Dual Conic Proxy for Semidefinite Relaxation of AC Optimal Power Flow

Guancheng Qiu; Mathieu Tanneau; Pascal Van Hentenryck

REVIEW 3 major objections 5 minor 18 references

Dual Conic Proxy for Semidefinite Relaxation of AC Optimal Power Flow

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper claims that a neural network can be trained to output verified lower bounds for AC optimal power flow from the semidefinite relaxation, without solving an SDP at run time.

desk verdict Genuinely new dual-feasible SDP proxy with clean eigenvalue-shift completion, but the main guarantee borrows unproven recovery rules from the SOC paper and the speedup claim lacks timing data. read the letter →

arxiv 2502.06978 v2 pith:CY2HSUTV submitted 2025-02-10 math.OC cs.LG

Guancheng Qiu , Mathieu Tanneau , Pascal Van Hentenryck This is my paper · ORCID

classification math.OCcs.LG MSC 90C2290C2690C90

keywords ACoptimalpowerflowconvexrelaxationsemidefiniteprogrammingdualconicproxyneuralnetworkself-supervisedlearningoptimalitycertificate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to make the semidefinite (SDP) relaxation of AC Optimal Power Flow fast enough to be used inside a learned predictor. It trains a neural network to output part of a solution to the dual of that relaxation, then fills in all remaining variables with fixed algebraic rules, including an eigenvalue-shift step that enforces the positive-semidefinite constraint. Because any feasible dual solution is a valid lower bound on the true optimal cost, every network output is a mathematical certificate, not a heuristic guess. Training is self-supervised: the network maximizes the dual objective directly, so no solver-generated data are needed. In experiments on grids up to 500 buses, the predicted bounds come orders of magnitude faster than an interior-point SDP solver and, on several systems, are tighter than the earlier second-order-cone proxy's bounds.

What carries the argument

The load-bearing object is Algorithm 1, the differentiable dual-completion layer for the dual SDP (called DSDP-OPF in the paper). The layer predicts $\lambda^p,\lambda^q$ and the $\nu$ flow variables, then recovers $\vec\lambda$ from the linear dual equalities, recovers the $\nu^s$ components from the second-order-cone constraints, recovers the generation-bound multipliers from the cost equalities, and finally sets $\hat S = -A_R(\lambda,\mu,\nu) - jA_I(\lambda,\mu,\nu)$, $S = \hat S - \delta I$ with $\delta = \min(0,\lambda_{\min}(\hat S))$, and $\bar\mu^w = 0$, $\underline\mu^w = -\delta$. The shift makes $S$ Hermitian positive semidefinite by construction, and the $\mu^w$ terms exactly compensate the shift in the linear equality, so constraints (6h) and (6k) hold by construction. Differentiability is what matters for training: the objective of the dual is used as the loss, and gradients flow through these recovery steps.

What would settle it

Run Algorithm 1 on a fresh test instance and check the output against every constraint of the dual SDP: the equalities (6b)-(6g), nonnegativity of the $\mu$ variables, membership in the second-order cones (6j), $S$ Hermitian positive semidefinite, and $A_R+jA_I+S=0$; a single violation, or a dual bound that exceeds the best-known AC-OPF objective, would refute the dual-feasibility guarantee.

Watch

Extended reading notes

Core claim

The paper's central claim is that a dual conic proxy can be built for the semidefinite relaxation of AC-OPF, not just for the weaker second-order-cone relaxation used before. The architecture predicts a subset of the dual variables, recovers the rest in closed form, and handles the positive-semidefinite constraint by computing an intermediate Hermitian matrix, shifting it by the negative part of its smallest eigenvalue, and absorbing that shift into the voltage-bound dual variables. The authors assert that this completion always produces dual-feasible solutions, so each forward pass yields a valid lower bound on the AC-OPF optimum. On the reported test systems the SDP proxy improves the dual gap over the SOC proxy on the 30-, 118-, and 500-bus cases, and on the 118-bus case it gives a bound better than the exact SOC relaxation bound.

Load-bearing premise

The load-bearing premise is that the closed-form recovery rules imported from the earlier SOC-relaxation proxy, which this paper does not re-prove, remain valid for the SDP dual's structure, so that the completion always returns a dual-feasible point.

Editorial extensions

If this is right

Every trained proxy output is a verified lower bound on the AC-OPF optimum, with no SDP solver run and no post-solve feasibility check at inference time.
Operators can use the much tighter SDP relaxation at the same latency as a learned heuristic, which matters for repeated real-time dispatches.
The self-supervised training loop removes the need to pre-compute optimal primal or dual solutions, so datasets cost only load perturbations, not solver calls.
On systems where the SDP relaxation is far tighter than SOC, the proxy can reach small dual gaps the SOC proxy cannot approach, and on one benchmark it beats the exact SOC bound.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the dual-feasibility guarantee holds under a rigorous floating-point check, this pattern could make SDP-based certificates practical in market-clearing contexts where a valid bound must accompany every bid or dispatch decision.
The size of $\delta = \min(0,\lambda_{\min}(\hat S))$ is a natural diagnostic for where the proxy is struggling: it measures how much of the semidefinite constraint is being satisfied by the shift rather than by the learned variables, and one could track it during training.
On large grids, the global eigenvalue shift will not scale; a chordal-sparsity-aware version would need to complete each clique's PSD block separately, which would change the shift from a scalar to block-local corrections.
Paired with a learned primal-feasible proxy, this dual proxy would yield an end-to-end learned optimality gap, letting a control system know how far its fast solution is from the true optimum without calling a solver.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the eigenvalue-shift trick is a legitimate new mechanism, and this is the first dual-feasible learning-based certificate for the SDP relaxation of AC-OPF. That is enough to take seriously. But the main guarantee leans on recovery rules imported from your SOC paper without stating them, and the headline speedup is not measured. Both are fixable before publication.

What’s new: Algorithm 1’s step 5, shifting the raw Hermitian completion by δI to enforce PSD, is a clean and correct construction. The proof that this yields S ⪰ 0 and satisfies (6h) is airtight. Training on the dual objective directly, with solver outputs used only for evaluation, is a clean setup with no circularity. On three of five benchmarks the SDP proxy beats the SOC proxy, and on ieee118 it beats the SOC relaxation itself, which is the point of switching to SDP.

The soft spots are real but not fatal. Steps 3 and 4 are the load-bearing unproven part: the paper says “the same argument as [6]” but does not give the formulas or a proof that the recovered νˢ satisfy the second-order cone constraints and that the μ multipliers are nonnegative and satisfy (6b)–(6c). Because [6] is the SOC dual, the transfer is plausible — the constraints have identical structure — but for a paper whose central claim is “always dual feasible,” the authors should spell out those closed forms or prove them in an appendix. The stress-test concern lands, but it is an omitted derivation, not an identified error.

Timing: the abstract and conclusion claim orders-of-magnitude speedups, but Section 4.3 only says “not reported for lack of space.” That is not acceptable for a practical-speed claim. A short table of wall-clock numbers would substantially raise the value of the paper.

The experiments are otherwise honest: two of five systems show the SDP proxy underperforming, and the paper explicitly says so. No code or hyperparameters are shipped, but that is common for this literature.

Who this is for: people working on learning-based optimization certificates for power systems, and anyone interested in dual Lagrangian learning for SDP. It deserves a serious referee, not a desk rejection. I would recommend minor-to-major revisions: make steps 3–4 explicit, add the timing table, and ideally release code. If those are supplied, this is a solid contribution.

Referee Report

3 major / 5 minor

Summary. The paper proposes a dual conic proxy (DCP) architecture for the semidefinite programming relaxation of AC optimal power flow (SDP-OPF). A neural network predicts a subset of dual variables, and a differentiable completion layer enforces all dual feasibility constraints of the conic dual DSDP-OPF, so that the network output is a certified lower bound on the optimal AC-OPF cost. The model is trained self-supervised by maximizing the dual objective, avoiding generation of ground-truth solutions. Numerical experiments on PGLib systems up to 500 buses compare the SDP proxy with a prior SOC-based proxy and with Mosek, reporting dual optimality gaps and gap-closed metrics. The paper claims several orders of magnitude speedups over interior-point SDP solvers, although timing results are not reported.

Significance. If the dual-feasibility guarantee is established, the paper makes a valuable contribution: it is the first DCP architecture for an SDP relaxation of AC-OPF, and the self-supervised training scheme avoids expensive data generation. The central mechanism — the eigenvalue-shift completion in step 5 of Algorithm 1 — is simple, elegant, and correctly enforces the PSD constraint (6h)-(6k) given a Hermitian completion. The experimental comparison against the SOC proxy is informative and shows that the stronger SDP relaxation can yield better bounds when the proxy is trained successfully. However, the main guarantee rests on recovery rules that are not specified or proved in this manuscript, and the headline speedup claim is not backed by any reported timing data. The paper is therefore promising but needs substantial revision before the central claims can be accepted.

major comments (3)

[Algorithm 1, steps 3–4 and Section 3] The paper's central claim — that the DCP architecture 'always outputs dual feasible solutions' — depends on the closed-form recovery rules in steps 3 and 4 of Algorithm 1. These steps are not specified in the manuscript; the text only says they 'use the same argument as [6]', where [6] addresses a second-order cone relaxation, not the SDP dual. The manuscript does not provide the formulas for recovering ⃗ν^s and ⃗ν^s from (6j), nor the formulas for recovering ¯µ^pg, ¯µ^pg, ¯µ^qg, ¯µ^qg from (6b)-(6c) together with the nonnegativity constraints (6i). Because (6b)-(6c) are equalities with nonnegative multipliers, there is a family of feasible choices, and the paper does not say which closed-form selection is used or why it always satisfies the sign constraints and the SOC cone constraints. The reader cannot verify the dual-feasibility guarantee from this manuscript alone. The authors should include the explicit recovery formulas and a proof (or a precise reduction to a lemma in [6] that maps every constraint to the corresponding SDP dual constraint) for both forward and reverse cones and for the generation-bound multipliers.
[Abstract, Section 4.3, and Conclusion] The abstract and conclusion state that the proposed SDP proxy achieves 'orders of magnitude of speedups' compared to interior-point solvers, but Section 4.3 says only that 'timing results, not reported for lack of space' support this claim. No timing measurements, hardware-specific runtime comparisons, or methodology for measuring inference time appear anywhere in the paper. Since the computational speedup is a primary motivation for the DCP approach and for the paper's title and abstract, this claim is unsupported as written. The authors should either include the timing results (e.g., median inference time per instance, including the eigenvalue decomposition and completion layer, versus Mosek solve time) or remove the speedup claim from the abstract and conclusion.
[Section 2.2, Model 3, constraint (6h)] The completion step 5 uses the matrix ˆS = -A_R(λ,μ,ν) - j A_I(λ,μ,ν) and sets S = ˆS - δ I, with δ = min(0, λ_min(ˆS)) ≤ 0, and then sets ¯µ^w = 0, ¯µ^w = -δ. This correctly enforces S ≽ 0 and constraint (6h) only if the definitions of A_R and A_I are consistent with the signs in (6h). However, Equation (4) contains a notational ambiguity: the term for the voltage-bound multipliers is printed as '¯µ^w_i − ¯µ^w_i', which appears to use the same symbol for the upper and lower bounds; the intended expression is presumably (¯µ^w_i − µ̲^w_i). This ambiguity should be fixed, because the sign of δ and the assignment of ¯µ^w and ¯µ^w depend on which multiplier is which. If the symbols are indeed meant to be identical, the expression is wrong and the completion step would not satisfy (6h).

minor comments (5)

[Section 3, after Algorithm 1] The sentence 'For a more detailed description of the DCP architecture, the reader may refer to [6, 8]' delegates a substantial part of the method description to previous work; the core novelty of this paper is precisely the SDP completion layer, so the main text should be self-contained for Algorithm 1.
[Section 4.2, Eq. (8)] The 'dual optimality gap' is defined relative to z*_AC, described as the best-known AC-OPF solution obtained by Ipopt. Since Ipopt may return a local optimum, the terminology 'gap' should be clarified to indicate that it is computed against a reference solution, not a proven global optimum; this does not affect the validity of the dual bound itself.
[Table 1] The table reports both mean and standard deviation, but for the 'gap closed' metric the arithmetic mean is used while the dual gaps use the geometric mean. This is stated, but the reader should be reminded that negative values of gap closed make geometric means undefined; the reporting choices are reasonable but worth a brief justification.
[Section 4.3] The discussion of the ieee30 system attributes the weaker performance to 'the higher output dimension of the SDP proxy' and 'the additional difficulty of the PSD constraint (6k)', but no diagnostic experiments (e.g., training curves, loss values, or ablations) are provided to support this explanation. This is a minor point, but a short analysis would strengthen the interpretation.
[Notation] The symbols ⃗ν and ⃗ν are used for forward and reverse dual SOC variables, and the same arrow convention is used for other flow variables; the paper should explicitly state the convention once in the notation section, since the distinction is critical for the cone constraints (6j).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the dual completion is an algebraic projection onto the dual feasible set and training maximizes the dual objective without fitting solver outputs.

full rationale

The derivation chain is not circular. Algorithm 1's completion layer is a sequence of algebraic projections onto the DSDP-OPF dual constraints: Step 2 substitutes predicted values into equalities (6d)-(6g); Step 3 chooses the minimum nu^s satisfying the second-order cone (6j), i.e. nu^s = ||(nu^p,nu^q)||_2; Step 4 chooses nonnegative generation multipliers as the positive and negative parts satisfying (6b)-(6c); Step 5 shifts hat S by -delta I and sets underline_mu^w = -delta to enforce (6h) and (6k). No step fits a parameter to the quantity it later predicts. Training is self-supervised with the dual objective (6a) as the loss, so maximizing it over the feasible dual set is legitimate and does not require solver targets; Mosek and Ipopt values are used only in test-time evaluation. The reference to the authors' prior work [6] for Steps 3-4 is a dependency on an earlier derivation, not a self-citation that carries the conclusion: the closed-form rules for cone recovery and multiplier splitting are elementary and are expressed in the present paper's own constraint set. Even if the correctness of importing SOC recovery rules to the SDP dual were questioned, that would be a proof-completeness or correctness risk, not a circularity in the derivation.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central theoretical result rests on standard duality, the known SDP relaxation, and an unproven transfer of recovery rules from the SOC setting. No new physical or mathematical entities are introduced; the delta shift is a construction, not an entity.

free parameters (1)

Neural network parameters (weights and biases)
Trained end-to-end via self-supervised objective (maximizing dual objective) on 90% of 20,000 perturbed-load instances per system; exact architecture and parameter count not reported.

assumptions (3)

standard math Weak duality between SDP-OPF (Model 2) and its conic dual DSDP-OPF (Model 3)
Used to claim any feasible dual solution yields a valid lower bound on AC-OPF; standard convex optimization result, not proven in the paper.
domain assumption SDP-OPF is a valid relaxation of AC-OPF via W = V V* and rank relaxation
Adopted from Bai et al. [11]; the paper does not re-derive the relaxation.
ad hoc to paper The dual recovery rules of steps 3-4 from [6] produce feasible dual variables for the SDP dual
Algorithm 1 steps 3-4 are stated by reference to [6] without proof; the paper's dual-feasibility guarantee depends on these rules holding.

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Cite this review

Pith. "Pith review of Dual Conic Proxy for Semidefinite Relaxation of AC Optimal Power Flow." pith.science (2026). https://pith.science/paper/CY2HSUTV

@misc{pith2026250206978,
  author       = {Pith},
  title        = {Pith review of: Dual Conic Proxy for Semidefinite Relaxation of AC Optimal Power Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CY2HSUTV}},
  note         = {Machine review of arXiv:2502.06978}
}

read the original abstract

The nonlinear, non-convex AC Optimal Power Flow (AC-OPF) problem is fundamental for power systems operations. The intrinsic complexity of AC-OPF has fueled a growing interest in the development of optimization proxies for the problem, i.e., machine learning models that predict high-quality, close-to-optimal solutions. More recently, dual conic proxy architectures have been proposed, which combine machine learning and convex relaxations of AC-OPF, to provide valid certificates of optimality using learning-based methods. Building on this methodology, this paper proposes, for the first time, a dual conic proxy architecture for the semidefinite (SDP) relaxation of AC-OPF problems. Although the SDP relaxation is stronger than the second-order cone relaxation considered in previous work, its practical use has been hindered by its computational cost. The proposed method combines a neural network with a differentiable dual completion strategy that leverages the structure of the dual SDP problem. This approach guarantees dual feasibility, and therefore valid dual bounds, while providing orders of magnitude of speedups compared to interior-point algorithms. The paper also leverages self-supervised learning, which alleviates the need for time-consuming data generation and allows to train the proposed models efficiently. Numerical experiments are presented on several power grid benchmarks with up to 500 buses. The results demonstrate that the proposed SDP-based proxies can outperform weaker conic relaxations, while providing several orders of magnitude speedups compared to a state-of-the-art interior-point SDP solver.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [6]

Dual conic proxies for ac optimal power ﬂow

Guancheng Qiu, Mathieu Tanneau, and Pascal V an Hentenry ck. Dual conic proxies for ac optimal power ﬂow. Electric Power Systems Research, 236:110661, 2024

work page 2024

[2] [1]

Mak, and Pascal V an H entenryck

Ferdinando Fioretto, Terrence W .K. Mak, and Pascal V an H entenryck. Predicting AC Optimal Power Flows: Combining Deep Learning and Lagrangian Dual Methods. Proceedings of the AAAI Conference on Artiﬁcial Intelligence, 34(01):630–637, Apr. 2020. doi: 10.1609/aaai.v34i01.54 03. REFERENCES 6

work page doi:10.1609/aaai.v34i01.54 2020

[3] [2]

DC3: A lea rning method for optimization with hard constraints

Priya L Donti, David Rolnick, and J Zico Kolter. DC3: A lea rning method for optimization with hard constraints. arXiv preprint arXiv:2104.12225 , 2021

arXiv 2021

[4] [3]

D eepOPF-V: Solving AC-OPF problems efﬁ- ciently

Wanjun Huang, Xiang Pan, Minghua Chen, and Steven H Low. D eepOPF-V: Solving AC-OPF problems efﬁ- ciently. IEEE Transactions on Power Systems, 37(1):800–803, 2021

work page 2021

[5] [4]

End-to-End Feasible Optimization Proxies for Large-Scale Economic Dispatch

Wenbo Chen, Mathieu Tanneau, and Pascal V an Hentenryck. End-to-End Feasible Optimization Proxies for Large-Scale Economic Dispatch. IEEE Transactions on Power Systems, 39(2):4723–4734, 2024

work page 2024

[6] [5]

Energy and operating reserve markets, 2022

MISO. Energy and operating reserve markets, 2022. Busin ess Practices Manual

work page 2022

[7] [7]

R. A. Jabr. Radial distribution load ﬂow using conic prog ramming. IEEE Transactions on Power Systems, 21(3): 1458–1459, Aug 2006. ISSN 0885-8950. doi: 10.1109/TPWRS.2 006.879234

doi:10.1109/tpwrs.2 2006

[8] [8]

Dual Lagrang ian Learning for Conic Optimization

Mathieu Tanneau and Pascal V an Hentenryck. Dual Lagrang ian Learning for Conic Optimization. In Advances in Neural Information Processing Systems , volume 37, 2024

work page 2024