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Differentiable Projection-based Learn to Optimize in Wireless Network-Part I: Convex Constrained (Non-)Convex Programming

Xiucheng Wang, Xuan Zhao, Nan Cheng

TL;DR

This paper proposes a projection-based method that projects any infeasible NN output onto the feasible domain, thus guaranteeing strict adherence to the constraints without compromising the NN's optimization capability.

Abstract

This paper addresses a class of (non-)convex optimization problems subject to general convex constraints, which pose significant challenges for traditional methods due to their inherent non-convexity and diversity. Conventional convex optimization-based solvers often struggle to efficiently handle these problems in their most general form. While neural network (NN)-based approaches offer a promising alternative, ensuring the feasibility of NN-generated solutions and effectively training the NN remain key hurdles, largely because finite-capacity networks can produce infeasible outputs. To overcome these issues, we propose a projection-based method that projects any infeasible NN output onto the feasible domain, thus guaranteeing strict adherence to the constraints without compromising the NN's optimization capability. Furthermore, we derive the objective function values for both the raw NN outputs and their projected counterparts, along with the gradients of these values with respect to the NN parameters. This derivation enables label-free (unsupervised) training, reducing reliance on labeled data and improving scalability. Experimental results demonstrate that the proposed projection-based method consistently ensures feasibility.

Differentiable Projection-based Learn to Optimize in Wireless Network-Part I: Convex Constrained (Non-)Convex Programming

TL;DR

This paper proposes a projection-based method that projects any infeasible NN output onto the feasible domain, thus guaranteeing strict adherence to the constraints without compromising the NN's optimization capability.

Abstract

This paper addresses a class of (non-)convex optimization problems subject to general convex constraints, which pose significant challenges for traditional methods due to their inherent non-convexity and diversity. Conventional convex optimization-based solvers often struggle to efficiently handle these problems in their most general form. While neural network (NN)-based approaches offer a promising alternative, ensuring the feasibility of NN-generated solutions and effectively training the NN remain key hurdles, largely because finite-capacity networks can produce infeasible outputs. To overcome these issues, we propose a projection-based method that projects any infeasible NN output onto the feasible domain, thus guaranteeing strict adherence to the constraints without compromising the NN's optimization capability. Furthermore, we derive the objective function values for both the raw NN outputs and their projected counterparts, along with the gradients of these values with respect to the NN parameters. This derivation enables label-free (unsupervised) training, reducing reliance on labeled data and improving scalability. Experimental results demonstrate that the proposed projection-based method consistently ensures feasibility.

Paper Structure

This paper contains 15 sections, 1 theorem, 19 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Related Works
  4. Differentiable Projection-Based L2O Method
  5. Convex Projection Theorem
  6. Projection-Based Feasibility Guarantee Method
  7. Unsupervised Training Method
  8. Applications on Beamforming Optimization
  9. Security Communication Rate Under Quality of Service Constraints
  10. Green Communication Optimization Under QoS Constraints
  11. Numerical Results
  12. Simulation Setting
  13. Eve Users' SINR Minimization
  14. Transmission Power Minimization
  15. Conclusion

Key Result

Lemma 1

For a closed convex set $\mathcal{C}\in\mathbb{C}^n$, and $y\in \mathbb{C}^n$ the projection Proj$(\bm{y})$ is unique.

Figures (7)

  • Figure 1: Illustration of projection-based L2O for convex constrained (non-)convex programming.
  • Figure 2: The comparison of different algorithms on $K$.
  • Figure 3: The comparison of different algorithms on $N$.
  • Figure 4: The comparison of different algorithms on $\gamma$.
  • Figure 5: The power cost comparison of different algorithms on $K$.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Lemma 1
  • proof