HoP: Homeomorphic Polar Learning for Hard Constrained Optimization

TL;DR

This paper tackles hard constrained optimization by introducing HoP, a learn-to-optimize framework that uses a homeomorphic mapping from neural network outputs in polar space to the feasible Cartesian domain. By enforcing a bijective, constraint-preserving transformation, HoP delivers end-to-end training with the objective as the loss and guarantees zero feasibility violations for star-convex constraints. The methodology includes 1-D and 2-D extensions and a semi-unbounded spherical mapping, plus a reconnection scheme to resolve radial stagnation during optimization. Across multiple synthetic benchmarks and a QoS-MISO WSR application, HoP achieves near-optimal solutions with zero constraint violations and substantial speedups over traditional solvers and penalty-based L2O methods, highlighting its practical impact for fast, reliable hard-constrained optimization.

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.

Figures (6)

• Figure 1: HoP is structured as follows: the gray box defines the problem considered in this paper, where the problem variable is expressed as $\mathbf{y}$, where $\mathbf{x}$ denotes problem parameters, and the objective function, respectively. The parameters $\mathbf{x}$ are fed into a NN, which contains a bounded activation function and produces a polar sphere vector comprising the direction vector $\mathbf{v}_{{\theta}}$ and the length scale $\bar{z}_r$. Using a homeomorphic mapping, the polar sphere vector is then transformed into Cartesian coordinates while strictly adhering to the original constraints. The warm-colored space on the right side represents the polar space corresponding to the polar sphere vector, while the green space is the Euclidean space for Cartesian coordinates. The loss function $f_\mathbf{x}(\mathbf{y})$ which can be trained end-to-end without requiring additional penalties or corrections.
• Figure 2: Illustration of the 2-D HoP principle: The larger green area represents the feasible region $\mathcal{Y}_\mathbf{x}$, while $\mathbf{y}_0\in\mathcal{Y}_\mathbf{x}$. NN in the HoP framework outputs the blue dot as an initial solution within unit circle, the yellow region, centered on $\mathbf{y}_0$ and constrained by a bounded activation function. The blue dot is then scaled along the direction specified by unit vector $\mathbf{v}_{\theta}$ to the red dot $\mathbf{\hat{y}}$. The scaling factor is defined as the ratio given in ${r\mathcal{R}(\mathbf{v}_{{\theta}},\mathcal{Y}_\mathbf{x})}$.
• Figure 3: Sketch of the spherical coordinate transformation for semi-unbounded constraints. The 2-D plane (green plane) is elevated to a higher-dimensional system, where the distance $\mathcal{R} (\mathbf{v}_{\theta},\mathcal{Y}_{\mathbf{x}})$ in direction $\mathbf{v}_{\theta}$ from $\mathbf{y}_0$ to boundary, is mapped as the boundary angle $\phi$. Then NN's output ratio $\bar{z}_r$ and $\phi$ is transformed by Eq. (\ref{['v_theta_transform']}) to $\psi$, the inclination angle of blue ray, in horizontal direction $\mathbf{v}_{\theta}$. Finally, $\mathbf{\hat{y}}$ is the intersection of the blue ray and green plane. Furthermore, points at infinity in the green space correspond to the angle on equator where $\psi = \frac{\pi}{2}$.
• Figure 4: Comparison of optimization trajectories in Cartesian and polar coordinates. The left column demonstrates disconnection issues in polar coordinates, while the right column shows improved behavior with reconnection strategies.
• Figure 5: Comparison of objective values across methods under different dimensional settings.

Theorems & Definitions (13)

• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Lemma 2.1: Tangent Space
• proof
• Lemma 2.2: Linear Independence of $\{\mathbf{v}_{\theta}, \mathbf{w}_1, \dots, \mathbf{w}_{d-1}\}$
• proof
• Theorem 2.3: Regularization of Jacobian Divergence
• proof