Feasibility-Aware Learning Framework

• Feasibility-aware learning frameworks are methodologies that embed hard and soft constraints directly into learning-based decision making.
• They utilize techniques like explicit constraint enforcement, dual optimization, and distribution matching to guarantee feasibility in complex systems.
• These frameworks enhance safety and efficiency in diverse applications such as robotics, power systems, combinatorial optimization, and control.

Feasibility-aware learning frameworks collectively refer to a family of methodologies and architectures designed to ensure, represent, or exploit the feasibility structure (i.e., constraint satisfaction under nontrivial hard or soft requirements) within learning-based decision-making and control processes. These frameworks systematically incorporate feasibility considerations into optimization, policy synthesis, supervised and reinforcement learning, and imitation protocols, addressing challenges in domains such as safe reinforcement learning, constrained control, combinatorial optimization, power systems, robot learning, and transfer learning. Approaches span explicit feasibility constraint learning, feasibility-driven loss formulations, dual optimization with sample-wise feasibility guarantees, calibration/tightening techniques, distribution-matching for feasible action sets, and statistical confidence-based constraint satisfaction.

1. Core Principles and Problem Formulations

Feasibility-aware learning frameworks depart from canonical empirical risk minimization and unconstrained policy learning by enforcing feasibility as a first-class objective or constraint:

• Explicit Feasibility Constraints in Optimization-based Control: In quadratic program (QP) frameworks for affine dynamical systems, constraints such as control barrier functions (CBFs) and high-order CBFs (HOCBFs) are incorporated to ensure safety and stabilization. Feasibility-aware learning strategies address the challenge posed by online QP infeasibility under tight bounds or high relative degree through state-dependent constraint learning (Xiao et al., 2023, Liu et al., 7 Dec 2025).
• Feasibility-oriented Representation and Policy Learning in RL: In safe RL with cost budgets, frameworks like Feasibility Consistent Safe Reinforcement Learning (FCSRL) learn latent representations and feasibility scores, using bootstrapped future cost metrics to separate feasible from infeasible regions for constrained policy optimization (Cen et al., 20 May 2024, Tao, 13 May 2025).
• Feasibility-centric Decision-focused Learning: Predict-then-optimize paradigms, when targeting constraint parameters, risk loss of feasibility due to prediction error. Feasibility-aware DFL simultaneously penalizes infeasibility of predicted solutions and suboptimality when the true optimizer is infeasible under predicted parameters, allowing tunable trade-off control (Mandi et al., 6 Oct 2025).
• Distribution-matching for Feasible Action Generation: Feasibility-aware RL splits policy learning into feasibility (learning a uniform generator over a feasible set via f-divergence minimization) and reward-optimization steps, permitting explicit, scalable coverage of disconnected feasible action sets (Theile et al., 2023).
• Feasible Learning vs. ERM: Feasible Learning (FL), or sample-centric feasibility-aware optimization, constrains the per-sample loss to remain below a fixed threshold $\epsilon$, inducing tail risk control and dynamic sample re-weighting through primal-dual optimization (Ramirez et al., 24 Jan 2025).

2. Methodologies for Feasibility Learning and Regularization

Learning-based Constraint Augmentation

• Sampling-based Constraint Learning: For each type of unsafe region (regular or irregular), the feasible region is empirically learned by sampling system states and solving the QP forward for $T$ steps, labeling the feasibility outcome, and training classifiers (such as SVMs) to define separating hyperplanes. The learned classifier $H_j(z)\geq0$ is enforced via an additional HOCBF in the QP, recursively improving feasibility coverage through feedback training until the online QP infeasibility rate falls below tolerance (Xiao et al., 2023).

Feasibility-consistent Representations and Scores

• Max-discounted Future Cost: The feasibility score $F^\pi(s,a)$ is defined as the expected maximum discounted cost across future trajectories, interpreted as the probability of constraint satisfaction over an infinite horizon. Representation learning employs self-supervised dynamics-consistency and KL-regression feasibility-consistency losses to anchor embeddings in both dynamics and safety information (Cen et al., 20 May 2024).
• Hamilton-Jacobi Reachability-inspired Feasibility Score: Offline RL approaches use reachability analysis to label (s,a) tuples by bounded future cost risk, forming the basis for CVAE training, safety classifier labeling, and pessimistic Q-value estimation that penalizes unsafe or out-of-distribution actions (Tao, 13 May 2025).

Dual/Penalized Optimization for Feasibility

• Primal-dual Sample-weighting: Feasible Learning reframes training as constrained optimization (per-sample loss $\leq\epsilon$), with dual ascent adjusting sample weights according to violation magnitude; optional slack variables yield relaxed formulations (Resilient FL) with explicit norm penalties and guarantees (Ramirez et al., 24 Jan 2025).
• Calibrated Constraint Tightening: For convex-constrained optimization problems solved via deep neural networks, preventive learning calibrates each training constraint by a margin $\eta$ computed via min-max optimization over the parameter set so that DNN-predicted solutions always satisfy the true constraints—universal feasibility holds if worst-case tightened-constraint violation does not exceed the calibration gap $\Delta$ (Zhao et al., 2021).
• Statistical Feasibility via Conformal Sets: In mixed-integer constraint learning, pointwise feasibility is replaced by conformal sets whose coverage is calibrated to guarantee ground-truth feasibility (with probability at least $1-\alpha$), yielding probabilistic guarantees and avoiding model exploitation errors (Ovalle et al., 4 Jun 2025).

Feasibility Regularization and Threshold-based Selection

• Feasibility Regularization in Graph-based Learning: In graph neural network surrogates for optimal power flow (OPF), a physics-aware feasibility penalty is added. Predicted quantities (generation, voltage, line flows) are projected or regularized to satisfy operational limits, achieving major reductions in constraint violation and scalability (Liu et al., 2022).
• Threshold-aware Coverage in Combinatorial Optimization: In ML-based MIP heuristics, coverage (fraction of integer variables fixed by the predictor) is learned and jointly optimized to exploit a sharp threshold interval for feasibility and solution quality, bridging the gap between classifier accuracy and problem-optimal bounds (Yoon et al., 2023).

3. Algorithms and Training Protocols

Feasibility-aware learning frameworks typically employ multi-stage or feedback training algorithms:

• Classifier-augmented QP Training: The recurrent algorithm iterates sampling, feasibility evaluation, classifier retraining, and stopping based on infeasibility rate convergence, yielding robust separators for feasible states (Xiao et al., 2023).
• Representation Update Loops: Feasibility-consistent RL alternates updates to the encoder via representation loss (dynamics and feasibility heads), with periodic soft updates to the momentum encoder and policy/value functions trained on stabilized embeddings (Cen et al., 20 May 2024).
• Two-phase Distribution-matching: Feasibility-polices are trained offline with simulated or oracle feasibility checks, then downstream RL/planning is restricted to safe latent indices (Theile et al., 2023).
• Adversarial Sample Aware Training: Feasibility-guaranteed networks are refined via adversarial sampling and retraining on near-violating parameter regions, preserving feasibility while boosting optimality (Zhao et al., 2021).
• Conformal Calibration and MIP Integration: Surrogate models for constraints are post-trained on calibration nonconformity scores, yielding quantile thresholds for conformal feasibility sets; these sets are embedded as constraints in subsequent mixed-integer programs (Ovalle et al., 4 Jun 2025).

4. Empirical Evaluations and Impact

Quantitative evaluation across domains reveals the operational and safety impact of feasibility-aware learning frameworks:

Framework	Application Domain	Feasibility Achieved	Additional Metric
CBF + ML constraint (Xiao et al., 2023)	Nonlinear robot control	Infeasibility < 0.6%	Safety, reachability, task success
FCSRL (Cen et al., 20 May 2024)	Safe RL (robotics, images)	Cost ≤ budget	Higher normalized reward
Threshold-aware MIP (Yoon et al., 2023)	Mixed-integer programs	OG 0.45% (workload)	10× gap reduction over SCIP
DeepOPF+ (Zhao et al., 2021)	DC optimal power flow	100% guarantee	<0.2% optimality loss, ×228 speedup
C-MICL (Ovalle et al., 4 Jun 2025)	Data-driven optimization	≥90%–95% feasi guar.	Modest objective gap, ×10–×100 faster
FABCO (Takahashi et al., 12 Mar 2025)	Imitation learning (robot)	93–97% w/feedback	Workload, robustness under covariate shift

These results demonstrate that feasibility-aware learning architectures deliver improved safety, constraint satisfaction, robustness, and efficiency while maintaining competitive objective or reward metrics.

5. Theoretical Guarantees and Interpretability

Feasibility guarantees span several mathematical regimes:

• Forward-invariance and recursive feasibility in QP-based optimal control (CBFs, HOCBFs) via constraints defined from learned classifiers and theoretical recursions (Xiao et al., 2023, Liu et al., 7 Dec 2025).
• Statistical coverage in constraint learning: conformal sets encode finite-sample, distribution-free guarantees on feasibility probability, supporting probabilistic decision processes (Ovalle et al., 4 Jun 2025).
• Calibration margin bounds: Explicit characterization of constraint tightening is proved to retain feasibility across the full input parameter domain, given sufficient model expressivity (Zhao et al., 2021).
• Risk and regret lower bounds: In transfer learning, formal existence theorems for feasible transfer mappings and risk quantification link feasibility directly to transfer regret (Cao et al., 2023, Cao et al., 2023).

These properties yield interpretable, demonstrably robust learning systems even in challenging, nonconvex, or high-dimensional domains.

6. Representative Domains and Extensions

Feasibility-aware learning is established across diverse domains:

• Control: Safety-constrained optimal control, real-time robot navigation under explicit unsafe set characterization, autonomous driving.
• Reinforcement Learning: Safe RL with explicit cost budgets, offline RL with pessimistic critics and feasibility-driven representation learning.
• Optimization: Mixed-integer programming, constraint-based combinatorial optimization with learned thresholds for feasibility.
• Imitation Learning: Observation-based and multi-agent demonstration filtering using feasibility metrics, weighted BC and state-only imitation.
• Power Systems: AC and DC optimal power flow with topology-adaptive, feasibility-regularized prediction via GNNs.
• Transfer Learning: Mathematical formulations resolving the existence and evaluation of feasible transfer procedures.

Extensions include probabilistic feasibility calibration, task-agnostic policy architectures, differentiable simulator integration, and curriculum- or schedule-based trade-offs in feasibility-optimality learning.

7. Open Problems and Future Directions

Emerging challenges and frontiers include:

• Scalability and Efficiency: While calibration and feedback methods guarantee feasibility, processing cost for nontrivial high-dimensional/online domains remains critical—integrating fast solvers, efficient uncertainty quantification, and adaptive data augmentation are active research areas.
• Generalization to Out-of-Distribution Scenarios: Incorporating explicit OOD detection and pessimistic estimation into safe RL frameworks widens robust applicability (Tao, 13 May 2025).
• Statistical Feasibility under Model Uncertainty: Data-driven and distributional frameworks (e.g., conformal prediction, f-divergence distribution matching) provide finite-sample guarantees crucial for safety in deployment.
• Balancing Suboptimality and Feasibility: Feasibility-aware decision-focused learning enables fine control over performance metrics as required by particular operational regimes (Mandi et al., 6 Oct 2025).

A plausible implication is that with further integration of machine learning and mathematical optimization, feasibility-aware methods will become foundational for deploying autonomous systems and data-driven decision architectures in safety-critical applications.

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Key references: (Xiao et al., 2023, Cen et al., 20 May 2024, Ramirez et al., 24 Jan 2025, Tao, 13 May 2025, Theile et al., 2023, Liu et al., 2022, Zhao et al., 2021, Yoon et al., 2023, Mandi et al., 6 Oct 2025, Ovalle et al., 4 Jun 2025, Liu et al., 7 Dec 2025, Takahashi et al., 12 Mar 2025, Cao et al., 2021, Cao et al., 2023, Cao et al., 2023, Kraipatthanapong et al., 21 Oct 2025).