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REVIEW 3 major objections 5 minor 52 references

Diffusion-based learning framework for Constrained Nonconvex Optimization with Weighted Bootstrapped Refinement

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A two-phase diffusion framework called DiOpt learns to map noise to near-optimal feasible points of constrained nonconvex optimization problems, where supervised diffusion solvers become exponentially infeasible with dimension.

desk verdict DiOpt gets real feasibility gains on nonconvex benchmarks with an interesting self-bootstrapping diffusion idea, but the theory is thinner than advertised and the key weight-clipping equivalence is asserted from the authors' own prior work. read the letter →

arxiv 2502.10330 v4 pith:5DJR2Q7M submitted 2025-02-14 cs.LG

Shutong Ding , Yimiao Zhou , Ke Hu , Xi Yao , Junchi Yan , Xiaoying Tang , Ye Shi This is my paper
classification cs.LG MSC 90C2668T0790C30
keywords diffusionmodelsconstrainedoptimizationnonconvexself-supervisedlearningbootstrappingfeasibilityoptimalpowerflowtooptimize
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

DiOpt claims that diffusion models can be made to solve constrained nonconvex optimization problems by replacing pure supervised training with a two-phase procedure: first learn near-optimal candidates, then keep retraining the model on its own most promising samples, weighted so that feasible points near the optimum dominate. The paper argues that supervised diffusion solvers are structurally doomed to violate constraints because their output distribution spreads like a Gaussian around the optimal point, while the feasible region occupies an exponentially small angular slice; Theorem 1 quantifies this as probability $\sim 1/2^d$ for a $d$-dimensional linear program. On QP, QPSR, CQP, ACOPF, and motion-retargeting benchmarks, DiOpt reports 70 to 100 percent feasibility with single-digit optimality gaps, where the model-based diffusion baseline often returns zero percent feasible samples. The result matters because it offers a self-supervised route to hard-constraint satisfaction without collecting labeled optimal solutions or hand-designed projection layers.

What carries the argument

The load-bearing object is the weighted bootstrapping update built around the target distribution $p(y;x) \propto \mathbb{1}_{C(x)}(y) \exp(-\beta f(y;x))$. The weight function is $\omega(y;x) = \exp(f^*(x) - f(y;x))$ for feasible candidates and $\omega(y;x) = -\sum_i \max(g_i(y;x), 0)$ for infeasible ones, and the training loss is the weighted noise-prediction loss $\mathbb{E}[\omega(y;x) \lVert \epsilon - \epsilon_\theta(y_t, x, t) \rVert^2]$. Because diffusion training requires positive weights, DiOpt uses the clipped mean-shifted weight $\tilde{\omega}(y;x) = \max(\omega(y;x) - \bar{\omega}, 0)$, citing a prior equivalence result to argue the diffusion model still converges to the same target distribution. The look-up table $B(x)$ stores the historically best candidate per problem instance so that a bad sampling round cannot regress training.

What would settle it

An experiment that would settle it: on the QP benchmark, replace the clipped weight $\tilde{\omega}$ with the raw signed weight $\omega$ during the bootstrapping phase; if feasibility and optimality gap stay essentially the same, the paper's convergence argument is not load-bearing, and if they collapse, the clipping is essential to the claimed behavior.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the failure of diffusion solvers on constrained problems is not a capacity problem but a target-distribution problem. Training a diffusion model to imitate optimal solutions forces it to place probability mass around a single point, and around that point the feasible set occupies only about $1/2^d$ of the local volume in $d$ dimensions, so high-dimensional solvers become exponentially infeasible. DiOpt instead trains toward the near-optimal feasible distribution $p(y;x) \propto \mathbb{1}_{C(x)}(y) \exp(-\beta f(y;x))$ by a bootstrapped loop: sample $K$ candidates, score them with a weight that rewards low objective inside the feasible region and penalizes violation outside it, clip negative weights via a mean-shift, and train on the best candidate via the weighted variational diffusion loss. A look-up table keeps historically best samples, and inference selects the highest-weight candidate from multiple draws. The empirical claim is that this loop drives feasibility from near zero percent to 70 to 100 percent while keeping optimality gaps around 0.2 to 7 percent, beating DC3 and model-based diffusion on the tested benchmarks.

Load-bearing premise

The whole self-supervised phase rests on the claim that the clipped, mean-shifted weight $\tilde{\omega}(y) = \max(\omega(y) - \bar{\omega}, 0)$ trains the diffusion model to the same distribution as the original signed weight $\omega$; that equivalence is imported from the authors' prior work rather than proven here for infeasible candidates with negative weights.

Editorial extensions

If this is right

  • If DiOpt is correct, supervised diffusion solvers will keep failing on high-dimensional constrained tasks, and any practical diffusion-based constrained solver needs a mechanism that actively reweights samples toward feasibility.
  • The same bootstrapped loop should transfer to other continuous constrained problems with smooth objectives and constraints, such as optimal power flow, trajectory optimization, and motion retargeting, without hand-designed projection layers.
  • With a supervised warm-start ratio $r_s$ around 0.2, the method reaches a feasibility-versus-optimality balance that neither pure supervised training nor pure bootstrapping achieves, so the ablation curves predict a sweet spot for this ratio.
  • Sampling multiple candidates and selecting by the weight function is what makes the final near-optimal feasible point reliable, meaning inference cost, the number of samples $K$, trades directly against solution quality.

Reading between the lines

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

  • The $1/2^d$ bound is proved only for linear programs with random hyperplane constraints; a natural test is whether the same exponential decay appears empirically for smooth nonlinear constraints in high dimensions, which would make the case for DiOpt stronger.
  • The mean-shift clipping $\tilde{\omega} = \max(\omega - \bar{\omega}, 0)$ is one choice among many positive reweightings; if the cited equivalence is fragile for negative weights, alternative transformations such as rank-based or sigmoid weights could be compared on the same benchmarks.
  • Because the target distribution is a Gibbs-type measure over the feasible region, DiOpt's training loop is effectively performing annealed self-distillation, so one could expect it to also produce information about which constraints are active near the optimum.
  • The equality-completion trick hides part of the difficulty: on problems where equality constraints cannot be solved for a few variables, the bootstrap would need an analogous mechanism, so the method's scope may be narrower than general constrained nonconvex optimization.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes DiOpt, a diffusion-based learning framework for constrained nonconvex optimization. DiOpt first trains a diffusion model with supervised learning on optimal solutions (warm-start), then switches to a weighted bootstrapping self-training phase: candidate solutions are sampled from the current model, weighted by a function that rewards feasibility and near-optimality, and the model is trained on a weighted variational loss. A look-up table stores historically optimal samples, and inference uses solution selection over multiple candidates. The paper includes a theoretical claim (Theorem 1) that supervised diffusion becomes exponentially infeasible with dimension, and evaluates DiOpt on QP, QPSR, CQP, ACOPF, and Motion Retargeting, reporting higher feasibility and lower optimality gaps than DC3, model-based diffusion, and supervised diffusion baselines.

Significance. The empirical results are the main strength: Table 1 shows large feasibility gains (e.g., 70-86% for DiOpt vs 0-37% for model-based diffusion on CQP, ACOPF57, ACOPF118) and consistently lower optimality gaps. The paper also provides detailed ablations (diffusion steps, supervised ratio, training samples, sampling samples, noise level), a public code repository, and an official project page, which are useful for reproducibility. If the bootstrapping mechanism provably converges to the target distribution p(y;x) ∝ I_C(x)(y) exp(-β f(y;x)), the framework would be a meaningful contribution to learning-based constrained optimization. However, the theoretical support for this convergence is currently too thin: the central equivalence behind the weighted loss is cited from the authors' own prior work and is not proven for the negative-weight case used here, and Theorem 1 analyzes uniform sampling from a small ball rather than the actual learned diffusion distribution. These gaps limit the strength of the paper's theoretical claims, though they do not invalidate the empirical findings.

major comments (3)
  1. [Section 5, Eqs. (9)-(13)] The claim that the clipped mean-shifted weight eω(y) = max(ω(y) - ω̄, 0) is 'equivalent to ω for diffusion training' is load-bearing for the bootstrapping phase, but it is not proven in this manuscript and is deferred to the authors' prior work [12]. This equivalence is nontrivial because Eq. (10) assigns negative weights to infeasible candidates and Eq. (13) trains only on the single argmax sample; the transformation is not a positive rescaling, and it changes the relative importance of low-weight and infeasible candidates. If the equivalence in [12] requires nonnegative weights or a fixed data distribution, then the self-supervised phase has no demonstrated convergence to the target distribution (8). Please either prove the equivalence for the case ω takes negative values and the argmax-selected sample, or provide empirical validation that the clipped weighting yields the intended target distribution.
  2. [Section 4 and Appendix A, Theorem 1] Theorem 1 states that a point uniformly sampled from a small ball around the LP optimum is feasible with probability about 1/2^d, but this does not directly quantify the behavior of a supervised diffusion model, whose output distribution is not uniform on such a ball. The proof in Appendix A appears to rely on symmetry of randomly drawn constraint normals, but in the theorem the constraints are fixed and only the point x is random; the feasible cone at a vertex of a fixed LP can occupy a very different fraction of the ball than 1/2^d. The paper also claims the toy example's 27.05% feasible rate matches 1/2^2, but that is a single 2D instance and does not substantiate the asymptotic claim. Please clarify the probabilistic model (over constraints or over x) and either adapt the theorem to the actual diffusion distribution or state it as a heuristic observation.
  3. [Section 5 and Appendix B, reset operation] The alternating reset between weight functions (10) and (11) is essential to avoid collapse to a Dirac distribution, but its convergence behavior is not analyzed. Appendix B shows that repeated weighting with (17) on a feasible-support distribution leads to collapse, and then asserts that resetting 'redistributes the probability density' with ρ1 ≈ ρ0; however, the reset operation is not proven to maintain the invariant that the model converges to the desired target distribution. Because the training objective (13) is non-stationary (the weight function changes each epoch, and the training samples come from the model's own evolving distribution), a convergence guarantee is missing. If the alternating schedule is a heuristic, the paper should say so explicitly and support it with additional empirical evidence rather than a formal claim.
minor comments (5)
  1. [Algorithm 1] The condition 'if t mod 2 = 0' uses a variable t that is not defined in the algorithm loop; it should presumably be the epoch index n. Please fix this notation.
  2. [Section 4, Figure 2] The sentence 'the number of blue points (feasible) is greater than that of the green points' is unclear from the figure because the color legend is not described in the caption; please specify which color denotes feasible and which denotes infeasible samples.
  3. [Abstract and Section 1] The abstract in the submitted text and the abstract at the beginning of the full paper are different: the former says 'DiOpt is the first successful integration of the diffusion solver in constrained nonconvex optimization,' while the latter is a longer description. Please harmonize the two versions.
  4. [Section 5, Eq. (8)] The target distribution is written as p(y;x) ∼ I_C(x)(y) exp(−βf(y;x)), but the proportionality constant and the role of β are not discussed. It would be helpful to state that β is an inverse-temperature parameter and how its value is chosen.
  5. [Table 1] The feasibility standard deviations are all reported as 0.00 for every method, which is suspicious and suggests that the reported mean and std may be computed over a single fixed seed or a different statistic than the reader would expect. Please clarify how the means and standard deviations are computed.

Circularity Check

1 steps flagged · score 4.0 of 10

Bootstrapping convergence rests on an unproved equivalence (eω ≡ ω) cited to the authors' own prior [12]; empirical benchmark results remain independent.

  1. self citation load bearing [Section 5, 'Training Diffusion with Bootstrapping', Eqs. (9)-(13) and Algorithm 1]
    "However, there is still a problem to be resolved in our weight function. As illustrated in [12], the weight in (9) must always be positive. Hence, we perform a modification on the final weight when there exists a candidate point with a negative weight. ... As shown in [12], eω is equivalent to ω for diffusion training, we can ensure the diffusion model converges to the target distribution with the modified weight eω."

    The theoretical guarantee that DiOpt's self-supervised phase converges to the target distribution p(y;x) of Eq. (8) is carried by the sentence 'As shown in [12], eω is equivalent to ω for diffusion training.' Reference [12] is prior work by the same authors (Shutong Ding, Ke Hu, Ye Shi), and the equivalence is not proved in this paper. The claim must cover exactly the new regime introduced here: Eq. (10) assigns negative weights to every infeasible candidate, and eω = max(ω − ω̄, 0) is not a positive rescaling of ω, so it changes the relative weighting of infeasible and below-mean feasible candidates. Additionally, Eq.

full rationale

The paper's main theoretical observation, Theorem 1, is derived from standard stochastic-geometry arguments (Cover-Efron and Wendel) and is not circular: it analyzes the feasibility of uniformly sampling a small ball around an LP vertex and gives a dimension-dependent probability. The empirical evaluation is also independent: DiOpt is tested on external benchmarks (ACOPF, Retargeting, constructed QP variants) against DC3, MBD, MLP, and supervised diffusion, so the reported feasibility and gap numbers are not forced by construction. The circularity is localized to the theoretical grounding of the bootstrapping mechanism: the convergence to the near-optimal feasible region is justified by asserting, with a citation to the authors' own prior work [12], that the clipped mean-shifted weight eω is equivalent to the signed weight ω for diffusion training. That equivalence is load-bearing, is not proved for negative weights and the argmax-selected single-sample loss in Eq. (13), and is not independently verified. Because the central algorithmic claim still has substantial empirical content, a moderate score of 4 is appropriate rather than a higher score reserved for results forced by self-citation or by definition.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The paper's core mechanism is a weighted bootstrapped diffusion training. The free parameters are training-stage hyperparameters (supervised ratio rs, diffusion steps T, candidate count K, training sample count Kt, noise coefficient eta) that are tuned per task and largely determine the reported feasibility/optimality balance. The theoretical axioms include standard diffusion-model assumptions and the geometric probability result behind Theorem 1. The most fragile entry is the asserted equivalence between the clipped weight function and the original weight, which is imported from the authors' own prior work [12] without proof; the negative weights assigned to infeasible candidates make that equivalence nontrivial. The reset operation's ability to avoid collapse to the optimal point is also only sketched.

free parameters (5)
  • Supervised ratio rs = 0.2 default (ablated 0.02, 0.05, 0.2)
    Controls the fraction of training epochs that use supervised learning; large enough to warm-start, small enough to show bootstrapping gains; tuned in Section D.2.
  • Diffusion steps T = 100 for QP/QPSR/CQP/ACOPF57; 5 for ACOPF118/Retargeting
    Larger T improves exploration and feasibility per Section D.1; adopted per task for best performance.
  • Candidate count K = 64 for most tasks; 32 in some ablation settings
    More candidates improve the chance of a feasible near-optimal selection at inference, at higher sampling cost (Section D.4).
  • Training samples Kt = 16
    Sets number of sampled candidates used in each bootstrap update; balances training cost and solution diversity (Section D.3).
  • Noise coefficient eta = 0 in many reported figures; unspecified for main Table 1
    eta=0 makes sampling deterministic, eliminating the diversity that solution selection relies on; the value used for headline results is not clearly stated, creating a reproducibility gap (Sections D.5, D.6).
assumptions (7)
  • standard math DDPM forward/reverse process and Gaussian noise assumption (Eqs. 2-6).
    Background for all diffusion training; standard from Ho et al. [20].
  • standard math Fundamental Theorem of Linear Programming: unique optimum of an LP occurs at a vertex, i.e. intersection of d linearly independent active constraints.
    Used in the proof of Theorem 1 in Appendix A; standard from Nocedal and Wright [33].
  • standard math Geometric result that d random hyperplanes through the origin in R^d partition the space into 2^d regions of equal measure.
    Core of Theorem 1's probability bound; cited to Cover and Efron [11] and Wendel [47].
  • ad hoc to paper The clipped weight function eomega is equivalent to omega for diffusion training.
    Assumed from authors' own prior work [12]; not proven for negative weights; underpins Eq. (13).
  • domain assumption Target distribution p(y;x) proportional to I_C(x)(y) exp(-beta f(y;x)) is the right objective for constrained optimization.
    Adopted from Liang and Chen [31]; justification is that it concentrates probability on feasible near-optimal points.
  • ad hoc to paper Alternating the weight function between (10) and (11) prevents collapse to a Dirac at the optimum.
    Appendix B gives only a heuristic sketch; no formal convergence guarantee.
  • domain assumption Equality constraints can be eliminated by generating a subset of variables and completing the rest with an equation solver.
    Standard practice from DC3 [15]; used for ACOPF and QP-family tasks.

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Cite this review

Pith. "Pith review of Diffusion-based learning framework for Constrained Nonconvex Optimization with Weighted Bootstrapped Refinement." pith.science (2026). https://pith.science/paper/5DJR2Q7M

@misc{pith2026250210330,
  author       = {Pith},
  title        = {Pith review of: Diffusion-based learning framework for Constrained Nonconvex Optimization with Weighted Bootstrapped Refinement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DJR2Q7M}},
  note         = {Machine review of arXiv:2502.10330}
}
read the original abstract

Recent advances in diffusion models show promising potential to accelerate nonconvex problem solving by leveraging their multimodality. However, most existing diffusion-based optimization approaches rely on supervised learning and lack a mechanism to enforce constraint satisfaction, which is required in real-world applications. In that case, we investigate and theoretically analyze the inherent problem of supervised diffusion solvers and identify the distributional misalignment problem, i.e., the generated solution distribution often exhibits low probability mass on the feasible region. To resolve this issue, we propose DiOpt, a new diffusion-based learning framework for constrained nonconvex optimization, which effectively learns the mapping from noise to the constraint region. Specifically, this framework operates in two distinct phases: an initial warm-start phase, implemented via supervised learning, followed by a bootstrapping training phase. This dual-phase architecture is designed to iteratively refine solutions, thereby improving the objective function with high constraint satisfaction. Finally, we also employ a solution selection technique in inference for better optimality. Notably, DiOpt is the first successful integration of the diffusion solver in constrained nonconvex optimization. Evaluations on diverse nonconvex tasks demonstrate the superiority of DiOpt in both optimality and constraint satisfaction. Our official page is released at https://dingsht.tech/diopt-webpage.

Figures

Figures reproduced from arXiv: 2502.10330 by the authors.

Figure 1
Figure 1. A schematic geometric interpretation of fea [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Comparison between supervised diffusion and DiOpt on a toy example. It can be observed that the distribution of diffusion trained in a supervised manner (a) approximates a Gaussian distribution centered around the optimal point, which leads to a low feasibility rate. For detailed settings of this toy example, please refer to the supplementary materials for more details. where y is the decision variable of the optimi… view at source ↗
Figure 3
Figure 3. Training and evaluating procedure of DiOpt. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Training progression across different optimization tasks, with each row representing a [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Effects of Different Diffuion Steps. In this experiment, we examine how varying the number of diffusion steps T affects performance. All other hyperparameters are fixed as follows: rs = 0.2, Kt = 16, and K = 32 for all values of T. 16 [PITH_FULL_IMAGE:figures/full_fig…
Figure 6
Figure 6. Figure 6: Effects of Different Supervised Ratio. In this experiment, we examine how varying the number of diffusion steps T affects performance. All other hyperparameters are fixed as follows: T = 5, Kt = 16, and K = 32 for all values of rs. 18 [PITH_FULL_IMAGE:figures/full_fig…
Figure 7
Figure 7. Figure 7: Effects of different Training Samples. In this experiment, we examine how varying the number of training samples Kt affects performance. All other hyperparameters are fixed as follows: rs = 0.2, T = 5, and K = 32 for all values of Kt. 20 [PITH_FULL_IMAGE:figures/full_…
Figure 8
Figure 8. Figure 8: Effects of number of sampling samples. In this experiment, we examine how varying the number of sampling points K affects performance. rs is set as 0.2 for all experiments and T = 5 for ACOPF118. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: Effects of number of sampling samples. In this experiment, we examine how varying the number of sampling points K affects performance. rs is set as 0.2 for all experiments and T = 5 for ACOPF118. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: Training Procedure of DiOpt and Diffusion. In this experiment, all hyperparameters are fixed as follows: rs = 0.2, T = 5, Kt = 16, and K = 32. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: Feasibility of DiOpt under different η with different T and K. (a) Diffusion 3.325 3.350 3.375 3.400 3.425 3.450 3.475 3.500 x 1.175 1.200 1.225 1.250 1.275 1.300 1.325 1.350 y DiOpt with 100.0% feasible rate for 10000 samples. (b) DiOpt [PITH_FULL_IMAGE:figures/full…
Figure 12
Figure 12. Figure 12: For the η = 0 configuration in the toy example, observations reveal that despite initial sampling starting from Gaussian-distributed random noise, all trajectories converge to nearly identical positions after denoising when η = 0. This demonstrates the necessity of no…

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Reference graph

Works this paper leans on

52 extracted references · 36 canonical work pages

  1. [12]

    Diffusion-based reinforcement learning via q-weighted variational policy optimization

    Shutong Ding, Ke Hu, Zhenhao Zhang, Kan Ren, Weinan Zhang, Jingyi Yu, Jingya Wang, and Ye Shi. Diffusion-based reinforcement learning via q-weighted variational policy optimization. arXiv preprint arXiv:2405.16173, 2024

    arXiv 2024

  2. [1]

    Learning warm-start points for ac optimal power flow

    Kyri Baker. Learning warm-start points for ac optimal power flow. In 2019 IEEE 29th International Workshop on Machine Learning for Signal Processing (MLSP), pages 1–6. IEEE, 2019

    work page 2019

  3. [2]

    Bellman, Rand Corporation, and Karreman Mathematics Research Collection

    R. Bellman, Rand Corporation, and Karreman Mathematics Research Collection. Dynamic Programming. Rand Corporation research study. Princeton University Press, 1957

    work page 1957

  4. [3]

    Anjos, and Sébastien Le Digabel

    Christian Bingane, Miguel F. Anjos, and Sébastien Le Digabel. Tight-and-cheap conic relaxation for the ac optimal power flow problem. IEEE Transactions on Power Systems, 33(6):7181–7188, 2018

    work page 2018

  5. [4]

    Diffusion policies for generative modeling of spacecraft trajectories

    Julia Briden, Breanna J Johnson, Richard Linares, and Abhishek Cauligi. Diffusion policies for generative modeling of spacecraft trajectories. In AIAA SCITECH 2025 Forum, page 2775, 2025

    work page 2025

  6. [5]

    Predict and constrain: Modeling cardinality in deep structured prediction

    Nataly Brukhim and Amir Globerson. Predict and constrain: Modeling cardinality in deep structured prediction. In International Conference on Machine Learning , pages 659–667. PMLR, 2018

    work page 2018

  7. [6]

    History of optimal power flow and formulations

    Mary B Cain, Richard P O’neill, Anya Castillo, et al. History of optimal power flow and formulations. Federal Energy Regulatory Commission, 1:1–36, 2012

    work page 2012

  8. [7]

    Neural networks for portfolio analysis with cardinality constraints

    Xinwei Cao and Shuai Li. Neural networks for portfolio analysis with cardinality constraints. IEEE Transactions on Neural Networks and Learning Systems, 35(12):17674–17687, 2024

    work page 2024

Show all 52 references
  1. [8]

    High- fidelity machine learning approximations of large-scale optimal power flow

    Minas Chatzos, Ferdinando Fioretto, Terrence WK Mak, and Pascal Van Hentenryck. High- fidelity machine learning approximations of large-scale optimal power flow. arXiv preprint arXiv:2006.16356, 2020

    2006 arXiv

  2. [9]

    Diffusion policy: Visuomotor policy learning via action diffusion

    Cheng Chi, Siyuan Feng, Yilun Du, Zhenjia Xu, Eric Cousineau, Benjamin Burchfiel, and Shuran Song. Diffusion policy: Visuomotor policy learning via action diffusion. arXiv preprint arXiv:2303.04137, 2023

    2023 arXiv

  3. [10]

    Constrained synthesis with projected diffusion models

    Jacob K Christopher, Stephen Baek, and Ferdinando Fioretto. Constrained synthesis with projected diffusion models. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024

    2024

  4. [11]

    Cover and Bradley Efron

    Thomas M. Cover and Bradley Efron. Geometrical probability and random points on a hyper- sphere. The Annals of Mathematical Statistics, 38(1):213–220, 1967

    1967

  5. [13]

    Reduced policy optimization for continuous control with hard constraints

    Shutong Ding, Jingya Wang, Yali Du, and Ye Shi. Reduced policy optimization for continuous control with hard constraints. Advances in Neural Information Processing Systems, 36, 2024. 10

    2024

  6. [14]

    Smart-pgsim: Using neural network to accelerate ac-opf power grid simulation

    Wenqian Dong, Zhen Xie, Gokcen Kestor, and Dong Li. Smart-pgsim: Using neural network to accelerate ac-opf power grid simulation. In SC20: International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1–15. IEEE, 2020

    2020

  7. [15]

    Dc3: A learning method for optimization with hard constraints

    Priya L Donti, David Rolnick, and J Zico Kolter. Dc3: A learning method for optimization with hard constraints. arXiv preprint arXiv:2104.12225, 2021

    2021 arXiv

  8. [16]

    Enhancing deep reinforcement learning: A tutorial on generative diffusion models in network optimization

    Hongyang Du, Ruichen Zhang, Yinqiu Liu, Jiacheng Wang, Yijing Lin, Zonghang Li, Dusit Niyato, Jiawen Kang, Zehui Xiong, Shuguang Cui, et al. Enhancing deep reinforcement learning: A tutorial on generative diffusion models in network optimization. IEEE Communications Surveys & ...

    2024

  9. [17]

    Lenssen, Christopher Morris, Jonathan Masci, and Nils M

    Matthias Fey, Jan E. Lenssen, Christopher Morris, Jonathan Masci, and Nils M. Kriege. Deep graph matching consensus. In International Conference on Learning Representations, 2020

    2020

  10. [18]

    Predicting ac optimal power flows: Combining deep learning and lagrangian dual methods

    Ferdinando Fioretto, Terrence WK Mak, and Pascal Van Hentenryck. Predicting ac optimal power flows: Combining deep learning and lagrangian dual methods. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pages 630–637, 2020

    2020

  11. [19]

    Learning human-to-humanoid real-time whole-body teleoperation

    Tairan He, Zhengyi Luo, Wenli Xiao, Chong Zhang, Kris Kitani, Changliu Liu, and Guanya Shi. Learning human-to-humanoid real-time whole-body teleoperation. arXiv preprint arXiv:2403.04436, 2024

    2024 arXiv

  12. [20]

    Denoising diffusion probabilistic models

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840–6851, 2020

    2020

  13. [21]

    Advancements and future directions in the application of machine learning to ac optimal power flow: A critical review

    Bozhen Jiang, Qin Wang, Shengyu Wu, Yidi Wang, and Gang Lu. Advancements and future directions in the application of machine learning to ac optimal power flow: A critical review. Energies, 17(6):1381, 2024

    2024

  14. [22]

    Erdos goes neural: an unsupervised learning framework for combinatorial optimization on graphs

    Nikolaos Karalias and Andreas Loukas. Erdos goes neural: an unsupervised learning framework for combinatorial optimization on graphs. Advances in Neural Information Processing Systems, 33:6659–6672, 2020

    2020

  15. [23]

    Cgd: Constraint-guided diffusion policies for uav trajectory planning

    Kota Kondo, Andrea Tagliabue, Xiaoyi Cai, Claudius Tewari, Olivia Garcia, Marcos Espitia- Alvarez, and Jonathan P How. Cgd: Constraint-guided diffusion policies for uav trajectory planning. arXiv preprint arXiv:2405.01758, 2024

    2024 arXiv

  16. [24]

    Equality constrained diffusion for direct trajectory optimiza- tion

    Vince Kurtz and Joel W Burdick. Equality constrained diffusion for direct trajectory optimiza- tion. arXiv preprint arXiv:2410.01939, 2024

    2024

  17. [25]

    Efficient and guaranteed-safe non-convex trajectory optimization with constrained diffusion model

    Anjian Li, Zihan Ding, Adji Bousso Dieng, and Ryne Beeson. Efficient and guaranteed-safe non-convex trajectory optimization with constrained diffusion model. arXiv e-prints, pages arXiv–2403, 2024

    2024

  18. [26]

    Amortized global search for efficient preliminary trajectory design with deep generative models

    Anjian Li, Amlan Sinha, and Ryne Beeson. Amortized global search for efficient preliminary trajectory design with deep generative models. CoRR, 2023

    2023

  19. [27]

    Learning to optimize

    Ke Li and Jitendra Malik. Learning to optimize. arXiv preprint arXiv:1606.01885, 2016

    2016 arXiv

  20. [28]

    Gauge flow matching for efficient constrained generative modeling over general convex set

    Xinpeng Li, Enming Liang, and Minghua Chen. Gauge flow matching for efficient constrained generative modeling over general convex set. In ICLR 2025 Workshop on Deep Generative Model in Machine Learning: Theory, Principle and Efficacy, 2025

    2025

  21. [29]

    From distribution learning in training to gradient search in testing for combinatorial optimization

    Yang Li, Jinpei Guo, Runzhong Wang, and Junchi Yan. From distribution learning in training to gradient search in testing for combinatorial optimization. Advances in Neural Information Processing Systems, 36, 2024

    2024

  22. [30]

    Fast t2t: Optimization consistency speeds up diffusion-based training-to-testing solving for combinatorial optimization

    Yang Li, Jinpei Guo, Runzhong Wang, Hongyuan Zha, and Junchi Yan. Fast t2t: Optimization consistency speeds up diffusion-based training-to-testing solving for combinatorial optimization. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024

    2024

  23. [31]

    Generative learning for solving non-convex problem with multi-valued input-solution mapping

    Enming Liang and Minghua Chen. Generative learning for solving non-convex problem with multi-valued input-solution mapping. In The Twelfth International Conference on Learning Representations, 2024

    2024

  24. [32]

    Learning to search in local branching

    Defeng Liu, Matteo Fischetti, and Andrea Lodi. Learning to search in local branching. In Proceedings of the aaai conference on artificial intelligence , volume 36, pages 3796–3803, 2022

    2022

  25. [33]

    Numerical optimization

    Jorge Nocedal and Stephen J Wright. Numerical optimization. Springer, 1999. 11

    1999

  26. [34]

    Model-based diffusion for trajectory optimization

    Chaoyi Pan, Zeji Yi, Guanya Shi, and Guannan Qu. Model-based diffusion for trajectory optimization. arXiv preprint arXiv:2407.01573, 2024

    2024 arXiv

  27. [35]

    Deepopf: A deep neural network approach for security-constrained dc optimal power flow

    Xiang Pan, Tianyu Zhao, Minghua Chen, and Shengyu Zhang. Deepopf: A deep neural network approach for security-constrained dc optimal power flow. IEEE Transactions on Power Systems, 36(3):1725–1735, 2020

    2020

  28. [36]

    Self-supervised primal-dual learning for constrained optimization

    Seonho Park and Pascal Van Hentenryck. Self-supervised primal-dual learning for constrained optimization. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 37, pages 4052–4060, 2023

    2023

  29. [37]

    Can push-forward generative models fit multimodal distributions? Advances in Neural Information Processing Systems, 35:10766–10779, 2022

    Antoine Salmona, Valentin De Bortoli, Julie Delon, and Agnes Desolneux. Can push-forward generative models fit multimodal distributions? Advances in Neural Information Processing Systems, 35:10766–10779, 2022

    2022

  30. [38]

    Scalable discrete diffusion samplers: Combinatorial opti- mization and statistical physics

    Sebastian Sanokowski, Wilhelm Berghammer, Martin Ennemoser, Haoyu Peter Wang, Sepp Hochreiter, and Sebastian Lehner. Scalable discrete diffusion samplers: Combinatorial opti- mization and statistical physics. arXiv preprint arXiv:2502.08696, 2025

    2025 arXiv

  31. [39]

    A diffusion model framework for unsupervised neural combinatorial optimization

    Sebastian Sanokowski, Sepp Hochreiter, and Sebastian Lehner. A diffusion model framework for unsupervised neural combinatorial optimization. arXiv preprint arXiv:2406.01661, 2024

    2024 arXiv

  32. [40]

    Global optimization for optimal power flow over transmission networks

    Ye Shi, Hoang Duong Tuan, Hoang Tuy, and S Su. Global optimization for optimal power flow over transmission networks. Journal of Global Optimization, 69:745–760, 2017

    2017

  33. [41]

    Denoising diffusion implicit models

    Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv:2010.02502, October 2020

    2010 arXiv

  34. [42]

    Score-based generative modeling through stochastic differential equations

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456, 2020

    2011 arXiv

  35. [43]

    Reinforcement learning for integer program- ming: Learning to cut

    Yunhao Tang, Shipra Agrawal, and Yuri Faenza. Reinforcement learning for integer program- ming: Learning to cut. In International conference on machine learning, pages 9367–9376. PMLR, 2020

    2020

  36. [44]

    On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming

    Andreas Wächter and Lorenz T Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical programming, 106:25–57, 2006

    2006

  37. [45]

    Towards one-shot neural combinatorial solvers: Theoretical and empirical notes on the cardinality-constrained case

    Runzhong Wang, Li Shen, Yiting Chen, Xiaokang Yang, and Junchi Yan. Towards one-shot neural combinatorial solvers: Theoretical and empirical notes on the cardinality-constrained case. In ICLR, 2023

    2023

  38. [46]

    Learning combinatorial embedding networks for deep graph matching

    Runzhong Wang, Junchi Yan, and Xiaokang Yang. Learning combinatorial embedding networks for deep graph matching. InProceedings of the IEEE/CVF international conference on computer vision, pages 3056–3065, 2019

    2019

  39. [47]

    J. G. WENDEL. A problem in geometric probability. Mathematica Scandinavica, 11(1):109– 111, 1962

    1962

  40. [48]

    Zamzam and Kyri Baker

    Ahmed S. Zamzam and Kyri Baker. Learning optimal solutions for extremely fast ac opti- mal power flow. In 2020 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm), pages 1–6, 2020

    2020

  41. [49]

    Learning to solve the ac optimal power flow via a lagrangian approach

    Ling Zhang and Baosen Zhang. Learning to solve the ac optimal power flow via a lagrangian approach. In 2022 North American Power Symposium (NAPS), pages 1–6, 2022

    2022

  42. [50]

    Diffusion models are evolutionary algorithms

    Yanbo Zhang, Benedikt Hartl, Hananel Hazan, and Michael Levin. Diffusion models are evolutionary algorithms. arXiv preprint arXiv:2410.02543, 2024

    2024 arXiv

  43. [51]

    Synergizing machine learning with acopf: A comprehensive overview, 2024

    Meng Zhao and Masoud Barati. Synergizing machine learning with acopf: A comprehensive overview, 2024

    2024

  44. [52]

    Diffusion

    Ray Daniel Zimmerman, Carlos Edmundo Murillo-Sánchez, and Robert John Thomas. Mat- power: Steady-state operations, planning, and analysis tools for power systems research and education. IEEE Transactions on Power Systems, 26(1):12–19, 2011. 12 A Proof for Theorem 1 Theorem 1 (...

    2011

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.