Solving Quadratic Programs via Deep Unrolled Douglas-Rachford Splitting

Akang Wang; Jiang Xue; Jinxin Xiong; Linxin Yang; Xiaodong Luo; Xi Gao

REVIEW 3 major objections 4 minor 18 references

Solving Quadratic Programs via Deep Unrolled Douglas-Rachford Splitting

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A four-layer unrolled Douglas-Rachford splitting network, trained to predict primal-dual QP solutions and used as a warm start for SCS, cuts solver iterations by up to 50% and solve time by up to 40% while the underlying modified algorithm

desk verdict Useful empirical unrolling-DR paper with a real but fixable gap in its convergence proof; the warm-start results are believable. read the letter →

arxiv 2508.11869 v1 pith:OIN5OF2K submitted 2025-08-16 math.OC

Jinxin Xiong , Xi Gao , Linxin Yang , Jiang Xue , Xiaodong Luo , Akang Wang This is my paper · ORCID

classification math.OC MSC 90C2090C2568T07

keywords convexquadraticprogrammingDouglas-Rachfordsplittingalgorithmunrollinglearningtooptimizewarm-startingoperatorfirst-ordermethodsneuralnetworks

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

This paper tries to make Douglas-Rachford splitting unrollable for convex quadratic programs. It replaces the expensive linear-system solve at each DR iteration with a single gradient descent step on the equivalent least-squares problem, proving that the resulting DR-GD algorithm still converges to the QP solution. Unrolling four iterations gives a small neural network trained with supervised regression to output warm-start primal-dual points. On synthetic and real QP benchmarks, these warm starts reduce SCS iteration counts by up to 50% and total solve time by up to 40%, with the largest gains on the largest instances. If correct, this offers a path from a classical splitting method to an interpretable learned solver that still carries a convergence guarantee.

What carries the argument

The load-bearing mechanism is the inexact-resolvent replacement: instead of computing $(I+M)^{-1}(w^k-q)$ exactly, Algorithm 2 takes one gradient step on the equivalent least-squares problem, producing the map $\Phi_k(w) = (I-\eta_k(I+M)^\top(I+M))\tilde{u}^k + \eta_k(I+M)^\top(w-q)$. The iteration is then an averaged operator $T_k = \tfrac12(\mathrm{Id} + C_{N_C}(2\Phi_k-\mathrm{Id}))$, so inexact DR convergence theory applies if the accumulated errors are summable. The same operator, with learnable matrices substituted for the fixed updates, is the neural network layer.

What would settle it

Run DR-GD on a family of convex QPs with increasing condition number of $I+M$ and record $\|w^{k+1}-w^k\|$. If for some instance the residual does not decay geometrically after an initial phase but instead oscillates or stalls, then the asserted geometric contraction step fails and Proposition 1 is not established for that instance. Equivalently, check directly whether the line search can satisfy both $\eta_k \ge (1-c_2)/\sigma_{\max}(I+M)$ and $\eta_k < \lambda_{\min}(I+M)/\lambda_{\max}^2(I+M)$; an empty admissible interval is a concrete witness.

Watch

Extended reading notes

Core claim

The paper's central claim is that a DR-GD algorithm—Douglas-Rachford splitting with the linear-system solve $(I+M)^{-1}(w^k-q)$ replaced by one gradient step on the least-squares objective $f(\tilde{u}) = \tfrac12\|(I+M)\tilde{u} - (w^k-q)\|^2$, with the step size chosen by a Wolfe line search—still converges to the solution of the convex QP (Proposition 1). The proof rewrites the iteration as an averaged monotone-operator update $w^{k+1}=T_k(w^k)$ with $T_k = \tfrac12(\mathrm{Id} + C_{N_C}(2\Phi_k-\mathrm{Id}))$ and applies existing inexact Douglas-Rachford convergence theory, provided the per-step errors are finitely summable. The same update, unrolled for four layers with learnable channe

Load-bearing premise

The proof needs the sequence of successive iterates to shrink geometrically once the algorithm is close to a fixed point, so the errors from the one-step gradient solves add up to a finite total; the proof asserts this shrinkage rather than deriving it from the stated lemmas, and the line search must land in an interval that can be empty for badly conditioned problems.

Editorial extensions

If this is right

A DR-GD Net with only four layers can produce, in milliseconds, warm-start points that save SCS a substantial fraction of its iterations, so the wall-clock benefit grows with problem size.
Because the unrolled network contains the DR-GD algorithm as a special parameter instantiation, with enough layers it can in principle recover optimal QP solutions, not just approximate ones.
Warm starts from DR-GD-NN cut SCS iterations by roughly 34% to over 50% on QPLIB and Portfolio instances and total solve time by up to about 56%, with inference overhead staying small relative to solver time.
Unrolling more gradient steps per layer does not improve results; solution quality and warm-start gains degrade beyond one or two inner steps, so the single-step design is both simpler and empirically better.
The method retains most of its speedup when test perturbations are 50% larger than training perturbations, suggesting the learned warm start is not tightly overfit to the training distribution.

Reading between the lines

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

The proof's geometric-contraction step is asserted rather than derived, so the most informative stress test is a condition-number sweep: for each QP, check whether the line search can satisfy both $\eta_k \ge (1-c_2)/\sigma_{\max}(I+M)$ and $\eta_k < \lambda_{\min}(I+M)/\lambda_{\max}^2(I+M)$; if the interval is empty for some ill-conditioned instance, the stated convergence guarantee stops applyi
Because the supervised training labels come from SCS itself, the method is not fully label-free; a hybrid loss that adds the Douglas-Rachford fixed-point residual to the regression loss would test whether the label cost can be reduced without the failure modes the paper reports for purely residual-based training.
The same gradient-replacement recipe should transfer to ADMM/OSQP, which is Douglas-Rachford splitting under a variable transformation; an unrolled OSQP with a one-step inner gradient update would test whether the benefit is specific to DR splitting or generic to operator-splitting unrolling.

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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: this is a solid empirical paper on unrolling Douglas-Rachford for QPs, with a real but fixable hole in its convergence proof. The empirical warm-start results—roughly 14–58% iteration cuts and 9–56% time cuts across synthetic, QPLIB, and portfolio sets, including 50% on QPLIB 8785—are believable and well reported. The idea of replacing the linear-solve in DR with a single learnable gradient step and unrolling it is genuinely new in the QP/L2O literature; the closest prior work unrolls PDHG, and the DR variant normally needs matrix factorization. The experiments are transparent: dataset generators, solver versions, hyperparameters, and code are all there. I trust the measured speedups.

The soft spot is Appendix A. Proposition 1 claims Algorithm 2 retains convergence guarantees, and the proof leans on Corollary 5 (Combettes) for inexact DR. To apply it, the one-step gradient errors must be absolutely summable. The argument tries to get that from Lemma 4's square summability plus an asserted eventual geometric decay ||w^{k+1}-w^k|| <= c||w^k-w^{k-1}||. Square summability does not imply geometric decay, and the step 'Phi_{k+1} approx Phi_k' is asserted, not derived. Lemma 4 also invokes Combettes with e_k=0 even though the resolvent evaluation is inexact. So the advertised convergence guarantee is not established. That said, this is a proof gap, not a sign the method is broken; the empirical claims are separate and hold up. Minor issues: no error bars or seeds, SCS settings tuned per family, labels come from the very solver being warmed up, and robustness is tested at only 1.5x the training perturbation. All addressable.

Who benefits: people working on learning-to-optimize, warm-starting, or operator-splitting for QPs. If I were an editor, I'd send it to review—the empirical core and the unrolling idea deserve referee time—but I'd push hard for either a repaired proof or an explicit downgrade of Proposition 1 to an empirical claim. My own verdict would be conditional acceptance.

Referee Report

3 major / 4 minor

Summary. The paper proposes an unrolled Douglas-Rachford splitting method for convex quadratic programs. Algorithm 2 replaces the exact linear-system solve in the classical DR iteration by a single gradient descent step with line search, and Proposition 1 claims that the resulting inexact iteration still converges to a KKT solution. The algorithm is unrolled into a four-layer network (DR-GD Net) trained in a supervised manner against SCS solutions; network outputs are used as warm starts for SCS. Experiments on synthetic QP(RHS), QP, QPLIB and Portfolio benchmarks report up to about 50% reduction in SCS iterations and up to about 40-50% reduction in solve time, including a 50.7%/49.7% improvement on the large QPLIB 8785 instance. The advertised theoretical guarantee is proved in Appendix A, and that proof is the main point of concern.

Significance. The contribution is potentially significant for learning-to-optimize: it provides a matrix-free unrolled DR scheme, public code, broad benchmarks, and an evaluation that is external to the fitted quantities (the speedup is measured by SCS solver iterations and solve time after warm starting). These are real strengths. However, the paper's second advertised contribution, the convergence guarantee of Algorithm 2, is not established by the proof in Appendix A. Since the theoretical claim is load-bearing for the paper's framing, the manuscript needs major revision. If the convergence proof can be repaired or the claim appropriately weakened, the empirical study would support publication.

major comments (3)

[Appendix A, proof of Proposition 1 (after Lemma 4)] The step upgrading Lemma 4's square summability to absolute summability is not justified. The proof asserts that eventually Phi_{k+1} approx Phi_k and ||w^{k+1}-w^k|| <= c||w^k-w^{k-1}|| with c<1; neither follows from ||w^{k+1}-w^k|| -> 0 or from square summability. Square-summable sequences need not be geometrically decaying, and Phi_k depends on \tilde u_k and eta_k, not only on w^k, so the eventual 'Phi_{k+1} approx Phi_k' is uncontrolled. Since this is the only route to sum_k ||epsilon_k|| < infinity needed to invoke Corollary 5, the central convergence guarantee is not established. Please provide a complete argument or replace the claim with a weaker, proven statement.
[Appendix A, Lemma 4] The lemma applies Combettes' Theorem 3.1 with e_k=0, although the algorithm's update is inexact: in the proof of Proposition 1 the error is e_k = epsilon_k, and the whole point is to show these errors are summable. As stated, Lemma 4 proves square summability only for a version in which the resolvent is evaluated exactly, not for Algorithm 2. In addition, the hypothesis \cap_k Fix T_k != empty is assumed rather than derived; because T_k depends on k through eta_k and \tilde u_k, a fixed point common to all operators requires justification. Without Lemma 4, the rest of the proof lacks its starting point.
[Appendix A, line search and contraction conditions] The proof requires eta_k to satisfy both the Wolfe-type lower bound eta_k >= (1-c_2)/sigma_max(I+M) and the contraction/strong-monotonicity bound eta_k < lambda_min(I+M)/lambda_max^2(I+M). These intervals can be disjoint for ill-conditioned QPs (when lambda_min is small relative to lambda_max^2/sigma_max), so the assumed line search need not exist. Please state explicit conditions on the spectrum of I+M under which the interval is nonempty, or modify the update so that the required step size always exists.

minor comments (4)

[Tables 3-5 and Figure 2] Averages over 100 test instances are reported without standard deviations, confidence intervals, or per-instance distributions (except for Figure 2b, which covers one dataset). Adding variability measures for the headline ratios would strengthen the empirical claims.
[Table 3 / Section 5.2] The text mentions the 'smallest case with 100 variables,' but Table 3 starts at n=200. Presumably this should be 200; please correct.
[Appendix A, Corollary 5] There is a typo: 'squence in H' should be 'sequence in H'. Also, the notation w^{(-1)} in the proof of Proposition 1 is introduced implicitly; a short clarification would help.
[Section 4.2] The claim that Algorithm 3 can emulate Algorithm 2 and hence recover optimal solutions is stated informally. Giving the parameter assignment or labeling the statement as a remark (rather than a direct consequence of Proposition 1) would make the inheritance of convergence guarantees precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence claim and warm-start benchmarks are externally grounded; the Appendix A gap is a proof correctness concern, not a circular reduction.

full rationale

The paper's central advertised result is Proposition 1, whose proof in Appendix A applies the inexact-DR machinery of Eckstein-Bertsekas/Combettes to Algorithm 2. The convergence claim is not circular: Proposition 1 is asserted for the deterministic DR-GD algorithm under explicit line-search assumptions, and the proof attempts to verify the required summability condition sum_k ||epsilon_k|| < infinity from the Wolfe conditions and Lemma 4. Whether that verification succeeds is a mathematical correctness question (the step 'let us further assume ... Phi_{k+1} approx Phi_k and ... ||w^{k+1}-w^k|| <= c||w^k - w^{k-1}||' upgrades only square summability to absolute summability without proof), but it is not a case of the target result being assumed as an input. The empirical warm-start claim is likewise externally grounded: the network is trained with an l2 regression loss against SCS solutions on a training split, and the reported iteration counts and solve times are measured by running SCS from the predicted points on a held-out test set; no fitted parameter is renamed as a prediction. Self-citations (Yang et al. 2024; Gao et al. 2024) appear in related-work and dataset-generation context and are not load-bearing for Proposition 1 or for the benchmark comparisons. I find no step in which a prediction reduces by construction to its training input, and no load-bearing self-citation chain. Score 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The empirical speedup rests on supervised labels from SCS, per-family training data (400 samples, perturbations U[0.9,1.1]), the 4-layer/128-embedding architecture, and SCS settings tuned per dataset (Table 6). The convergence theorem rests on standard monotone operator theory, inexact DR convergence from Combettes 2004 and Eckstein-Bertsekas 1992, a line-search window that must simultaneously satisfy Wolfe and contraction bounds, and the unproven eventual contraction of the operators Phi_k. No new entities (particles, forces, dimensions) are postulated.

free parameters (4)

Layer step-size priors eta_l = 0.05 (QPLIB), 0.1 (other datasets)
Hand-chosen per dataset family; Section 5, paragraph before 5.1.
Network parameters Theta = U, V, W matrices, biases b_eta, final map P_L^u; 4 layers, embedding size 128
Fitted by Adam (lr 1e-5, batch 2, early stopping) against SCS-computed optimal solutions; Section 4.3 and Section 5.
SCS solver configurations per dataset family = Table 6: data scaling False for QP(RHS)/QP, True for QPLIB/Portfolio; dual scale factor, alpha, acceleration differ
Settings were explored per dataset rather than fixed prospectively; this tuning affects the reported baseline iteration and time counts.
Training hyperparameters = lr 1e-5, batch 2, patience 10, layers 4, embedding 128
Reported without sensitivity analysis; ablations only cover number of layers and number of inner GD steps.

assumptions (6)

standard math Maximal monotonicity of the normal cone N_C and the projection/resolvent identity Pi_C = (Id + N_C)^{-1}
Lemma 2 in Appendix A; the standard bridge between projections and operator splitting.
domain assumption Classical inexact DR convergence: Algorithm 1 converges if inner linear-system errors are summable (Eckstein & Bertsekas 1992; Combettes 2004, Thm 3.1 / Cor 5.2)
Invoked in Lemma 4 and the proof of Proposition 1 as the convergence engine.
ad hoc to paper Line search can satisfy both Wolfe conditions (eta_k >= (1-c_2)/sigma_max(I+M)) and the contraction bound eta_k < lambda_min(I+M)/lambda_max^2(I+M)
Lemma 4 and Prop 1 require both; feasibility of the intersection is asserted and can be a thin interval for ill-conditioned QPs.
ad hoc to paper Eventual operator stationarity and contraction: there exists K such that for all k > K, Phi_{k+1} approx Phi_k and ||w^{k+1} - w^k|| <= c||w^k - w^{k-1}|| with c < 1
The load-bearing unproven step in the proof of Proposition 1 (Appendix A) that upgrades square summability to the absolute summability of GD errors.
domain assumption Ground-truth optimal solutions from SCS at tolerance 1e-4 are adequate labels and adequate references for solution-quality metrics
Used for the supervised loss (Section 4.3) and the reported l2 norms (Appendix F).
standard math sigma_min(I+M) >= 1
Used in the tau contraction bound; holds because the symmetric part of I+M is I + [[P,0],[0,0]] (their stated eigenvalue justification is incomplete but the conclusion is correct).

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Cite this review

Pith. "Pith review of Solving Quadratic Programs via Deep Unrolled Douglas-Rachford Splitting." pith.science (2026). https://pith.science/paper/OIN5OF2K

@misc{pith2026250811869,
  author       = {Pith},
  title        = {Pith review of: Solving Quadratic Programs via Deep Unrolled Douglas-Rachford Splitting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIN5OF2K}},
  note         = {Machine review of arXiv:2508.11869}
}

read the original abstract

Convex quadratic programs (QPs) are fundamental to numerous applications, including finance, engineering, and energy systems. Among the various methods for solving them, the Douglas-Rachford (DR) splitting algorithm is notable for its robust convergence properties. Concurrently, the emerging field of Learning-to-Optimize offers promising avenues for enhancing algorithmic performance, with algorithm unrolling receiving considerable attention due to its computational efficiency and interpretability. In this work, we propose an approach that unrolls a modified DR splitting algorithm to efficiently learn solutions for convex QPs. Specifically, we introduce a tailored DR splitting algorithm that replaces the computationally expensive linear system-solving step with a simplified gradient-based update, while retaining convergence guarantees. Consequently, we unroll the resulting DR splitting method and present a well-crafted neural network architecture to predict QP solutions. Our method achieves up to 50% reductions in iteration counts and 40% in solve time across benchmarks on both synthetic and real-world QP datasets, demonstrating its scalability and superior performance in enhancing computational efficiency across varying sizes.

Figures

Figures reproduced from arXiv: 2508.11869 by the authors.

**Figure 2.** (a) Validation loss and iteration improvement ratio on the dataset [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Improvement ratios for iteration count (3a) and solve time (3b) on the QP(RHS) validation sets, [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 14 canonical work pages

[1]

fk is Lipschitz continuous with constantL =σmax(I +M), as∥∇fk(x)−∇fk(y)∥≤ L∥x−y∥

∇fk(˜uk) 2 . fk is Lipschitz continuous with constantL =σmax(I +M), as∥∇fk(x)−∇fk(y)∥≤ L∥x−y∥. Therefore − (∇fk(˜uk+1)−∇fk(˜uk))⊤∇fk(˜uk) ≤| (∇fk(˜uk+1)−∇fk(˜uk))⊤∇fk(˜uk)| ≤∥ (∇fk(˜uk+1)−∇fk(˜uk))∥∥∇fk(˜uk)∥ ≤ηkL∥∇fk(˜uk)∥2, which implies that ηk≥− (∇fk(˜uk+1)−∇fk(˜uk))⊤∇fk(˜uk) L∥∇fk(˜uk)∥2 ≥ (1−c2) ∇fk(˜uk) 2 L∥∇fk(˜uk)∥2 = 1−c2 L . Therefore f...

work page 2025
[2]

is an averaged operator. As (5) taking the form of Algorithm 1.2 in Combettes (2004) withλk = 1,ek = 0 and αk = 1 2, according to Theorem 3.1 and Remark 3.4 in Combettes (2004) we can conclude that∑ k∈N∥Tk(wk)−wk∥2 < +∞, and thus∥wk+1−wk∥→ 0 Corollary 5 (Corollary 5.2 Combettes (2004)). Let γ∈ (0,∞), let{νk} be a sequence in(0, 2), and let {ak} and{bk} be...

work page 2004
[3]

A single iteration in Algorithm 2, which mapswk to wk+1, can be expressed by the operatorTk as follows: wk+1 =Tk(wk) =wk + [1 2(Id +CNC(2Φk−Id))wk−wk ] , where Φk(w) = ( I−ηk(I +M)⊤(I +M) ) ˜uk +ηk(I +M)⊤(w−q) and CNC is the Cayley operator associated with the normal coneNC. Proof. The proof proceeds by direct substitution. The Cayley operatorCNC is defin...

work page 2025
[4]

Inf. Time(s)

on validation set of QP(RHS) and observed a clear trade-off. From the table, we observe that the most significant solution time reductions are achieved with 1 to 2 GD steps per layer and solution qual- ity deteriorates as step count increases, evidenced by rising maximum violation values and growingℓ2-error norms. These results demonstrate that our curren...

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Obj.”), feasibility satisfaction (“Max Viol

On average, the SCS solver required 1,559 iterations and 14.474 seconds to solve this instance. However, by using the solution from our DR-GD Net as a warm start, the iteration count and solve time for SCS were reduced to 711 and 7.025 seconds, respectively. This corresponds to an improvement of 50.7% in iterations and 49.7% in solve time. These results d...

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Therefore, ∀k > K,∃0 < c <1, such that∥wk+1−wk∥≤ c∥wk−wk−1∥ and thus∑ k=K∥wk+1−wk∥ < +∞

Let us further assume thatηk also satisfies the assumption in Lemma 4, then∃K, such that∀k>K Φk+1≈ Φk and 2Φk−Id is contractive. Therefore, ∀k > K,∃0 < c <1, such that∥wk+1−wk∥≤ c∥wk−wk−1∥ and thus∑ k=K∥wk+1−wk∥ < +∞. As∑ k∈N∥wk+1−wk∥2 < +∞,∑K−1 k=0 ∥wk+1−wk∥isbounded. Wecanconcludethat ∑ k∈N∥wk+1−wk∥< +∞. Defining τk = fk(˜uk+1) fk(˜uk) and taking 0<c 1 ...

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

fk is Lipschitz continuous with constantL =σmax(I +M), as∥∇fk(x)−∇fk(y)∥≤ L∥x−y∥

∇fk(˜uk) 2 . fk is Lipschitz continuous with constantL =σmax(I +M), as∥∇fk(x)−∇fk(y)∥≤ L∥x−y∥. Therefore − (∇fk(˜uk+1)−∇fk(˜uk))⊤∇fk(˜uk) ≤| (∇fk(˜uk+1)−∇fk(˜uk))⊤∇fk(˜uk)| ≤∥ (∇fk(˜uk+1)−∇fk(˜uk))∥∥∇fk(˜uk)∥ ≤ηkL∥∇fk(˜uk)∥2, which implies that ηk≥− (∇fk(˜uk+1)−∇fk(˜uk))⊤∇fk(˜uk) L∥∇fk(˜uk)∥2 ≥ (1−c2) ∇fk(˜uk) 2 L∥∇fk(˜uk)∥2 = 1−c2 L . Therefore f...

work page 2025

[2] [2]

is an averaged operator. As (5) taking the form of Algorithm 1.2 in Combettes (2004) withλk = 1,ek = 0 and αk = 1 2, according to Theorem 3.1 and Remark 3.4 in Combettes (2004) we can conclude that∑ k∈N∥Tk(wk)−wk∥2 < +∞, and thus∥wk+1−wk∥→ 0 Corollary 5 (Corollary 5.2 Combettes (2004)). Let γ∈ (0,∞), let{νk} be a sequence in(0, 2), and let {ak} and{bk} be...

work page 2004

[3] [3]

A single iteration in Algorithm 2, which mapswk to wk+1, can be expressed by the operatorTk as follows: wk+1 =Tk(wk) =wk + [1 2(Id +CNC(2Φk−Id))wk−wk ] , where Φk(w) = ( I−ηk(I +M)⊤(I +M) ) ˜uk +ηk(I +M)⊤(w−q) and CNC is the Cayley operator associated with the normal coneNC. Proof. The proof proceeds by direct substitution. The Cayley operatorCNC is defin...

work page 2025

[4] [4]

Inf. Time(s)

on validation set of QP(RHS) and observed a clear trade-off. From the table, we observe that the most significant solution time reductions are achieved with 1 to 2 GD steps per layer and solution qual- ity deteriorates as step count increases, evidenced by rising maximum violation values and growingℓ2-error norms. These results demonstrate that our curren...

work page 2025

[5] [7]

Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

Diederik P Kingma. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980,

arXiv

[6] [12]

doi: 10.1109/NAPS56150.2022.10012237. 14 Published in Transactions on Machine Learning Research (08/2025) A Proof of Proposition 1 Our goal is to prove that the sequence generated by Algorithm 2 converges to the solution of problem (2). The proof will contain two parts: • First, we will prove that Algorithm 2 is convergent. • Second, we will prove that Al...

arXiv 2022

[7] [13]

Obj.”), feasibility satisfaction (“Max Viol

On average, the SCS solver required 1,559 iterations and 14.474 seconds to solve this instance. However, by using the solution from our DR-GD Net as a warm start, the iteration count and solve time for SCS were reduced to 711 and 7.025 seconds, respectively. This corresponds to an improvement of 50.7% in iterations and 49.7% in solve time. These results d...

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[8] [16]

Therefore, ∀k > K,∃0 < c <1, such that∥wk+1−wk∥≤ c∥wk−wk−1∥ and thus∑ k=K∥wk+1−wk∥ < +∞

Let us further assume thatηk also satisfies the assumption in Lemma 4, then∃K, such that∀k>K Φk+1≈ Φk and 2Φk−Id is contractive. Therefore, ∀k > K,∃0 < c <1, such that∥wk+1−wk∥≤ c∥wk−wk−1∥ and thus∑ k=K∥wk+1−wk∥ < +∞. As∑ k∈N∥wk+1−wk∥2 < +∞,∑K−1 k=0 ∥wk+1−wk∥isbounded. Wecanconcludethat ∑ k∈N∥wk+1−wk∥< +∞. Defining τk = fk(˜uk+1) fk(˜uk) and taking 0<c 1 ...

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