Feasibility & Optimality-Aware Reinforcement Learning

• FOARL is a reinforcement learning framework that enforces strict safety constraints while optimizing performance.
• It integrates statewise and regionwise methods to maintain feasibility throughout both learning and deployment.
• The approach delivers robust theoretical guarantees and scalable solutions for safety-critical domains like robotics and combinatorial optimization.

Feasibility-and-Optimality-Aware Reinforcement Learning (FOARL) refers to a class of reinforcement learning (RL) methodologies, algorithms, and theoretical frameworks that explicitly unify two requirements: (1) all learned or deployed policies must satisfy feasibility—meaning strict adherence to problem-specific constraints or safety requirements—and (2) the resulting policy should be as close to optimal as possible with respect to the quantitative performance objective. FOARL spans safety-critical RL, constrained policy improvement, safe exploration, efficient policy transfer, and multi-agent planning, and incorporates advances in both theoretical guarantees and empirical algorithms that address the unique dynamical aspects of RL compared to purely supervised learning or static planning settings.

1. Conceptual Foundations of Feasibility and Optimality

The dual concern of feasibility and optimality in RL is driven by real-world demands: many applications (e.g., robotics, cyber-physical systems, combinatorial optimization, operations research) impose stringent constraints that must never be violated, while simultaneously seeking to maximize (or minimize) a task-specific objective (reward, cost, efficiency, utility).

Feasibility encompasses ensuring that policy executions remain within state–action regions admissible under problem constraints. Examples include hard safety constraints in robotics, action admissibility in combinatorial settings, or temporal logic obligations in formal RL formulations.

Optimality, in the context of FOARL, refers to achieving performance—reward, average gain, minimal cost, or specified satisfaction objectives—that is as close as possible to the best attainable policy, subject to the feasibility constraints.

Historically, most RL frameworks optimized only for expected cumulative reward or return, often disregarding feasibility except when encoded as “soft” penalties. FOARL formalizes and operationalizes the requirement to treat feasibility as a first-class, strictly enforced property.

Key distinctions of FOARL from traditional RL include:

• Explicit state and policy feasibility characterization—often through feasibility functions, constraint value functions, or controlled invariant sets (Yang et al., 15 Apr 2024).
• Strict enforcement of feasibility in both learning and deployment, not just at convergence (Ma et al., 2021, Yang et al., 2023).
• Monotonic expansion or tracking of the feasible policy region throughout training (Yang et al., 15 Apr 2024, Yang et al., 2023).
• Rigorous theoretical containment and optimality guarantees, adapted to the Markovian, iterated nature of RL (Yang et al., 15 Apr 2024).
• Modular separation of feasibility-preserving “filters” from optimal exploration and exploitation, enabling safe exploration (Theile et al., 2023, Chen, 2023).

2. Mathematical Frameworks and Algorithmic Designs

FOARL approaches span several algorithmic forms and theoretical tools, including:

a) Statewise and Regionwise Safe Policy Optimization

• Statewise Lagrangian Methods: The Feasible Actor-Critic (FAC) approach introduces state-dependent Lagrange multipliers, enforcing safety constraints at each initial state, yielding the objective

$$\mathcal{L}_{\text{stw}}(\pi, \lambda) = \mathbb{E}_{s\sim d_0} \left[ -v^\pi(s) + \lambda(s)\left(v^{\pi}_C(s) - d\right) \right]$$

where $v^{\pi}_C(s)$ is the expected cost, $d$ is the threshold, and $\lambda(s)$ adapts per state (Ma et al., 2021).

• Regionwise Policy Iteration: Feasible Policy Iteration (FPI) decouples learning into safe (feasible) region identification (via a constraint decay function, or CDF), and uses a region-specific policy improvement process to maximize value within and expand the feasible region (Yang et al., 2023, Yang et al., 15 Apr 2024). Iteratively, CDFs contract toward zero (signaling safety), and value functions improve monotonically inside the feasible region.

b) Distribution Matching and Feasibility Decoupling

• Action Mapping Layer: In high-dimensional or disconnected action spaces, feasibility is enforced by learning a surjective mapping from latent variables to the feasible action set, trained via distribution matching (e.g., minimizing f-divergences between the generated and uniform feasible distributions), so all output actions obey the feasibility model (Theile et al., 2023).

c) Multi-objective Formulation and Lagrangian Dynamics

• Zero-Sum Multichain Lagrangian Formulation: In safety-constrained Markov Decision Processes (CMDPs) with probabilistic safety requirements, feasibility and optimality are encoded in a multi-objective Lagrangian game; value iteration is performed on the statewise Lagrangian, and modified off-policy Q-learning algorithms are constructed that incorporate log-barrier or dual variables for constraint satisfaction (Misra et al., 2023).

d) Temporal Logic, Average Reward, and Formal Specification

• Optimality-Preserving Reductions: Temporal specifications (in LTL or ω-regular form) are optimally translated to average reward formulations, using finite-memory reward machines. This allows generic RL methods to enforce complex constraint satisfaction and optimal behavior with average reward guarantees (Le et al., 16 Oct 2024).

3. Analysis of Feasibility Regions and Policy Improvement

The exploitation of policy improvement in FOARL is critically linked to the characterization of feasible regions under both optimal and intermediate policies. The paper of feasible regions is formalized through notions such as:

• Initial Feasibility Region (IFR): The set of states from which there exists a policy that locally satisfies the virtual-time (planning horizon) constraint.
• Endless Feasibility Region (EFR): The set of states from which safety can be maintained over an infinite horizon under some policy.
• Relationship Hierarchy: For any arbitrary RL policy $\pi$,

$X(\pi) \subseteq X_e(\pi) \subseteq X^*$

where $X(\pi)$ is the IFR, $X_e(\pi)$ is the EFR, and $X^*$ is the maximal endlessly feasible region (Yang et al., 15 Apr 2024).

Designing virtual-time constraints (including CBFs, reachability functions, or cost value functions) to ensure that $X(\pi)$ and $X_e(\pi)$ align with $X^*$ enables monotonic expansion of the safe region and ensures the optimal policy achieves the largest feasible region.

As RL algorithms progress, the feasible region of intermediate (non-optimal) policies is tracked to prevent catastrophic violations, and learning procedures are crafted to enlarge these regions iteratively toward $X^*$.

4. Integration with Deep RL, Representation Learning, and Applications

Recent FOARL techniques integrate feasibility awareness with modern function approximation and representation learning:

• Feasibility-Consistent Representation Learning: FCSRL learns state embeddings predictive of both dynamics and long-horizon safety via auxiliary feasibility consistency losses, which estimate a “smooth” feasibility score (such as a maximum discounted future cost) rather than relying on sparse constraint violations. The embedding is jointly trained to provide safety-awareness to the upper policy layers and enhances constraint estimation and sample efficiency (Cen et al., 20 May 2024).
• Application Domains:
  • Robotics: FOARL frameworks enforce kinematic, geometric, and safety constraints in mobile manipulation and navigation, using dense feasibility rewards and modular decoupling of feasibility and task objectives (Honerkamp et al., 2021).
  • Combinatorial Optimization: LLMs, after supervised fine-tuning, are further refined via FOARL to eliminate constraint violations in complex combinatorial problems, leveraging composite reward functions that combine feasibility and optimality measures (Jiang et al., 21 Sep 2025).
  • Automata and Formal Language Theory: Supervisory control and automaton-derived specifications are used to constrain RL exploration dynamically, guaranteeing that optimality is preserved despite restriction to specification-compliant behaviors (Chen, 2023).

5. Theoretical Guarantees and Trade-Offs

FOARL methods provide various forms of theoretical guarantees:

• Strict Constraint Satisfaction: By embedding feasibility directly in RL objectives, many FOARL approaches ensure that all iterates, not just the final policy, avoid constraint violations in feasible regions (Ma et al., 2021, Yang et al., 2023).
• Optimality in Safe Regions: Geometric convergence of policy/value iterates to the optimal safe policy in the maximal feasible region $X^*$ is established under contraction mappings or carefully designed BeLLMan operators (Yang et al., 2023, Yang et al., 15 Apr 2024).
• Policy Improvement Properties: Monotonic expansion of feasible regions and non-decreasing value inside these regions are guaranteed by regionwise update rules and properly designed virtual-time constraints.
• Feasibility-Induced Performance Bounds: When constraints induce infeasibility at some states, FOARL methods can mathematically distinguish feasible and infeasible states, back off to “least unsafe” behaviors, or deliver local guarantees (Ma et al., 2021).
• Sample Complexity and Scalability: By decoupling feasibility from optimality, or by leveraging focused rollouts (e.g., FSSS in MCTS planning), computational and sample complexity can be made independent of the full state space size (Asmuth et al., 2012).

Trade-offs arise between strictness of feasibility enforcement, computational overhead, convergence speed, and the size of the achievable region for optimal behavior, especially in high-dimensional or stochastic environments.

6. Extensions and Future Directions

FOARL remains a rapidly developing paradigm. Open problems and extensions include:

• Efficient Constraint Representation: Further development of feasibility functions, scalable to high-dimensional settings and nontrivial constraint sets, is needed. This touches on automated discovery of control barrier functions, reachability sets, or learning feasibility from data.
• Safe Exploration: Methods that guarantee exploration does not leave the feasible region, leveraging formal automata or model-based techniques, are an ongoing research area.
• Policy Transfer and Generalization: Leveraging successor features, convex coverage sets, and generalized policy improvement supports zero-shot optimal transfer of feasible policies to new objectives, provided the set of considered constraints is encoded (Alegre et al., 2022).
• Temporal Properties and Formal Guarantees: There is increasing interest in methods that guarantee temporal logic properties via optimality-preserving reductions to reward-based RL, making high-level specification-driven RL more practical (Le et al., 16 Oct 2024).
• Integration with Representation Learning: Advances in feasibility-consistent representation learning may offer new avenues to handle high-dimensional sensory states without sacrificing constraint satisfaction (Cen et al., 20 May 2024).

A plausible implication is that as FOARL techniques are deployed in larger and more complex systems (multi-agent, partially observable, or real-world safety-critical domains), rigorous feasibility-aware learning will become an essential component of RL pipelines—driving both formal safety guarantees and scalable optimality in practice.