Convex-SKM-Net: Low-Latency Block Halfspace Corrections for Neural Convex Constraint Satisfaction

Motivation

• Constraint layers use feasible-set parameterizations, optimization, or iterative repair, whose applicability may depend on interior points or component projections and whose repeated corrections can be costly.Repair direction also matters because it changes the predicted decision and can affect the final objective.Source: §1 · p. 1

Benchmark results

• 80.7–144.3× faster inference than fixed-400-step Πnet on SOC-MPC, while 300-bus semidefinite-relaxation postprocessing is 7.7× faster than LMI-Net with a lower mean gap on identical raw predictions.The 300-bus comparison uses a fully finetuned predictor.Source: §1 · p. 2

Learnable mobility

• 22.4% lower five-seed mean gap on the affine task with learned mobility, which also reduces matched mean gaps by 11.0% and 8.1% on 300-bus conic and semidefinite relaxations.A positive-definite mobility changes directional correction costs while preserving the feasible set.Source: §1 · p. 2

Contributions

• Convex-SKM-Net forms linearization halfspaces from residual and (sub)gradient oracles, avoiding constraint-specific inequality projectors while supporting postprocessing and joint training.Selected-block minimum-displacement updates use fixed-depth dual PGD and GPU-friendly matrix operations.Source: §1 · pp. 1–2

Theoretical contribution

• Fixed-depth block corrections use dual PGD, with outer-iterate convergence and linear convergence in distance under Slater’s condition and bounded subgradients.The updates avoid per-block factorizations.Source: §1 · p. 2

Feasible-set parameterization

• Feasible-set parameterizations include generators, gauge maps, decision rules, calibrated constraints, restricted network maps, and interior-point or invertible transformations.RAYEN uses an interior point and ray boundaries, computing the point offline for fixed constraints [Tordesillas et al., 2026].Source: §2 · p. 2

Differentiable solver layers

• Differentiable solver layers span affine repair, optimization layers, entropy-regularized dual iterations, and operator splitting methods such as Πnet and LMI-Net [Tang et al., 2026].Dykstra’s algorithm and alternating projections also support constrained prediction.Source: §2 · p. 2

Residual repair and SKM methods

• Residual-based layers include DC3’s equality completion and violation gradients and FSNet’s violation minimization, alongside adaptive, damped, optimality-based, and regularized-Newton updates [Donti et al., 2021; Nguyen & Donti, 2025; Lastrucci & Schweidtmann, 2025; Goertzen et al., 2026; Iftakher et al., 2026; Chu et al., 2026].SKM methods repair sampled violated rows, while related work also studies linear block projections and convex-inequality subgradient, parallel, and minibatch methods [De Loera et al., 2017; Zhang et al., 2023; Zhu et al., 2026; Zhang et al., 2026; Needell & Tropp, 2014; Zhang & Li, 2021; 2023; Briskman & Needell, 2015; dos Santos, 1987; Kiwiel, 1995; Bauschke & Combettes, 2017; Necoara & Nedi´c, 2021].Source: §2 · p. 2

3 METHOD: CONVEX-SKM-NET

• For a prediction and prescribed positive-definite mobility, the layer projects onto affine equalities and restricts subsequent corrections to their nullspace before iterating halfspace repairs.Postprocessing supplies mobility directly; joint training learns it with the predictor.Source: §3 · p. 3

3.1 HALFSPACE BLOCK CORRECTIONS

• Each correction linearizes convex inequalities into halfspaces containing the feasible set, then selects a block by normalized violation scores and solves a minimum-displacement problem over their intersection.For a single row, the block problem gives projection onto the selected halfspace.Source: §3.1 · pp. 3–4

3.2 DUAL PGD

• Fixed-depth dual PGD updates selected blocks using GPU-friendly matrix operations without per-block factorizations or linear solves.It reuses the block Gram matrix across inner steps; fixed block shapes support GPU batching.Source: §3.2 · p. 4
• Unlike boundary fitting or squared-positive-residual descent, the block problem minimizes displacement subject to halfspace inequalities.This distinguishes the correction objective from fitting a boundary hyperplane or minimizing violation magnitude.Source: §3.2 · p. 4

3.3 LEARNABLE MOBILITY

• A positive-definite mobility sets directional displacement costs and can be learned jointly with the predictor; identity mobility recovers Euclidean repair.The feasible set is unchanged, and a coordinate transformation reuses the Euclidean solver.Source: §3.3 · p. 4

Assumptions

• The convergence analysis assumes a nonempty feasible set, finite convex inequalities, bounded violated-row subgradients on the iterate region, exact arithmetic, and no outer cap.It uses zero initialization, 1/LB steps, full displacements, and fixed mobility transformed to Euclidean coordinates.Source: §4 · p. 5

Finite-depth progress

• For a block with positive violation, Theorem 4.1 establishes progress toward any feasible point for every dual depth T ≥ 1.If the block has no violation, the update is zero, so fixed T requires no inner convergence test.Source: §4 · p. 5

Convergence and linear rate

• With fixed S and T ≥ 1, outer iterates converge to a feasible point; under Slater’s condition, distance to the feasible set converges linearly.The stated rate also relies on bounded subgradients and the theorem’s setup assumptions.Source: §4 · p. 5

5.1 QUADROTOR SOC-MPC

• All 3,000 SOC-MPC outputs meet the 10^-3 tolerance, with mean gaps below 0.240% and 0.94–3.13 ms inference, yielding 80.7–144.3× speedups over fixed-400-step Πnet.Πnet has a lower gap at H = 20, while Convex-SKM-Net has lower gaps at H = 40 and H = 80; its small mean gaps can precede full feasibility for Πnet.Source: Table 2 · p. 6, §5.1 · p. 6

5.2 POWER-SYSTEM CONVEX RELAXATIONS

• All three variants meet the 10^-3 tolerance on six ACOPF benchmarks; on SDP300, full FT post has a lower mean gap than LMI-Net with a 7.7× inference speedup.Joint training with learned mobility matches or improves on head FT post in mean gap on every case.Source: §5.2 · p. 6, Table 3 · p. 7

5.3 NONCONVEX OPTIMIZATION WITH AFFINE CONSTRAINTS

• Learning mobility reduces the affine-task five-seed mean gap from 0.245% to 0.190%, with improvements in four seeds.Against fully feasible K = 40 Πnet, Convex-SKM-Net has a lower gap at comparable single-instance latency: 0.523 versus 0.533 ms.Source: §5.3 · p. 8, Table 4 · p. 8

5.4 CONVEX QUADRATICALLY CONSTRAINED OPTIMIZATION

• With joint training, Convex-SKM-Net is feasible on all 833 QCQP test inputs and 7.7× faster than SnareNet with a lower mean gap.It also improves on fully feasible K = 100 Πnet in mean gap (0.012% versus 0.040%) and latency (2.469 versus 3.175 ms).Source: §5.4 · p. 8, Table 5 · p. 8

5.5 BLOCK CORRECTION ABLATIONS

• Dual PGD is fastest across all six block-correction groups: 1.06–1.17× over Violation-GD, 4.88–19.56× over M-SSP, and 7.82–27.35× over single-row repair.The comparisons use paired timings; Violation-GD and dual PGD match block sizes and iteration budgets.Source: §5.5 · p. 9, Table 6 · p. 9

6 CONCLUSION

• Convex-SKM-Net unifies convex constraint repair through residual and (sub)gradient oracles, with GPU-friendly fixed-depth dual PGD and jointly learned mobility.The conclusion also reports low inference cost across control, power-system, and public optimization tasks.Source: §6 · p. 9

A FULL THEORETICAL ANALYSIS

• The analysis assumes a nonempty feasible set, finite convex constraints, available subgradients, and bounded subgradients on violated rows in the iterate region.For fixed positive-definite mobility, the analysis first transforms correction coordinates to Euclidean coordinates.Source: A FULL THEORETICAL ANALYSIS · p. 13, §A.1 · p. 13

A.1 HALFSPACE CONTAINMENT AND BLOCK NOTATION

• Each constraint’s residual and subgradient define a halfspace containing the feasible set, so selected halfspace intersections remain feasible-set supersets.A violated row cannot have a zero subgradient when the feasible set is nonempty.Source: §A.1 · p. 13

A.2 DUAL PROGRESS FOR PGD ALONE

• For a stable PGD step size, dual progress is nondecreasing across iterations, and a block with no positive residual returns zero.The block update also gives distance descent relative to any feasible point.Source: §A.2 · p. 14

A.3 STANDARD STEP SIZE AND FINITE DUAL DEPTH

• For a fixed block, the guaranteed progress fraction approaches exact projection as dual depth grows, but the finite-depth error bound is relative to the linearized block.It does not bound error relative to projection onto the original convex feasible set.Source: §A.3 · p. 15

A.4 STEP-SIZE CHOICE AND AMPLIFIED-STEP FINITE-DEPTH BOUNDS

• With ω = 1.8, the base guarantee is cB,T ≥ 0.36EB/ℓB; uniform convergence constants require ω to stay away from zero and two.The step size can be recomputed per block while remaining fixed within each local solve.Source: §A.4 · p. 16

A.5 MULTIPLE BLOCK STEPS UNDER A FROZEN LINEARIZATION

• Under stable positive-depth block updates, multiple corrections under a frozen linearization preserve the original feasible set and accumulate progress.The first inner residuals come from the true nonlinear constraints; later steps use frozen affine residuals.Source: §A.5 · p. 17

A.6 OUTER CONVERGENCE AND BLOCK-DEPENDENT RATES

• With zero-initialized PGD, full steps, fixed top-b selection, and bounded block step sizes, the best-iterate bound is O(K^-1/2) and the full sequence converges to feasibility.A normalized-residual error bound yields a linear rate; Slater’s condition and bounded violated-row subgradients provide such a bound in the stated setting.Source: §A.6 · pp. 17–18

A.7 EXACT BLOCK PROJECTION AND PARALLEL CORRECTION

• Exact block projection decreases distance to every feasible point, and its displacement is bounded above by distance to the feasible set and below by EB/ρB.The appendix also relates averaged violated-row corrections to an adaptive extrapolation method.Source: §A.7 · p. 19

A.8 LOCAL RATE FOR EXACT BLOCK CORRECTIONS

• Near a feasible point, exact full-step projection onto a block containing all active constraints gives a local quadratic decrease in infeasibility under the stated smoothness and local error-bound assumptions.The argument uses bounded constraint Hessians and strict slack for inactive constraints.Source: §A.8 · p. 19

B TRAINING AND IMPLEMENTATION DETAILS

• The benchmarks use shared task-specific splits, with ACOPF cases built from fixed topologies and randomly split load scenarios.The QCQP data use SnareNet’s released 100-variable, 100-inequality, 50-equality dataset; ACOPF scenarios are filtered by original residuals before selection and splitting.Source: B TRAINING AND IMPLEMENTATION DETAILS · pp. 19, 23
• Each block starts dual variables at zero and uses a fixed number of PGD steps; tolerance masking leaves already-satisfied samples unchanged.Gradients follow the finite unroll, holding top-b indices and Boolean branches fixed locally; one-step power estimates at ACOPF and QCQP inference may not meet the theoretical step-size condition.Source: B TRAINING AND IMPLEMENTATION DETAILS · p. 20
• Training protocols span 100 epochs for SOC-MPC, 1,500 for convex QCQP, and 1,900-epoch ACOPF pretraining followed by 100-epoch fine-tuning.SOC-MPC training uses no solution labels; ACOPF pretraining minimizes task MSE, and its fine-tuning variants either update the full predictor or selected heads.Source: B TRAINING AND IMPLEMENTATION DETAILS · pp. 21, 22, 24
• Batch-512 SOC-MPC training at H = 80 encountered out-of-memory errors with the default XLA allocator on a 24 GB RTX 4090.Source: B TRAINING AND IMPLEMENTATION DETAILS · p. 21
• 200/200 SOCP57, 353/400 SOCP118, 175/200 SDP57, and 105/400 SDP118 outputs were feasible; reported gaps and latencies include infeasible outputs.Timing covers compiled predictor-and-repair inference but excludes compilation, input transfer, and independent metric evaluation.Source: B TRAINING AND IMPLEMENTATION DETAILS · p. 25

B.1 MOBILITY PARAMETERIZATION AND IMPLEMENTATION

• Learnable mobility changes directional correction costs while leaving the constraint set unchanged.The positive diagonal mobility is fixed during each repair; zero-output initialization gives identity mobility.Source: §B.1 · p. 25
• Factoring Pz=RR^T turns the minimum-displacement correction into a Euclidean block problem; transformed-row norms drive top-b selection, and updates map back to the original coordinates.At Pz=I, the correction reduces to Euclidean repair.Source: §B.1 · pp. 25–26

B.2 COMPUTATIONAL COMPLEXITY

• Dual PGD needs no matrix factorization or linear-system solves: it uses matrix products, reductions, and elementwise operations, with fixed block shapes enabling batched GPU execution.Each block forms and reuses GB=UU^T for Tdual steps and maintains nonnegative dual variables by elementwise clipping.Source: §B.2 · p. 26
• O(b^2q) forms each block Gram matrix and O(Tdual b^2) runs dual PGD; each outer iteration reuses one linearization for S inner block updates.The outer-iteration cost also includes residual queries, gradient evaluation, coordinate transforms, normalization, top-b selection, and O(mq) affine-residual updates.Source: §B.2 · p. 26
• Block workspace stores O(bq+b^2) values, while linearization storage adds O(mq+m); with learned mobility, block-operation and storage costs replace q by n.Positive-diagonal column scaling implements the transformed gradients without forming Pz^-1 or taking a Cholesky factorization; coordinate transforms belong to Clin.Source: §B.2 · p. 26

C.1 CONVEX RESIDUAL PRIMITIVES

• Constraint families supply residual and subgradient queries, while block selection and dual PGD avoid projectors onto their full feasible sets.The PSD residual still requires a minimum-eigenpair computation; synthetic families use their corresponding primitives.Source: §C.1 · p. 26

C.2 SOC-MPC OBJECTIVE AND CONSTRAINTS

• SOC-MPC uses state and control decisions, affine dynamics equalities, and 2H nonlinear residuals, with instance-specific initial states, targets, and tube centers.The Πnet adapter receives the same affine SOC maps, and the objective evaluator retains target-dependent constants.Source: §C.2 · p. 27

C.3 CONVEX QCQP OBJECTIVE AND CONSTRAINTS

• The convex QCQP has a 50-dimensional equality right-hand side and 100-dimensional decision vector, with fixed objective and constraint coefficients across instances.Both quadratic matrices are positive semidefinite; feasibility is checked in original coordinates at 10^-4.Source: §C.3 · p. 27

C.4 ACOPF OBJECTIVE AND CONSTRAINTS

• The ACOPF relaxations minimize generator cost using lifted voltage and flow variables, affine bounds and balances, and nonlinear SOCP constraints.The SDP version additionally imposes chordal PSD-group residuals and consistency equalities for shared and duplicate lifted variables.Source: §C.4 · pp. 27–28

C.5 SCALING AND NUMERICAL CONVENTIONS

• Normalized residuals guide block selection and correction, but feasibility is audited using native-coordinate residuals.Synthetic selected rows and Torch SOC-MPC use a 10^-12 norm-denominator floor; ACOPF cone helpers use the dtype’s smallest positive normal value.Source: §C.5 · p. 28

D.1 ADDITIONAL SOC-MPC CONTROLS

• 3.7%, 69.6%, and 82.9% lower mean relative Gap with learned versus identity mobility at H20, H40, and H80, respectively, but latency is higher.All configurations are 100% feasible; H20’s common-rate control has 0.33924% Gap, above identity, and the H20 mean raw objective does not improve.Source: §D.1 · p. 28, Table 11 · p. 29
• K = 400 is the first evaluated fixed Πnet depth making all 1,000 test cases feasible at each horizon; validation-selected depths 200/300/400 yield 97.8%/99.9%/100% test feasibility.Figure 2 groups horizons by column and shows objective Gap and maximum constraint violation in its top and bottom rows.Source: §D.1 · p. 28, Table 12 · p. 29
• Calibration makes SnareNet test outputs feasible at both horizons, with more repair iterations and higher latency.The calibrated settings use internal threshold 10^-4 and cap 500; the original settings’ lower mean Gap includes infeasible outputs.Source: §D.1 · pp. 29–30, Table 13 · p. 30

D.2 NONCONVEX OPTIMIZATION WITH AFFINE CONSTRAINTS: DEPTH AND TRAINING CONTROLS

• 0.190 ± 0.027% mean Gap for learned mobility versus 0.245 ± 0.020% for identity mobility across five seeds; fixed-50-step Πnet is 0.331 ± 0.191%.JAXopt is 0.388 ± 0.194%, and SnareNet is 6.169 ± 1.723%.Source: §D.2 · p. 31, Table 15 · p. 31
• At K = 30, four Πnet outputs miss tolerance; at K = 40, all are feasible, and further iterations leave Gap essentially unchanged while reducing maximum violation.The table reports Gap and latency averaged over five seeds and maximum violation across 5,120 outputs; Gap includes infeasible outputs.Source: §D.2 · p. 31, Table 16 · p. 31
• Tighter test tolerance substantially increases latency, while gaps are similar across training tolerances.Mean 50-epoch training times are 14.35 seconds for Πnet, 23.36 seconds for identity mobility, and 37.75 seconds for learned mobility.Source: §D.2 · p. 31, Table 17 · p. 32
• With AdaRel enabled, extending SnareNet training from 50 to 100 epochs lowers mean Gap from 6.169% to 4.723%, using additional training budget.The corrected objective omits a violation penalty, and every configuration in the relaxation and duration controls is 100% feasible at 10^-3.Source: §D.2 · pp. 31–32, Table 18 · p. 32

D.3 ACOPF TRAINING SCOPE AND MOBILITY

• 11.0% and 8.1% relative Gap improvements with learned mobility on SOCP300 and SDP300, respectively, while 57-bus differences are small.Adding learned mobility lowers Gap in all six matched cases, but each ACOPF case uses one training seed.Source: §D.3 · p. 32
• Learned-metric condition numbers rise from 1.31/1.28 on SOCP/SDP57 to 15.77/15.66 on SOCP/SDP300, while 57-bus metrics stay near identity.The 57-bus metrics also vary little across inputs, consistent with nearly tied Gap controls.Source: §D.3 · p. 32
• In seven baseline configurations, every output missing 10^-3 tolerance reached the 5,000-iteration cap rather than stopping at an internal threshold.Reported Gap and latency include these failed outputs.Source: §D.3 · p. 33

D.4 DC3 ACOPF ATTEMPTS

• All four DC3 cases remained infeasible on test after 2,000 epochs at the increased penalty; the original penalty also caused one run to stop on a nonfinite gradient.The attempts do not cover 300 buses.Source: §D.4 · p. 33

D.5 MOSEK REFERENCE AND TIMING AUDIT

• All 3,200 MOSEK reference solves returned OPTIMAL, passed the native 10^-3 check, and reproduced the FP64 labels elementwise.Timing includes backend setup, solve, and extraction, but excludes JuMP model construction, process start, warm-up, file I/O, and external checks.Source: §D.5 · p. 34

• The ablation covered nine deterministic methods on two scales, three convex families, and five paired problem seeds, with timings including oracle, correction, scheduling, and stopping costs.Methods used identity mobility, FP64, block size 128, and native maximum-violation tolerance 10^-3; runs stopped at tolerance, 1,024 outer iterations, or a soft 0.5-second budget.Source: §D.6 · p. 35
• Block variants include nonlinear SKM row refresh, selected-equation least-squares fitting, violated-row pseudoinverse fitting, violation-gradient steps, and principal-pivoting solves of the dual-PGD halfspace QP.BPP-4 and BPP-16 cap each block solve at four and sixteen pivots; M-SSP samples 128 rows with replacement and refreshes true subgradients each step.Source: §D.6 · p. 35
• Both Equality-LS variants failed to reach tolerance in all six groups; BPP-4 timed out on four small squared-affine instances, while single-row SKM hit the iteration cap in eight medium instances.All remaining method–instance combinations reached tolerance; success counts use returned outputs.Source: §D.6 · p. 35
• Reported group summaries use medians across instances, while paired speedups are medians of per-instance ratios only when both methods succeed on all five instances.A dagger denotes stopping time on failures rather than time-to-feasibility.Source: §D.6 · p. 35
• All 180 paired M-SSP/dual-PGD trials reached tolerance, with maximum native inequality violation 9.9125 × 10^-4 and equality residual below 10^-14.M-SSP averages row corrections rather than computing a joint minimum-norm block correction.Source: §D.6 · p. 35

• The seven configurations evaluated all 833 test instances; joint repairs train predictors through their repairs, whereas the three postprocessors share predictor weights and raw outputs.Main-table results use seed 456 and the final epoch-1,500 checkpoint.Source: §D.7 · p. 38
• The main-text Gap averages nonnegative per-instance relative differences, and feasibility is checked using original quadratic-inequality and affine-equality residuals at 10^-4.SnareNet may stop on small updates or its iteration cap, so its internal 10^-6 threshold does not guarantee that original-residual tolerance.Source: §D.7 · p. 38
• Calibration reduced infeasible outputs from 31 to zero for the final DC3 checkpoint, but mean Gap remained about 66.9%.The calibrated extra correction averaged 7.94 steps and took at most 709, in addition to the first ten fixed steps.Source: §D.7 · p. 38
• At joint Πnet K = 5, all outputs were infeasible despite a clipped mean Gap of zero; post K = 5 and 10 were fully feasible but had large Gap.Feasibility was nonmonotone in subsequent iterations, so objective quality must be read together with original residuals.Source: §D.7 · p. 38
• All main-table configurations were 100% feasible at 10^-4, while all reported absolute objective gaps were positive.The table reports absolute-gap geometric mean and maximum over all test instances.Source: Table 26 · p. 39