Learn Feasibility Once, Optimize All Objectives: Derivative-Free Diffusion Models for Chance-Constrained Programming

Ziwen Liu; Yan Liu; Congying Han; Tiande Guo; Yao Yan; Weichen Zhao

cs.LGOct 2, 2026

Learn Feasibility Once, Optimize All Objectives: Derivative-Free Diffusion Models for Chance-Constrained Programming

Authors: Ziwen Liu, Yan Liu, Congying Han, Tiande Guo, Yao Yan, Weichen Zhao

Organizations: School of Mathematical Sciences, Beijing University of Posts and Telecommunications · School of Statistics and Data Science, Nankai University · School of Mathematical Sciences, University of the Chinese Academy of Sciences · School of Statistics and Data Science, Southwest University of Finance and Economics

Abstract

Chance-constrained programs (CCPs) optimize decisions under uncertainty by limiting the probability of constraint violation. Despite advances in traditional and learning-based approaches, optimizing non-convex or non-smooth objectives and adapting to different objectives under fixed chance constraints remain challenging. In this paper, we propose a Derivative-free Diffusion-based framework that Disentangles constraint modeling from objective optimization, termed \textbf{D $^3$ Opt}. We learn the chance-feasible structure once, independently of any particular objective, by training a risk-conditioned diffusion model solely on constraint-filtered decisions and freezing it as a reusable prior for post-specified objectives. At inference time, we propose an annealed, particle-based Feynman--Kac correction along the frozen reverse diffusion process to optimize post-specified objectives using only function evaluations. This enables derivative-free optimization of non-convex and non-smooth objectives without objective-specific retraining. We prove that the correction preserves feasibility when this property holds for the frozen prior, and derive an optimization-error bound separating learned-prior coverage, finite-particle approximation, and finite-temperature effects. Experiments on linear Gaussian CCPs, objective-transfer tasks, and chance-constrained economic dispatch demonstrate effective optimization across smooth and non-smooth objectives, including non-convex cases, and objective generalization under fixed chance constraints without retraining.

Figures & tables

Method	Repeat	FvalMean $\downarrow$	FvalStd $\downarrow$	FvalMedian $\downarrow$
SOC-CVX ( Grant et al., 2008 ) (Global Reference)	1	$-0.6586$	$0$	$-0.6586$
SAA-CVaR ( Nemirovski and Shapiro, 2007 )	100	$-0.5893$	$0.0248$	$-0.5869$
SAA-MIP ( Pagnoncelli et al., 2009 )	100	$-0.6281$	$0.0157$	$-0.6318$
SAA-PDCA ( Wang et al., 2023 )	100	$-0.6389$	$0.0314$	$-0.6408$
SAA-SNSCO ( Zhou et al., 2024 )	100	$0.8051$	$3.4014$	$-0.6371$
GGDOpt First-order ( Zhang et al., 2025 )	100	$-0.6483$	$0.0051$	$-0.6488$

Table 1: Comparison on the linear Gaussian chance-constrained problem equation 11 with

n=8

and

\rho=0.1

. Lower objective values are better.

Task	Method	FvalMean $\downarrow$	FvalStd $\downarrow$	FvalBest $\downarrow$	Optimality Gap (%) $\downarrow$
ASQ	SOC-CVX ( Grant et al., 2008 )	0.0979	–	0.0979	0.0000
	GGDOpt ( Zhang et al., 2025 ) First-order ( $\beta=0.1$ )	0.2026	0.0630	0.1055	106.9530
	GGDOpt ( Zhang et al., 2025 ) Second-order ( $\beta=0.1,\sigma=0.1$ )	0.1994	0.0625	0.1055	103.6460
	DiffOPT ( Kong et al., 2025 )	0.1313	0.0081	0.1092	34.0980
	D 3 Opt w/o FKC	0.8549	0.2086	0.3209	773.1010
	D 3 Opt	$\mathbf{0.1035}$	$\mathbf{0.0018}$	$\mathbf{0.0999}$	$\mathbf{5.7040}$

Table 2: Objective-transfer results on ASQ and MW4 under the same linear Gaussian chance constraint. Each stochastic method is evaluated over 100 independent runs. Lower objective values and gaps are better.

Method	Cost Mean $\downarrow$	Cost Std $\downarrow$	Cost Best $\downarrow$	Mean Probability
DE + exact projection	$104399.055$	$5.2\times 10^{-8}$	$104399.055$	$0.9000$
PWL-MILP + exact projection	$104399.103$	–	$104399.103$	$0.9000$
95% CI-SAA-DE	$104400.092$	$\approx 0$	$104400.092$	$0.9003$
GGDOpt ( Zhang et al., 2025 ) (Vanilla)	Not directly applicable
GGDOpt ( Zhang et al., 2025 ) + VPE adaptation	$108821.619$	$1389.720$	$106540.523$	$0.9522$
DiffOPT ( Kong et al., 2025 )	$105678.450$	$163.418$	$105240.143$	$0.9210$

Table 3: CCED-VPE-10 results at

\rho=0.10

over

100

runs, except PWL-MILP (one solve). Mean Probability is the average analytic satisfaction probability.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Unit $i$	$P_{i}^{\min}(MW)$	$P_{i}^{\max}(MW)$	$a_{i}$	$b_{i}$	$c_{i}$	$d_{i}$	$e_{i}$
1	10	55	1000.403	40.5407	0.12951	33	0.0174
2	20	80	950.606	39.5804	0.10908	25	0.0178
3	47	120	900.705	36.5104	0.12511	32	0.0162
4	20	130	800.705	39.5104	0.12111	30	0.0168
5	50	160	756.799	38.5390	0.15247	30	0.0148
6	70	240	451.325	46.1592	0.10587	20	0.0163

Appendix

Table 4: Generator output limits and valve-point cost coefficients for CCED-VPE-10.

$\rho$	Method	Cost Mean $\downarrow$	Cost Std $\downarrow$	Cost Best $\downarrow$	Mean Probability
$0.05$	DE + exact projection	$104689.448$	$0.000$	$104689.448$	$0.9500$
	PWL-MILP + exact projection	$104689.483$	–	$104689.483$	$0.9500$
	95% CI-SAA-DE	$104717.493$	$0.000$	$104717.493$	$0.9537$
	GGDOpt ( Zhang et al., 2025 ) + VPE adaptation	$108951.188$	$1344.889$	$106546.719$	$0.9729$
	DiffOPT ( Kong et al., 2025 )	$105986.707$	$166.335$	$105554.119$	$0.9500$
	D 3 Opt	$\mathbf{105111.845}$	$\mathbf{112.923}$	$\mathbf{104831.074}$	$0.9500$

Appendix

Table 5: Risk-level adaptation on CCED-VPE-10 at

\rho\in\{0.05,0.15,0.20\}

. We report mean

\pm

sample standard deviation and best cost over

100

runs, except PWL-MILP, which uses one solve per risk level. Bold costs indicate the best available learning-based results.

$\beta_{\max}$	$M$	FvalMean $\downarrow$	FvalStd $\downarrow$	FvalBest $\downarrow$
10	64	$105161.669$	$243.555$	$104691.425$
10	256	$104887.677$	$126.620$	$104593.105$
10	512	$104823.577$	$102.065$	$104582.444$
10	1024	$104759.778$	$90.430$	$104549.575$
10	2048	$\mathbf{104707.598}$	$\mathbf{74.224}$	$\mathbf{104521.263}$
0	512	$107183.587$	$268.987$	$106146.715$

Appendix

Table 6: Ablation experiments on CCED-VPE-10 at

\rho=0.1

, varying particle count

M\in\{64,256,512,1024,2048\}

and terminal inverse temperature

\beta_{\max}\in\{0,5,10,20,50\}

. Results are reported over

100

runs per setting.

$\rho$	Method	Power Mean $\downarrow$	Power Std $\downarrow$	Power Best $\downarrow$	$P_{1}/P_{2}/P_{3}$
$0.05$	Sphere Bounding ( Wang et al., 2014 )	$0.1374$	–	–	$(0.99;0.99;0.99)$
	Bernstein-type Inequality ( Wang et al., 2014 )	$0.1260$	–	–	$(0.97;0.98;0.95)$
	GGDOpt First-order ( Zhang et al., 2025 )	$0.127992$	$0.000205$	$0.127339$	$(0.99;0.98;0.95)$
	GGDOpt Second-order ( Zhang et al., 2025 )	$0.127989$	$0.000206$	$0.127335$	$(0.98;0.98;0.95)$
	DiffOPT ( Kong et al., 2025 )	$0.246433$	$0.000565$	$0.246110$	$(0.99;0.98;0.97)$
	D 3 Opt	$\mathbf{0.127732}$	$0.000209$	$\mathbf{0.127275}$	$(0.97;0.95;0.96)$

Appendix

Table 7: Robust waveform results with

N_{t}=16

and

K=3

. Learning-based power statistics are from 100 runs. The final column reports per-user chance-constraint satisfaction probabilities on the returned decisions, averaged across runs for learning-based methods. For a traditional baseline it refers to its single solution. Bold power values are the lowest mean and best among learning-based methods.

Chance-Constrained Optimization

Chance-constrained optimization chooses decisions that optimize performance while limiting the probability of violating uncertain constraints.

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Optimization for One, Feasibility for Most: Decoupling Optimization from Feasibility in Chance-Constrained Programs

Justin Dumouchelle, Shunyu Yao, Beste Basciftci, Thiago Serra

University of Calgary · University of Iowa

Chance-Constrained Optimization Stochastic Optimization Adaptive Sampling

Open paper AskPDF arXiv

Chance-constrained stochastic programs (CCSPs) require solutions that satisfy uncertain constraints with high probability. Standard approaches either embed scenarios directly into optimization by considering sample average approximation (SAA)-based reformulations, growing prohibitively expensive and often failing to find feasible solutions; or employ conservative reformulations that guarantee feasibility but sacrifice objective quality by design. We propose ACCESS (Adaptive Chance-Constrained Exploration via Structured Sampling), a structured sampling framework that decouples optimization from feasibility evaluation. First, high-quality candidate solutions are generated by solving many inexpensive single-scenario subproblems, avoiding sampling directly in the constrained high-dimensional solution space. Second, feasibility is assessed separately via large-scale batched simulation over a massive number of scenarios, which is naturally amenable to GPU parallelism. That reduces the search from a high-dimensional solution space to a two-dimensional parameter space controlling conservatism and scenario diversity; while concentrating on candidates near the feasibility boundary, where high-quality feasible solutions lie. We also introduce a variant, ACCESS(

δ

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1-δ

. Experiments across three benchmark classes and five scenario distributions show that ACCESS matches or exceeds the best objective found by any method in nearly every setting, while ACCESS(

δ

) is empirically feasible on every instance. The conservative reformulations remain several percent worse in objective, and SAA is frequently infeasible while taking an order of magnitude longer on the harder benchmarks.

Abstract

Chance-Constrained Optimization

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