inequalities, branch-and-check, incumbent posting, and variation across seeds.

S3 The campaign tables, seed by seed

  • Each campaign run is repeated with five Gurobi seeds, and each reported row selects the run with the median certificate, breaking ties by solve time.Where all five runs close, the selection is the run with the median certificate; rows are individual runs, not column-wise averages.Source: S3 The campaign tables, seed by seed · p. 35
  • The seed-by-seed tables reproduce monolithic scalability, band inequalities inside the search, branch-and-check, and incumbent posting results.The root-relaxation study is deterministic and is not repeated by seed.Source: S3 The campaign tables, seed by seed · p. 35, Table S1 · p. 35, Table S2 · p. 37
  • Posting the exact incumbent wins 19 of 25 instance-seed pairs, ties 5, and loses 1 under the stated time-to-proof or final-gap comparison.Differences below 0.5 s or 0.01 gap points count as ties.Source: Table S4 · p. 41

S3.5 Dispersion across seeds

  • Dispersion is reported per instance across five seeds for certificates or gaps, proof times, and the branch-and-check incumbent’s exact-energy spread.For branch-and-check, the certificate range is a final-gap range; a lower bound indicates at least one run ended without an incumbent.Source: Table S5 · p. 42

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