HoP: Homeomorphic Polar Learning for Hard Constrained Optimization
Ke Deng, Hanwen Zhang, Jin Lu, Haijian Sun
TL;DR
HoP introduces a novel neural network approach with homeomorphic mapping to efficiently solve hard-constrained optimization problems, ensuring solutions are both near-optimal and feasible without additional penalties.
Contribution
The paper presents Homeomorphic Polar Learning (HoP), a new neural network method embedding homeomorphic mapping for improved constrained optimization.
Findings
HoP outperforms existing L2O methods in solution quality.
HoP strictly maintains feasibility across tested tasks.
HoP achieves solutions closer to the optimum in diverse applications.
Abstract
Constrained optimization demands highly efficient solvers which promotes the development of learn-to-optimize (L2O) approaches. As a data-driven method, L2O leverages neural networks to efficiently produce approximate solutions. However, a significant challenge remains in ensuring both optimality and feasibility of neural networks' output. To tackle this issue, we introduce Homeomorphic Polar Learning (HoP) to solve the star-convex hard-constrained optimization by embedding homeomorphic mapping in neural networks. The bijective structure enables end-to-end training without extra penalty or correction. For performance evaluation, we evaluate HoP's performance across a variety of synthetic optimization tasks and real-world applications in wireless communications. In all cases, HoP achieves solutions closer to the optimum than existing L2O methods while strictly maintaining feasibility.
Peer Reviews
Decision·Submitted to ICLR 2026
* The proposed method is strong in premise and innovative, providing a clean and cost-effective way to enable feasibility enforcement in neural networks (for certain classes of constraint sets) by leveraging homeomorphic mappings. * The method is very well-presented, with helpful pedagogical examples for the 1D and 2D case before extending to the semi-unbounded and high-dimensional cases. In general, the writing is strong. * The trick to resolve stagnation of gradient descent with polar coordin
Major: * The method is limited to optimization problems with convex constraint sets, or specialized nonconvex structure (ray intersects the boundary exactly once), but this limitation is not stated up-front. The scope of the method should be stated much more clearly early on. (Appendix G does provide some exploration into the non-convex case, but it is very initial.) * Some of the offline costs associated with the framework are not transparently disclosed. This also affects what sets of parametr
The strengths of the paper are listed below. - The paper considers the important and fundamental problem of enforcing hard constraints for L2O. - The proposed use of polar coordinates to reformulate the neural network outputs is interesting. - The paper includes experimental evaluations on multiple benchmark problems.
The weaknesses are given below. - The proposed method is limited to problems with convex constraints. However, learning-to-optimize (L2O) approaches are often expected to handle broader classes of non-convex and complex optimization problems, which restricts the general applicability of this work. - The approach requires redefining the semantics of neural network outputs through coordinate transformation and mapping, making it difficult to integrate with existing neural network architectures. Co
The paper addresses the important problem of learning to optimize under constraints. The experimental results, on synthetic problems and a communications application, are positive. The idea of transforming a constrained problem into an unconstrained one using a radial (polar) mapping is mathematically elegant.
Despite the strengths described above, the paper raises several concerns that I believe must be addressed before publication. I have reviewed a previous version of this paper, and while it has been moderately revised, many of the major concerns raised earlier remain unresolved. **(1) Contribution and Novelty:** The ideas presented in this paper appear closely related to Liang et al. (2023). Although the authors cite this prior work, the paper lacks an explicit and careful discussion of how th
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Taxonomy
TopicsMachine Learning and ELM · Domain Adaptation and Few-Shot Learning