Feasibility-Net: Constraint-Aware Neural Optimization

• Feasibility-Net is a framework for designing neural architectures that explicitly assess and enforce constraint satisfaction in optimization and planning problems.
• It employs modules like differentiable repair (FSNet) and feasibility tensor prediction to refine candidate solutions and guide robotic locomotion planning.
• The approach offers strong theoretical guarantees and empirical evidence of runtime improvements and robustness in both simulated and real-world applications.

Feasibility-Net denotes neural architectures and computational frameworks developed to assess, predict, or enforce feasibility—adherence to problem-specific constraints—in machine learning-driven optimization, planning, or control scenarios. The term has emerged in multiple research subdisciplines, notably constrained optimization for continuous variables and feasibility-aware planning in robotic locomotion. Central to these approaches is the explicit prediction, correction, or estimation of constraint satisfaction, either for individual solution candidates or within state-action spaces. Prominent instantiations include FSNet, an integrated feasibility-seeking neural optimization architecture (Nguyen et al., 31 May 2025), and the feasibility tensor predictor for locomotion policy planning (Luo et al., 8 Feb 2026).

1. Fundamental Problem Settings for Feasibility Networks

Feasibility-Net architectures address constrained optimization or planning problems where the notion of feasibility is explicitly defined by task-specific constraints, typically:

• Continuous Constrained Optimization: For parameters $x \in \mathbb{R}^d$, seek $y \in \mathbb{R}^n$ minimizing $f(y; x)$ subject to constraints $g(y; x) \leq 0$, $h(y; x) = 0$, with the feasible set $C(x) = \{ y \mid g(y; x) \leq 0,\, h(y; x) = 0 \}$ (Nguyen et al., 31 May 2025).
• Robotic Planning: For grid-based terrain planning, feasibility refers to the robot’s ability to execute a locally feasible trajectory under a specific locomotion policy for each candidate state and motion direction, as encoded in feasibility tensors $F(x, y, j) \in [0, 1]$ (Luo et al., 8 Feb 2026).

The common feature across domains is that feasibility is neither guaranteed by neural prediction nor easily enforced using standard penalty methods alone. Instead, dedicated network modules or subproblems are introduced to predict or enforce feasibility as a differentiable, learnable process.

2. Architectural Principles and Variants

Two distinct Feasibility-Net paradigms dominate current literature:

A. Feasibility-Seeking Optimization Networks (FSNet):

• Structure: Composed of an initial neural network predictor $y_0 = y_\theta(x)$ followed by a differentiable feasibility-seeking “repair” module $FS(y_0; x)$.
• Repair Process: Minimizes a violation metric, $$\phi(s; x) = \lVert h(s; x) \rVert_2^2 + \lVert g^+(s; x) \rVert_2^2$$, via $K$ steps of gradient-based unconstrained optimization, with $g^+(·) = \max(g(·), 0)$ applied elementwise.
• Inference: Outputs the repaired candidate $\hat{y} = s_K$, which (empirically) achieves near-zero constraint violation.

B. Feasibility Prediction for Planning (Locomotion Feasibility-Net):

• Structure: A two-headed model: a Variational Autoencoder (VAE) branch encodes local heightmaps and provides an out-of-distribution (OOD) signal, while a Multi-Layer Perceptron (MLP) “feasibility head” predicts a feasibility score $\hat{f} \in [0,1]$ for a given terrain patch and motion direction.
• Output Tensor: For discretized spatial locations $(x, y)$ and directions $j$, assembles $f_{x,y,j}$ into a feasibility tensor $F$ used for planning and policy selection (Luo et al., 8 Feb 2026).

This diversity in implementation reflects the broad applicability of the feasibility-seeking principle: enforce or estimate constraint attainability directly within an end-to-end neural pipeline.

3. Training Objectives and Loss Functions

Feasibility-Net models are trained via joint objectives combining solution quality with constraint satisfaction:

FSNet (Nguyen et al., 31 May 2025):

• Unconstrained Repair Loss: For repair variable $s$, solve $\min_s \phi(s; x)$ initialized at $y_0$.
• End-to-End Loss: $$F(y_0, \hat{y}; x) = f(\hat{y}; x) + \frac{\rho}{2} \| y_0 - \hat{y} \|_2^2$$, with empirical training objective $$L(\theta) = \frac{1}{S} \sum_{i=1}^S F\bigl(y_\theta(x^{(i)}), FS(y_\theta(x^{(i)}); x^{(i)})\bigr)$$. The $\rho$ term incentivizes the predictor to output near-feasible warm-starts.

Locomotion Feasibility-Net (Luo et al., 8 Feb 2026):

• MLP Head Loss: $\ell_\text{feas} = \| \hat{f} - r_v \|_2^2$, where $$r_v = \exp(-\lVert v_\text{actual} - v_\text{cmd} \rVert_2 / \sigma)$$ with $\sigma=0.25$.
• VAE Regularization: $$L_\text{VAE}(H) = L_\text{recon}(H) + \beta \,\text{KL}[q(z|H) \| \mathcal{N}(0, I)]$$ with $$L_\text{recon}(H) = \mathbb{E}_{z \sim q} \| H - \hat{H}(z) \|_2^2$$.
• Total Objective: $$L_\text{feas} = \ell_\text{feas} + \alpha L_\text{VAE}(H)$$, controlling for both ground-truth reward prediction and latent encoding quality.

This dual focus accommodates both the informative prediction of feasibility and robust policy learning or solution inference.

4. Integration into Inference and Planning Pipelines

Feasibility-Net components directly inform downstream inference or planning:

FSNet: The repaired output $\hat{y}$, created via an iterated, differentiable feasibility-seeking process, can be used directly as a decision variable for applications requiring strict constraint adherence. During training, the entire prediction-and-repair sequence is unrolled and backpropagated through, allowing the network to continually adapt its warm-start distribution for efficient repair (Nguyen et al., 31 May 2025).

Locomotion Feasibility-Net: Per-policy feasibility tensors are constructed over discretized spatial grids and headings. Their outputs are incorporated into classical planning via

• OOD-weighting: Using the VAE reconstruction loss to attenuate feasibility scores on out-of-distribution terrain.
• Policy Fusion: For $n$ policies, taking the pointwise maximum of weighted feasibility tensors.
• Graph Edge Costs: Edge cost $$c(x,y \to x',y') = 1 / (\hat{F}_\text{max}(x, y, j) + \epsilon)$$, so low predicted feasibility increases traversal cost.
• Path Computation: Classical graph search (Dijkstra’s algorithm) is used to find the minimum-cost route, with per-step policy assignment determined by $\arg\max_{i} \hat{F}_i(x, y, j)$ (Luo et al., 8 Feb 2026).

This structure ensures that only regions and transitions predicted to be feasible (by the underlying policy and task) are prioritized during planning.

5. Theoretical Guarantees and Performance Analysis

Feasibility-Net methods provide explicit theoretical and empirical guarantees:

FSNet:

• Repair Step Convergence: Under $L_\phi$-smoothness and Polyak–Łojasiewicz condition, gradient descent for the violation metric $\phi$ converges linearly. For a feasible solution $s^*$, $\phi(s_k) \to 0$ exponentially fast (Nguyen et al., 31 May 2025).
• NN Training Convergence: Stochastic gradient descent over the full pipeline achieves $L(\theta_T)$ with gradient norm converging at $O(T^{-1/2})$, with bias decaying exponentially in the number of unrolled repair steps $K$.
• Stationarity and Optimality: At convergence, warm-starts $y_0$ satisfy stationarity of $F(y_0, FS(y_0))$. In the limit $\rho \to \infty$, the solution approaches the original constrained optimum.
• Truncated Backpropagation: If only $K' < K$ repair steps are unrolled during backprop, solution quality and training bias degrade exponentially slowly in $K'$.

Locomotion Feasibility-Net:

• Sample Efficiency and Adaptivity: As specialized policies evolve, the predicted feasibility tensors track skill improvements, yielding success rates above 99% in all specialized terrain tasks and 98.6% in composite maps, compared to 0%–98.7% for a monolithic policy.
• Transfer to Real Robots: Real-world runs (e.g., Unitree A1 with LiDAR) realize segment success rates between 85–100% and an overall mixed traversal success of 70% versus 0% for the general policy (Luo et al., 8 Feb 2026).

This suggests that integrating feasibility prediction not only enhances constraint-satisfaction but yields robust performance in challenging, high-dimensional environments and real-world settings.

6. Applications and Empirical Results

Method-specific applications include:

Domain	Feasibility-Net Variant	Key Results
Constrained Optimization	FSNet (Nguyen et al., 31 May 2025)	Feasibility violations $10^{-4}$–$10^{-5}$, speedups 30×–3,000× over solvers, negative average gap versus IPOPT on nonconvex tasks
Robotic Planning	Locomotion Feasibility-Net (Luo et al., 8 Feb 2026)	Fused planners: $\geq$99% success on all terrain types, 70% real-world traversal success
Hierarchical Control	Feasibility-guided policy switching	Enables reliable and interpretable multi-policy fusion over heterogeneous terrains

In constrained optimization, FSNet provides feasible, high-quality solutions with orders-of-magnitude runtime improvements relative to classical solvers. In robotic planning, Feasibility-Net enables safe and performant traversal in environments requiring policy specialization across terrain types, with data-driven feasibility maps grounding the planning cost structure and enabling robust, interpretable expert switching.

7. Contextual Significance and Extensions

The Feasibility-Net paradigm marks a departure from penalty-centric or black-box approaches to constraint handling in neural inference. By embedding feasibility prediction or correction as a first-class, learnable component, these networks deliver provable guarantees on constraint satisfaction and facilitate modular, interpretable integration with legacy solvers and planning frameworks. A plausible implication is that feasibility modules can be increasingly used to connect learning-based predictors with model-based optimization or planning elements, acting as a precise interface between data-driven skill and logic-based task specification.

Research continues to address the generalization of Feasibility-Net architectures to broader classes of constraints (e.g., nonconvex, nonsmooth, hybrid systems) and to settings involving discrete variables or hierarchical multi-agent coordination. The combination of theoretical convergence, flexible enforcement via unconstrained surrogates, and seamless integration into classical inference pipelines distinguishes Feasibility-Net as an influential design principle for next-generation machine learning in constrained environments (Nguyen et al., 31 May 2025, Luo et al., 8 Feb 2026).