Tight-and-cheap conic relaxation for the AC optimal power flow problem

Christian Bingane; Miguel F. Anjos; S\'ebastien Le Digabel

REVIEW 2 major objections 3 minor 38 references

Tight-and-cheap conic relaxation for the AC optimal power flow problem

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper introduces a conic relaxation for AC optimal power flow that is stronger than the second-order cone relaxation, nearly as tight as the semidefinite relaxation, and much faster to solve than the chordal relaxation on large…

desk verdict Solid new conic relaxation for ACOPF with correct proofs, but the abstract overstates tightness and the speed claims rest on a generic CVX baseline for chordal SDPs. read the letter →

arxiv 1908.02319 v1 pith:U544HM32 submitted 2019-08-06 math.OC

Christian Bingane , Miguel F. Anjos , Sébastien Le Digabel This is my paper · ORCID

classification math.OC MSC 90C2290C26

keywords ACoptimalpowerflowconicrelaxationsemidefiniteprogrammingsecond-orderconereformulation-linearizationtechniquechordalconvexsystems

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 proposes a new convex relaxation, the tight-and-cheap relaxation (TCR), for the alternating-current optimal power flow problem, a nonconvex and NP-hard problem central to power-system operations. TCR is built by combining semidefinite constraints with reformulation-linearization (RLT) inequalities on voltage magnitudes, and it is shown to be stronger than the standard second-order cone relaxation and nearly as tight as the full semidefinite relaxation. The authors prove that a closely related formulation is exactly equivalent to the standard SDP relaxation, so TCR inherits a rigorous lower-bounding property. On standard test networks with up to 6515 buses, TCR's optimality gap is close to SDR's, and its solve time is about 30 times lower than the chordal relaxation on large instances and 55 times lower on the largest ones. A strengthened variant, STCR, is tighter still and coincides with SDR when the network minus the reference bus is acyclic.

What carries the argument

The load-bearing object is the per-branch $3\times3$ semidefinite block $$ \begin{bmatrix} 1 & v_k^* & v_m^* \\ v_k & V_{kk} & V_{km} \\ v_m & V_{km}^* & V_{mm} \end{bmatrix} \succeq 0, $$ imposed for every branch $(k,m)$. This block simultaneously encodes the second-order cone relation $|V_{km}|^2 \le V_{kk}V_{mm}$ and couples the complex voltage variables $v$ to $V$. The RLT ingredient is a secant lower bound on the concave square root: Lemma 1 gives $\sqrt{y} \ge (y + \sqrt{\ell u})/(\sqrt{\ell}+\sqrt{u})$ on $\ell \le y \le u$, applied to $V_{11}=|v_1|^2$ to yield the reference-bus inequality $\mathrm{Re}(v_1) \ge (V_{11}+\underline{v}_1\overline{v}_1)/(\underline{v}_1+\overline{v}_1)$. That inequality is what lets the proof recover $v$ from $V$, making the equivalence between nSDR and SDR go through.

What would settle it

Run a timing comparison on the 6515-bus test network using a semidefinite solver that exploits sparsity, including the time to construct the chordal extension; if the sparsity-exploiting relaxation then solves in time comparable to or less than TCR while giving the same bound, the paper's central speed claim is refuted.

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Extended reading notes

Core claim

The central discovery is that the full positive semidefinite constraint $V \succeq 0$ in the standard SDP relaxation can be replaced, without losing tightness, by the combination of $V \succeq vv^H$ (where $v$ is the complex voltage vector) and a single RLT inequality at the reference bus. Proposition 2 constructs, for any feasible $V$, the vector $v = Ve_1/\sqrt{V_{11}}$ and uses Schur-complement reasoning to show this pair satisfies the new constraints, so the new semidefinite relaxation nSDR is exactly equivalent to SDR. Replacing the global constraint $V \succeq vv^H$ by per-branch $3\times3$ semidefinite blocks on $(1, v_k, v_m)$ yields TCR, which dominates the second-order cone relaxation and is dominated by SDR. The strengthened STCR, which uses the $3\times3$ principal submatrices on the reference bus together with each branch, is strictly stronger than TCR and is proven equivalent to SDR when the network graph with the reference bus removed has no cycles. On unmodified test cases up to 6515 buses, TCR's optimality gaps are close to SDR's while its solve times are about 30 times lower than the chordal relaxation on large instances.

Load-bearing premise

The speed advantage rests on comparing TCR against a sparsity-exploiting semidefinite relaxation that is solved with a generic solver and timed without counting the cost of building the sparsity structure; a purpose-built solver could shrink that advantage.

Editorial extensions

If this is right

On large networks, TCR gives a lower bound within a few hundredths of a percent of the SDP bound while solving roughly 30 times faster than the chordal relaxation, making SDP-quality bounds practical at scales where the full SDP was too slow.
Because TCR uses only per-branch $3\times3$ semidefinite blocks, it avoids storing or factoring an $n\times n$ positive semidefinite matrix, which is the main source of SDP's computational cost.
STCR is strictly tighter than TCR and is equivalent to SDR for networks whose graph minus the reference bus has no cycles, so SDP-quality bounds can be obtained cheaply on such topologies.
As a valid relaxation, TCR provides a certificate (lower bound) on the global optimum of ACOPF, so it can be used to bound the suboptimality of any local solution found by nonlinear solvers.

Reading between the lines

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

The same construction could be applied at other buses: adding RLT cuts on more voltage magnitudes, or on branch current magnitudes, would likely tighten TCR further while preserving its local structure; the paper only uses the reference bus.
The equivalence strategy of Proposition 2 — global PSD replaced by $V \succeq vv^H$ plus one scalar RLT cut at an anchor variable — may transfer to other complex quadratically constrained programs with bounded magnitudes, not just power flow.
The speed comparison uses a generic solver for the sparsity-exploiting relaxation and does not count the time to build its chordal structure; a purpose-built solver could narrow TCR's advantage, so the 'cheap' part should be revalidated in production-grade software.

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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

Here's my take on 1908.02319. The useful new thing is the nSDR/TCR construction: adding an explicit voltage vector v with the 3x3 PSD constraints (8) gives a relaxation that provably sits between SOCR and SDR. Proposition 2 (nSDR equivalent to SDR) is neat and correct, and Proposition 4 gives a clean exactness condition when the network without the reference bus is a forest. The proofs are self-contained and I found no circularity.

The computational study is honest in several ways: unmodified MATPOWER data, both cost and loss objectives, and they visibly report STCR's stronger tightness. But the abstract's 'nearly as tight as SDR' is too strong. Their own Table I shows case5 with TCR gap 12.75% vs SDR 5.22%, and ACTIV_SG_500 with 4.39% vs 2.11%. Those are not 'nearly as tight' in any meaningful sense; the right statement is 'sometimes close, sometimes meaningfully looser.'

The bigger soft spot is the timing comparison. TCR is compared against CHR using CVX with MOSEK, which treats the chordal PSD blocks as generic constraints. The specialized chordal implementations cited in the paper (Molzahn et al., Andersen et al.) exploit the clique tree and are likely much faster than the CVX baseline. The claimed order-of-magnitude speedup may shrink substantially against a proper chordal solver. The exclusion of chordal extension construction time is actually a small concession to CHR, so that's not the problem. The problem is the generic baseline. I also would have liked a direct comparison with Kocuk et al.'s strong SOCP relaxations, which is the natural baseline for the tight-and-cheap trade-off.

None of this undermines the theoretical result. The relaxation is new, the proofs check out, and for practitioners using CVX/MOSEK it is indeed faster and tighter than SOCR on these instances. But the paper should be revised to soften the abstract, add the missing baseline, and ideally benchmark with a chordal solver.

I'd send it to peer review. The math deserves referee time and the computational claims can be tested. I'd bring it to reading group and likely cite it.

Referee Report

2 major / 3 minor

Summary. The paper proposes a new convex relaxation for the AC optimal power flow problem, obtained by combining semidefinite constraints with reformulation-linearization inequalities on voltage magnitudes. The authors first introduce nSDR, a reformulation of the standard SDP relaxation with additional voltage variables and an RLT constraint at the reference bus, and prove in Proposition 2 that nSDR is equivalent to SDR. They then drop the global positive-semidefinite constraint to local 3-by-3 blocks, obtaining the tight-and-cheap relaxation (TCR), and a further strengthened variant (STCR) using 3-by-3 principal submatrices anchored at the reference bus. Propositions 3 and 4 establish that STCR is stronger than TCR and, for networks whose graph becomes acyclic after removing the reference bus, equivalent to SDR. Computational experiments on MATPOWER instances up to 6515 buses compare TCR and STCR with SOCR, CHR, and SDR in terms of optimality gap and solver time. The paper claims that TCR is stronger than SOCR, nearly as tight as SDR, and up to an order of magnitude faster than the chordal relaxation.

Significance. The theoretical hierarchy in the paper is clean and the proofs of Propositions 2-4 are self-contained and correct; the nSDR/TCR construction is a legitimate and potentially useful way to localize the semidefinite constraint in OPF relaxations. The empirical study is also carefully done in one respect: the authors use unmodified MATPOWER cases and report both cost and loss objectives, with no fitted parameters. However, two load-bearing applied claims are not fully supported by the evidence as presented. First, the abstract's 'nearly as tight as SDR' is contradicted by the paper's own tables on instances where the SDR bound is not already near-exact. Second, the speed comparison against the chordal relaxation is made with a generic CVX/MOSEK interface that does not exploit chordal structure, so the reported speedups may not represent a state-of-the-art chordal SDP solver. If these computational claims are hardened or appropriately qualified, the paper would be a solid contribution to the OPF relaxation literature.

major comments (2)

[Abstract and Section V, Tables I and II] The claim that TCR is 'nearly as tight as the standard semidefinite relaxation' is not supported on instances where SDR itself has a non-negligible gap. In Table I, case5 has a TCR optimality gap of 12.75% versus 5.22% for both SDR and CHR, and case_ACTIV_SG_500 has a TCR gap of 4.39% versus 2.11% for SDR/CHR. The paper acknowledges these discrepancies in Section V, bullet 3, but the abstract and conclusion state the tightness claim without qualification. The authors should either restrict the closeness claim to instances where the SDR bound is already tight, or add a quantitative statement about the worst-case or instance-dependent gap between TCR and SDR.
[Section V, first paragraphs and Tables I and II] The central speed claim, that TCR is up to an order of magnitude faster than the chordal relaxation, is measured by solving every relaxation with CVX 2.1 and MOSEK 8.0.0.60 under default settings. CVX passes each positive-semidefinite block to the solver as a generic constraint and does not exploit the clique tree that makes chordal SDP relaxations scalable, while the paper cites specialized chordal implementations in [25]-[27]. The CHR times in Tables I and II are therefore likely not representative of a properly implemented chordal relaxation, and the abstract's speedup claim needs to be either benchmarked against a chordal-exploiting solver or explicitly limited to generic SDP solvers.

minor comments (3)

[Model 2] The initialization line contains a doubled word: 'LG + I|N| ≻ 0, where LG is the Laplacian matrix of G and and I|N| is the identity matrix of size |N|.' Please fix the typo.
[Section V, paragraph before Tables I and II] The sentence 'Both objective functions of test cases from [36] are the same' is ambiguous. Please clarify that for the RTE/PEGASE cases the generation-cost coefficients already encode active loss, which explains why the large-scale rows of Tables I and II are identical.
[Section IV, Model 5] It would help the reader to state explicitly that constraint (8) implies the usual 2-by-2 SOCR constraints by taking the leading 2-by-2 principal submatrix, rather than leaving the domination argument entirely to the phrase 'Clearly TCR dominates SOCR.'

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the relaxations are derived and proved from stated assumptions, and the computational claims are benchmarked on external unmodified MATPOWER cases.

full rationale

The paper's derivation chain is self-contained. The new relaxations are built from the SDP relaxation by explicit RLT inequalities and a positive-semidefinite block constraint; Proposition 2 proves equivalence of nSDR and SDR using a constructive argument (Lemma 1 and Lemma 2), and Propositions 3 and 4 prove the dominance and exactness results from the given constraints and chordal graph theory. No parameter is fitted to the test data and no 'prediction' is defined in terms of the quantity it is said to predict. The empirical speed comparisons use unmodified MATPOWER test cases and report solver times from MOSEK, which is an external benchmark. The caveat that CHR is solved via CVX's generic interface rather than a specialized chordal-exploiting implementation affects the fairness of the speed comparison but is not a circularity in the derivation. There are no load-bearing self-citations: the cited completion theorem of Grone et al. is an external mathematical result. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters; all constraints are derived from first principles. Standard convex analysis and chordal graph theory are used as axioms. The only domain assumptions are the standard ACOPF model assumptions: positive voltage bounds, convex generation costs, and the reference bus angle fixed.

assumptions (4)

standard math Positive semidefinite matrices and Schur complement properties
Used throughout the proofs of Propositions 2 and 3.
standard math Chordal graph theory: PSD of partial matrices equivalent to clique PSD on chordal extension
Invoked in Section III-B and in the proof of Proposition 4.
domain assumption Voltage magnitude bounds are strictly positive (vk > 0)
Stated in Section II after constraint (1h); needed for the construction of v in Proposition 2.
domain assumption Generation cost functions are convex
Assumed in Section II for the ACOPF model.

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Cite this review

Pith. "Pith review of Tight-and-cheap conic relaxation for the AC optimal power flow problem." pith.science (2026). https://pith.science/paper/U544HM32

@misc{pith2026190802319,
  author       = {Pith},
  title        = {Pith review of: Tight-and-cheap conic relaxation for the AC optimal power flow problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U544HM32}},
  note         = {Machine review of arXiv:1908.02319}
}

read the original abstract

The classical alternating current optimal power flow problem is highly nonconvex and generally hard to solve. Convex relaxations, in particular semidefinite, second-order cone, convex quadratic, and linear relaxations, have recently attracted significant interest. The semidefinite relaxation is the strongest among them and is exact for many cases. However, the computational efficiency for solving large-scale semidefinite optimization is lower than for second-order cone optimization. We propose a conic relaxation obtained by combining semidefinite optimization with the reformulation-linearization technique, commonly known as RLT. The proposed relaxation is stronger than the second-order cone relaxation and nearly as tight as the standard semidefinite relaxation. Computational experiments using standard test cases with up to 6515 buses show that the time to solve the new conic relaxation is up to one order of magnitude lower than for the chordal relaxation, a semidefinite relaxation technique that exploits the sparsity of power networks.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [25]

Molzahn, Jesse T

Daniel K. Molzahn, Jesse T. Holzer, Bernard C. Lesieutr e, and Christo- pher L. DeMarco. Implementation of a large-scale optimal po wer ﬂow solver based on semideﬁnite programming. IEEE Transactions on Power Systems, 28(4):3987–3998, 2013

work page 2013

[2] [27]

Madani, M

R. Madani, M. Ashraphijuo, and J. Lavaei. OPF solver. https://lavaei.ieor.berkeley.edu/Software.html, 2014

work page 2014

[3] [1]

Carpentier

J. Carpentier. Contribution to the economic dispatch pr oblem. Bulletin de la Soci´ et´ e franc ¸aise des ´ electriciens, 3(8):431–447, 1962

work page 1962

[4] [2]

Cain, Richard P

Mary B. Cain, Richard P . ONeill, and Anya Castillo. Histo ry of optimal power ﬂow and formulations. Federal Energy Regulatory Commission , pages 1–36, 2012

work page 2012

[5] [3]

Optimal power ﬂow: A bibliographic survey I

Stephen Frank, Ingrida Steponavice, and Steffen Rebenn ack. Optimal power ﬂow: A bibliographic survey I. Energy Systems , 3(3):221–258, 2012

work page 2012

[6] [4]

Optimal power ﬂow: A bibliographic survey II

Stephen Frank, Ingrida Steponavice, and Steffen Rebenn ack. Optimal power ﬂow: A bibliographic survey II. Energy Systems , 3(3):259–289, 2012

work page 2012

[7] [5]

Power Grid Security Analysis: An Optimization Approach

Abhinav V erma. Power Grid Security Analysis: An Optimization Approach. Columbia University, 2010

work page 2010

[8] [6]

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Karsten Lehmann, Alban Grastien, and Pascal V an Hentenr yck. AC- feasibility on tree networks is NP-hard. IEEE Transactions on Power Systems, 31(1):798–801, 2016

work page 2016