Feasibility-Guided Exploration

• Feasibility-Guided Exploration is defined as a family of strategies that prioritize safety, validity, and constraint satisfaction over pure objective optimization.
• It employs explicit feasibility estimation, frontier targeting, and conservative sampling to safely navigate hierarchical and constraint-rich search spaces.
• FGE integrates methods like safe classifiers, Bayesian models, and sequential design to expand feasible zones and enhance robust exploration in various domains.

Searching arXiv for papers on Feasibility-Guided Exploration and closely related formulations. Feasibility-Guided Exploration (FGE) denotes a family of exploration strategies in which feasibility—safety, validity, non-crash operation, or constraint satisfaction—is the primary signal used to allocate samples, resets, or evaluations. The term is used in several related senses across the literature: as a named reinforcement-learning method for parameter-robust avoid problems with unknown feasibility, as a feasible-first strategy for constrained deployment optimization, as a sequential design method for valid/invalid simulation regions, as a Bayesian feasibility-learning procedure for expensive constraints, and as an equilibrium-oriented safe-exploration framework based on maximal feasible zones and uncertain models (So et al., 17 Feb 2026, Lysenstøen, 27 Apr 2026, Heese et al., 2019, Rahat et al., 2020, Yang et al., 31 Jan 2026). Across these formulations, the common premise is that objective-only or return-only optimization can be systematically misaligned with the problem of discovering, certifying, or enlarging the feasible part of a search space.

1. Conceptual scope and motivating problems

FGE arises when the dominant difficulty is not merely optimizing a scalar objective, but identifying where optimization is even admissible. In robust avoid control, the mismatch is between expected-return RL and the worst-case reach-avoid objective of safety analysis; standard deep RL can be unsafe on low-probability initial states that still lie within the operating domain. In crash-prone deployment optimization, the search space is hierarchical and mixed-variable, and many configurations are invalid because they crash, exceed memory limits, or violate latency constraints. In simulation-driven design, regions of parameter space can yield non-convergent or unphysical outcomes. In expensive constrained search, feasibility is only observable through costly simulations. In safe exploration, restricting actions to a feasible zone is only the first step; the central question becomes how the maximum feasible zone can be enlarged without violating constraints (So et al., 17 Feb 2026, Lysenstøen, 27 Apr 2026, Heese et al., 2019, Rahat et al., 2020, Yang et al., 31 Jan 2026).

A recurring feature of FGE is that feasibility is treated as structurally different from ordinary objective information. Unsafe or invalid outcomes are often asymmetric: a successful safe trajectory can certify feasibility, whereas a failed trial may be ambiguous because failure can result either from intrinsic infeasibility or from an immature policy, a poor surrogate, or insufficient model knowledge. This asymmetry motivates conservative classifiers, feasibility-first curricula, boundary-focused acquisition functions, or worst-case forward-invariance computations rather than direct maximization of expected objective value alone (So et al., 17 Feb 2026, Heese et al., 2019, Rahat et al., 2020).

The literature does not use the label uniformly. In some works, FGE is the name of a specific algorithm; in others, it is an organizing principle instantiated by a concrete procedure such as Thermal Budget Annealing (TBA), Probability of Boundary and Entropy (PBE), or Safe Equilibrium Exploration (SEE). This suggests that FGE is best understood as a methodological pattern rather than a single canonical algorithm (Lysenstøen, 27 Apr 2026, Rahat et al., 2020, Yang et al., 31 Jan 2026).

2. Mathematical representations of feasibility

The formal object called “feasible” varies by domain, but it is always defined before exploration proceeds.

In parameter-robust avoid RL, deterministic discrete-time dynamics are written as

$$x_{t+1} = f(x_t, u_t; \theta), \qquad x_0 = s_0(\theta),$$

with a safety function $h_\theta : \mathcal{X} \to \mathbb{R}$ whose strict zero-superlevel set

$A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$

is the unsafe set. Feasibility for an initial condition $(x_0,\theta)$ means that there exists $\pi_\theta$ such that $x_t \notin A_\theta$ for all $t \ge 0$. The Hamilton–Jacobi-style value

$$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$

and the RL-compatible avoid value

$$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$

have coincident zero-level sets for feasible initial conditions. The robust problem is therefore reformulated as maximizing the size of a feasible subset $\Theta' \subseteq \Theta$ on which safety can be maintained for all time, with coverage

$h_\theta : \mathcal{X} \to \mathbb{R}$0

used to quantify how much of the parameter domain is solved (So et al., 17 Feb 2026).

In constrained deployment optimization, the search space is hierarchical and mixed-variable:

$h_\theta : \mathcal{X} \to \mathbb{R}$1

with activation conditions $h_\theta : \mathcal{X} \to \mathbb{R}$2 determining the active set

$h_\theta : \mathcal{X} \to \mathbb{R}$3

Hard constraints are written as $h_\theta : \mathcal{X} \to \mathbb{R}$4, crashes are indicated by $h_\theta : \mathcal{X} \to \mathbb{R}$5, and the feasible set is

$h_\theta : \mathcal{X} \to \mathbb{R}$6

Constraint violation is summarized by

$h_\theta : \mathcal{X} \to \mathbb{R}$7

This makes feasibility explicitly combinatorial, hardware-dependent, and cost-asymmetric because wall-clock evaluation cost $h_\theta : \mathcal{X} \to \mathbb{R}$8 can vary widely across trials (Lysenstøen, 27 Apr 2026).

In the chemical-process setting, feasibility is binary: a point is valid if the simulation converges and is physically reasonable, and invalid otherwise. The method therefore learns a kernel SVM classifier with RBF kernel and Platt-calibrated probability

$h_\theta : \mathcal{X} \to \mathbb{R}$9

then uses this probability both for boundary localization and for entropy-based uncertainty sampling (Heese et al., 2019).

In Bayesian feasible-space learning under expensive constraints, the feasible set is

$A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$0

with each constraint modeled by an independent GP regression surrogate. For

$A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$1

the per-constraint feasibility probability is $A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$2, and the joint feasibility probability is

$A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$3

Here feasibility is probabilistic, but the target remains the geometry of the feasible subset rather than direct objective optimization (Rahat et al., 2020).

In SEE, feasibility is defined on a set-valued uncertain model $A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$4. A zone $A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$5 is feasible if $A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$6 and

$A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$7

The maximal feasible zone under the current uncertain model is

$A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$8

so feasibility is expressed as worst-case forward invariance rather than as a binary label or probability (Yang et al., 31 Jan 2026).

Setting	Feasibility object	Representative formulation
Parameter-robust avoid RL	Safe-for-all-time initial conditions	$A_\theta = \{x \in \mathcal{X} : h_\theta(x) > 0\}$9 and $(x_0,\theta)$0
Deployment optimization	Non-crash, constraint-satisfying configurations	$(x_0,\theta)$1
Industrial simulation	Valid vs invalid runs	Platt-calibrated $(x_0,\theta)$2
Bayesian feasible-space learning	Joint satisfaction of multiple constraints	$(x_0,\theta)$3
Safe exploration	Forward-invariant safe zone under uncertainty	$(x_0,\theta)$4

3. Recurrent algorithmic structure

Despite substantial domain differences, several algorithmic motifs recur across FGE formulations.

The first motif is explicit feasibility estimation. In robust avoid RL, FGE learns a classifier $(x_0,\theta)$5 for parameter feasibility using a mixture model that exploits asymmetric labels: safe observations are reliable positives, whereas unsafe observations may reflect policy suboptimality. Thresholding $(x_0,\theta)$6 yields a conservative classifier with zero false positives on infeasible $(x_0,\theta)$7 and controlled false negatives on feasible $(x_0,\theta)$8. In the chemical-process method, the kernel SVM provides a feasibility boundary and calibrated probabilities for entropy-based acquisition. In Bayesian feasibility search, GP posteriors over constraints induce $(x_0,\theta)$9 directly. In SEE, the uncertain model itself is the feasibility object, and model refinement corresponds to shrinking transition sets while preserving well-calibration (So et al., 17 Feb 2026, Heese et al., 2019, Rahat et al., 2020, Yang et al., 31 Jan 2026).

The second motif is frontier targeting rather than average-case sampling. Robust avoid FGE rejection-samples from parameters satisfying $\pi_\theta$0, thereby probing the currently estimated infeasible set to discover harder but feasible conditions. The 2019 sequential design method uses Shannon entropy of the calibrated feasibility probability,

$\pi_\theta$1

plus a repulsion term

$\pi_\theta$2

to concentrate samples near uncertain boundaries while maintaining diversity. PBE similarly multiplies a boundary term and an entropy term:

$\pi_\theta$3

TBA does not use a probabilistic feasibility boundary in Phase 1, but its acceptance rule explicitly minimizes violation $\pi_\theta$4 before any objective-driven acceptance is allowed (So et al., 17 Feb 2026, Heese et al., 2019, Rahat et al., 2020, Lysenstøen, 27 Apr 2026).

The third motif is separation or alternation between feasible-set expansion and objective exploitation. In robust avoid RL, policy learning is coupled with best-response parameter selection and a rehearsal buffer over previously discovered hard-but-feasible parameters. In deployment optimization, Phase 1 performs crash-aware simulated annealing to map feasible regions, and Phase 2 warm-starts constrained TPE with all Phase 1 trial data, including infeasible and crashed outcomes. In SEE, the alternation is formal: each iteration computes the maximum feasible zone under the current uncertain model and then computes the least uncertain model consistent with newly collected safe data (So et al., 17 Feb 2026, Lysenstøen, 27 Apr 2026, Yang et al., 31 Jan 2026).

A fourth motif is conservatism in the face of asymmetric error costs. False positives in feasibility prediction can be safety-critical or budget-wasting, so several FGE methods are designed to avoid them. The RL variant states zero false positives on infeasible parameters under its thresholded classifier. TBA never accepts crashed configurations as the current state and uses timeouts plus temporary subspace blacklisting to suppress repeatedly failing categorical values. SEE guarantees zero constraint violations by executing actions only within the current feasible zone, which is forward invariant under a well-calibrated uncertain model (So et al., 17 Feb 2026, Lysenstøen, 27 Apr 2026, Yang et al., 31 Jan 2026).

4. Major instantiations

Domain	Instantiation	Core mechanism
Robust avoid control	FGE	Feasibility-aware curriculum, conservative classifier, saddle-point PPO/FTRL
ML deployment optimization	TBA as FGE	Crash-aware simulated annealing, timeouts, blacklisting, warm-started c-TPE
Chemical-process simulation	Sequential design through feasibility modeling	SVM entropy, KRR objective surrogate, distance repulsion
Expensive constrained search	PBE	Joint feasibility boundary probability times GP entropy
Safe exploration	SEE as FGE	Risky Bellman feasible-zone expansion and Lipschitz graph refinement

In robust avoid control, FGE is designed for parameter-robust avoid problems when the feasibility of the parameter set is unknown. Its central optimization is

$\pi_\theta$5

which simultaneously enlarges a feasible subset and learns a policy safe for all time on that subset. The practical implementation uses PPO with reward

$\pi_\theta$6

a feasibility-aware reset mixture over base, exploration, and rehearsal distributions, and an approximate saddle-point update in which best-response parameters are selected from the estimated feasible set via a policy-conditioned classifier $\pi_\theta$7 (So et al., 17 Feb 2026).

In deployment optimization, TBA operationalizes FGE as a feasible-first front end for hostile hierarchical spaces. Phase 1 uses crash-aware simulated annealing with an adaptive temperature schedule

$\pi_\theta$8

structural mutation schedule

$\pi_\theta$9

timeouts at $x_t \notin A_\theta$0, and temporary blacklisting after 3 consecutive crash or infeasible outcomes with 8-trial cooldown. Phase 2 hands all history to constrained TPE, whose acquisition is described schematically as

$x_t \notin A_\theta$1

The method is specifically intended for small budgets and hidden crash zones, where cold-start TPE can prematurely exploit a narrow structural subset (Lysenstøen, 27 Apr 2026).

The 2019 simulation-driven method exemplifies FGE through a tri-component utility

$x_t \notin A_\theta$2

where $x_t \notin A_\theta$3 is feasibility entropy, $x_t \notin A_\theta$4 is a min-max-scaled objective surrogate from kernel ridge regression, and $x_t \notin A_\theta$5 is feature-space distance repulsion. The next sample is chosen by

$x_t \notin A_\theta$6

with differential evolution used for global optimization of the utility. This formulation makes feasibility uncertainty primary, but allows optional steering toward high-value feasible regions through the weight vector $x_t \notin A_\theta$7 (Heese et al., 2019).

PBE instantiates FGE in Bayesian feasible-space learning by constructing an acquisition that is high only when a point is both near the boundary between feasible and infeasible sets and jointly uncertain across constraints. The boundary probability is

$x_t \notin A_\theta$8

and the entropy term is

$x_t \notin A_\theta$9

Their product defines the acquisition. This is explicitly a feasibility learner rather than a constrained optimizer: its purpose is to delineate $t \ge 0$0 under tight evaluation budgets (Rahat et al., 2020).

SEE casts FGE as a fixed-point problem between feasibility and model uncertainty. The optimal constraint-decay function $t \ge 0$1 satisfies the risky Bellman equation

$t \ge 0$2

and the maximal feasible zone is the zero-level set

$t \ge 0$3

Model refinement is implemented through a graph whose nodes are candidate transitions $t \ge 0$4; nodes are pruned either by direct data consistency inside the feasible zone or by Lipschitz inconsistency, approximated through a sufficient neighbor-color condition. The alternating updates converge, in finite discrete spaces, to an equilibrium $t \ge 0$5 satisfying

$t \ge 0$6

(Yang et al., 31 Jan 2026).

5. Empirical behavior and evaluation protocols

Empirical evaluation of FGE is highly domain-specific, but the reported outcomes consistently measure how much useful feasible coverage is obtained under limited budget.

In robust avoid RL, benchmarks include ToyLevels, Dubins driving, MuJoCo mjx Hopper with floating platforms, MuJoCo mjx Cheetah on a moving platform, an F-16 extended-trail formation-keeping task, and Kinetix Lunar Lander variants with RGB image observations. Reported metrics include safety rate conditioned on feasibility, coverage gain versus domain randomization, coverage loss versus domain randomization, and unique coverage. The method is reported to learn policies with over 50% more coverage than the best existing methods on challenging initial conditions across MuJoCo tasks, with the largest coverage gains and minimal coverage loss. Case studies attribute this to concentration of samples on rare but feasible conditions, including “overtake” scenarios in driving and gap regions in Hopper; once other regions are solved, one-third of samples are allocated to the gap. The method is also reported to scale to RGB observations and a 9D parameter space in LanderHard (So et al., 17 Feb 2026).

In deployment optimization, synthetic experiments on Crashy Branin at budget $t \ge 0$7 report best feasible objective values of $t \ge 0$8 for hybrid TBA+TPE, $t \ge 0$9 for TPE, $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$0 for random search, and $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$1 for pure simulated annealing. Wasted budget fractions are reported as 23–34% for the hybrid, 29–54% for TPE, and 63–67% for random search. On DeployBench with edge-tight constraints and budget 25, TBA+TPE improves discovery of the globally best model family, vit_tiny, notably 8/10 versus 3/10 on RTX 5080 and 5/5 versus 3/5 on H100. Timeouts reportedly save 7–47 seconds per seed on T4 and 6–25 seconds per seed on H100, with savings concentrated on int8_dynamic and large-batch configurations (Lysenstøen, 27 Apr 2026).

In the chemical-process case study on pressure-swing distillation, the method is evaluated by the relative success rate $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$2, false-positive rate $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$3, false-negative rate $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$4, validity ratio $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$5, and best target value $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$6. For $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$7 and weights $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$8, the reported values are $$V_{\text{reach}}(x_0;\theta) \coloneqq \min_{\pi_\theta} \max_{t \ge 0} h_\theta(x_t)$$9, $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$0, $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$1, and $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$2. When the objective term is included with $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$3, $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$4 decreases to $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$5, while $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$6 increases to $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$7 and $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$8 reaches $$V(x_0;\theta) \coloneqq \max_{\pi_\theta} J(\pi_\theta,\theta), \qquad J(\pi_\theta,\theta) \coloneqq \sum_{t=0}^{T_{\theta,x_0}-1} \mathbf{1}\{h_\theta(x_t) > 0\}$$9, close to the MISQP benchmark $\Theta' \subseteq \Theta$0 obtained with many more runs. These results are contrasted with grid and Latin hypercube baselines using 256 runs, for which $\Theta' \subseteq \Theta$1 is approximately $\Theta' \subseteq \Theta$2 and $\Theta' \subseteq \Theta$3, respectively (Heese et al., 2019).

In Bayesian feasibility learning, PBE is compared on CEC2006 constrained problems using informedness on 10,000 random validation points, 21 repeated runs, and budgets of $\Theta' \subseteq \Theta$4 evaluations after $\Theta' \subseteq \Theta$5 LHS initial points. Reported median informedness for PBE is 99.99% on G4, 100% on G8 despite feasible volume $\Theta' \subseteq \Theta$6, and 99.71% on G24. On G9, where $\Theta' \subseteq \Theta$7, the Echard criterion $\Theta' \subseteq \Theta$8 is reported as best with 97.95% median informedness, while PBE reaches 81.24%. The paper also reports that Knudde’s acquisition can underperform both PBE and, in some cases, LHS (Rahat et al., 2020).

In SEE, experiments measure feasible-zone recall, number of iterations to equilibrium, and uncertainty degree $\Theta' \subseteq \Theta$9. On the double integrator with default $h_\theta : \mathcal{X} \to \mathbb{R}$00 and $h_\theta : \mathcal{X} \to \mathbb{R}$01, SEE converges in 8 iterations with 100% feasible-zone recall and zero violations. On the pendulum with default $h_\theta : \mathcal{X} \to \mathbb{R}$02 and $h_\theta : \mathcal{X} \to \mathbb{R}$03, it converges in 14 iterations with approximately 52.05% recall and zero violations. On the unicycle task with obstacle avoidance, it converges in 10 iterations and achieves 95.78% feasible-zone recall, with the $h_\theta : \mathcal{X} \to \mathbb{R}$04 slice matching the true maximal feasible region. Sensitivity studies show that overly large Lipschitz constants can sharply reduce expansion; for the double integrator, recall drops to approximately 1.08% at $h_\theta : \mathcal{X} \to \mathbb{R}$05 and 0.84% at $h_\theta : \mathcal{X} \to \mathbb{R}$06 (Yang et al., 31 Jan 2026).

6. Limitations, misconceptions, and research directions

A common misconception is to treat FGE as synonymous with conservative objective optimization. The surveyed literature does not support that reduction. Some FGE methods are explicit optimizers over feasible regions, but others are feasibility learners whose primary purpose is boundary identification or safe set expansion. Another misconception is that FGE must be probabilistic. The literature includes probabilistic classifiers and GP posteriors, but also worst-case set-valued models and exact forward-invariance computations (Rahat et al., 2020, Yang et al., 31 Jan 2026).

Several limitations recur. In robust avoid RL, classifier false negatives are possible early in training, and the hyperparameters $h_\theta : \mathcal{X} \to \mathbb{R}$07, $h_\theta : \mathcal{X} \to \mathbb{R}$08, and $h_\theta : \mathcal{X} \to \mathbb{R}$09 control the conservativeness of exploration. Rejection sampling can require careful tuning when feasible regions are extremely sparse or nonstationary. Theoretical guarantees rely on convex–concave losses and exact best responses, assumptions that do not fully hold in control, and the framework currently assumes deterministic dynamics; extension to stochastic safety and chance constraints is identified as a promising direction (So et al., 17 Feb 2026).

In deployment optimization, reported limitations include hardware and domain scope, small budgets and seed counts, residual traps in the most hostile environments, and the possibility that the early-stopping factor $h_\theta : \mathcal{X} \to \mathbb{R}$10 the latency constraint can miss “slow-but-feasible” regimes. The hybrid’s mean objective often ties TPE or random search; its reported advantage is chiefly discovery consistency and waste reduction. Proposed extensions include explicit learned feasibility models $h_\theta : \mathcal{X} \to \mathbb{R}$11, multi-fidelity exploration, adaptive annealing and handoff criteria, and integration with other constrained Bayesian optimization methods (Lysenstøen, 27 Apr 2026).

In simulation-driven FGE, high-dimensional spaces remain difficult for both SVM/KRR and GP-based variants. The 2019 work does not provide formal convergence or regret guarantees, and it suggests dimensionality reduction, dynamic weighting of exploration and optimization terms, multi-class invalidity handling, and multi-objective extensions as future directions. Its comparison with Kriging-based exploration also highlights a substantive issue: when only binary validity labels are available and no continuous violation magnitude exists, GP regression on $h_\theta : \mathcal{X} \to \mathbb{R}$12 can overestimate the feasible region early and produce a high false-positive rate unless modified with a distance-based continuous target (Heese et al., 2019).

In Bayesian feasible-space learning, the main limitations are the independence assumption across constraints, cubic GP training cost, difficulty of acquisition optimization in high dimensions, and weaker behavior when the feasible volume is extremely small, as illustrated by G9. Extensions discussed in the literature include multi-output GPs for correlated constraints, sparse or scalable GP approximations, batch Bayesian search, and use of feasibility learning as a precursor to constrained optimization (Rahat et al., 2020).

In SEE, the discrete-state formulation, dependence on a conservative Lipschitz constant, and practical need for a non-empty initial feasible zone constrain applicability. Overestimation of $h_\theta : \mathcal{X} \to \mathbb{R}$13 can stall expansion, whereas underestimation is unsafe. Proposed extensions include function approximation for the constraint-decay function and set-membership predictor, adaptive discretization near the frontier, active sampling within the current feasible zone, and integration with model-free or model-based policy optimization restricted to that zone (Yang et al., 31 Jan 2026).

Taken together, these results indicate that FGE is most useful when infeasibility is not an occasional nuisance but a first-order structural property of the problem. In such settings, the central design choice is not only how to optimize, but how to represent feasibility, how conservatively to estimate it, and how to allocate exploration budget between frontier discovery and exploitation of already mapped feasible regions.