Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Gradient Guided Diffusion Framework for Chance Constrained Programming

Boyang Zhang, Zhiguo Wang, Ya-Feng Liu

TL;DR

The paper introduces GGDOpt, a gradient-guided diffusion framework for solving chance-constrained programming under unknown uncertainty distributions. It reformulates CCP as sampling from the product distribution Qβ(x|ρ) ∝ r(x|ρ) Bβ(x) and leverages diffusion-based score modeling with gradient guidance to sample asymptotically optimal solutions. The authors establish convergence guarantees and practical error bounds, and validate the approach on synthetic problems and robust waveform design, achieving higher-quality solutions with substantial computational savings. By integrating feasibility-aware data generation, classifier-free conditional score training, and first- and second-order gradient guidance, GGDOpt extends diffusion-based optimization to a broad class of CCPs without explicit distributional knowledge.

Abstract

Chance constrained programming (CCP) is a powerful framework for addressing optimization problems under uncertainty. In this paper, we introduce a novel Gradient-Guided Diffusion-based Optimization framework, termed GGDOpt, which tackles CCP through three key innovations. First, GGDOpt accommodates a broad class of CCP problems without requiring the knowledge of the exact distribution of uncertainty-relying solely on a set of samples. Second, to address the nonconvexity of the chance constraints, it reformulates the CCP as a sampling problem over the product of two distributions: an unknown data distribution supported on a nonconvex set and a Boltzmann distribution defined by the objective function, which fully leverages both first- and second-order gradient information. Third, GGDOpt has theoretical convergence guarantees and provides practical error bounds under mild assumptions. By progressively injecting noise during the forward diffusion process to convexify the nonconvex feasible region, GGDOpt enables guided reverse sampling to generate asymptotically optimal solutions. Experimental results on synthetic datasets and a waveform design task in wireless communications demonstrate that GGDOpt outperforms existing methods in both solution quality and stability with nearly 80% overhead reduction.

A Gradient Guided Diffusion Framework for Chance Constrained Programming

TL;DR

The paper introduces GGDOpt, a gradient-guided diffusion framework for solving chance-constrained programming under unknown uncertainty distributions. It reformulates CCP as sampling from the product distribution Qβ(x|ρ) ∝ r(x|ρ) Bβ(x) and leverages diffusion-based score modeling with gradient guidance to sample asymptotically optimal solutions. The authors establish convergence guarantees and practical error bounds, and validate the approach on synthetic problems and robust waveform design, achieving higher-quality solutions with substantial computational savings. By integrating feasibility-aware data generation, classifier-free conditional score training, and first- and second-order gradient guidance, GGDOpt extends diffusion-based optimization to a broad class of CCPs without explicit distributional knowledge.

Abstract

Chance constrained programming (CCP) is a powerful framework for addressing optimization problems under uncertainty. In this paper, we introduce a novel Gradient-Guided Diffusion-based Optimization framework, termed GGDOpt, which tackles CCP through three key innovations. First, GGDOpt accommodates a broad class of CCP problems without requiring the knowledge of the exact distribution of uncertainty-relying solely on a set of samples. Second, to address the nonconvexity of the chance constraints, it reformulates the CCP as a sampling problem over the product of two distributions: an unknown data distribution supported on a nonconvex set and a Boltzmann distribution defined by the objective function, which fully leverages both first- and second-order gradient information. Third, GGDOpt has theoretical convergence guarantees and provides practical error bounds under mild assumptions. By progressively injecting noise during the forward diffusion process to convexify the nonconvex feasible region, GGDOpt enables guided reverse sampling to generate asymptotically optimal solutions. Experimental results on synthetic datasets and a waveform design task in wireless communications demonstrate that GGDOpt outperforms existing methods in both solution quality and stability with nearly 80% overhead reduction.
Paper Structure (43 sections, 125 equations, 10 figures, 11 tables, 2 algorithms)

This paper contains 43 sections, 125 equations, 10 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Related works
  4. Our contributions
  5. Organization
  6. Problem reformulation via sampling
  7. Diffusion models
  8. Gradient guidance
  9. GGDOpt for CCP
  10. Stage 1: data generation
  11. Stage 2: diffusion and learning
  12. Stage 3: sampling with guidance
  13. Convergence analysis
  14. Theoretical convergence
  15. Practical error bound
  16. ...and 28 more sections

Figures (10)

  • Figure 1: A framework of GGDOpt. (1) Generate a training set of points satisfying the chance constraint by solving a deterministic restricted problems. (2) Train a diffusion model with classifier-free guidance to learn the score of the conditional distribution. (3) Perform the reverse diffusion process with additional gradient guidance to sample from the product of the data distribution and the Boltzmann distribution.
  • Figure 2: A sketch map for U-Net structure of GGDOpt
  • Figure 3: The forward process of GGDOpt.
  • Figure 4: The sampling process of GGDOpt
  • Figure 5: Sampling process visualization of GGDOpt with median and quantiles
  • ...and 5 more figures