Improving Feasibility via Fast Autoencoder-Based Projections
Authors: Maria Chzhen
Priya L. Donti
Venue: ICLR 2026 Poster (2026)
Keywords: amortized optimization
nonconvex optimization
surrogate models
feasibility
Primary area: optimization
TL;DR: We propose a data-driven amortized approach that uses a trained autoencoder as an approximate projector to provide fast corrections to infeasible predictions.

Abstract: Enforcing complex (e.g., nonconvex) operational constraints is a critical challenge in real-world learning and control systems. However, existing methods struggle to efficiently enforce general classes of constraints. To address this, we propose a novel data-driven amortized approach that uses a trained autoencoder as an approximate projector to provide fast corrections to infeasible predictions. Specifically, we train an autoencoder using an adversarial objective to learn a structured, convex latent representation of the feasible set. This enables rapid correction of neural network outputs by projecting their associated latent representations onto a simple convex shape before decoding into the original feasible set. We test our approach on a diverse suite of constrained optimization and reinforcement learning problems with challenging nonconvex constraints. Results show that our method effectively enforces constraints at a low computational cost, offering a practical alternative to expensive feasibility correction techniques based on traditional solvers.

Published as a conference paper ICLR 2026
IMPROVING FEASIBILITY VIA FAST AUTOENCODER-BASED PROJECTIONS
Maria Chzhen
Priya L. Donti
University of Toronto
Massachusetts Institute of Technology Cambridge, MA, USA donti ierimmmiteeei
Toronto, ON, Canada
maria. wwwwhnemmil.lccmcmmmmmmmm www.mnatu.ooootoooooommm
ABSTRACT
Enforcing complex (c.g., nonconvex) operational constraints is a critical challenge in real- world learning and control systems. However, existing methods struggle to efficiently enforce general classes of constraints. To address this, we propose a novel data-driven amortized approach that uses trained autoencoder fast corrections to infeasible predictions. Specifically, we train an autoencoder using an adversarial objective learn structured, convex latent representation of the feasible set, This enables rapid correction of neural network outputs by projecting their associated latent representations onto a simple convex shape before decoding into the original feasible set. We test our approach on diverse suite of constrained optimization and reinforcement learning problems with challenging nonconvex constraints. Results show that our method effectively enforces constraints at a low computational cost, offering practical alternative to expensive feasibility correction techniques based on traditional solvers
approximate projector provide
INTRODUCTION
Many learning and control systems, across such as roboties, energy systems, and industrial automa- tion, must produce outputs that respect difficulttto-satisfyy (often nonconvex) constraints. Enforcing these constraints reliably and efficiently is crucial for both safety and real-world usability. A number of ap- proaches have been proposed to address this challenge within learning-based systems, including penalty methods (Fioretto al. 2021; Stooke al. 2020). differentiable projection and correction methods (Chen et al. 2021 Pan et al. 2022), and post-hoc repair algorithms (Zamzam & Baker 2020 Nocedal & Wright 2006). However, each these approaches comes with distinct trade-offs in terms of reliability, generality, computational cost, and solution quality- for instance, failing to satisfy constraints in practice, not handling general classes of constraints, being slower than desired (and potentially prohibitively slow) to deploy real-world systems, and/or degrading end-to-end model performance.
To address this challenge, we propose a data-driven amortized alternative to traditional constraint enforce- ment algorithms: trained autoencoder that acts as an approximate projector to provide fast corrections to infeasible predictions. Specifically, we train an autoencoder to learn a structured, convex latent repre- sentation of the feasible set, in way that enables rapid mapping from infeasible to feasible points. This autoencoder can then be leveraged as a polug-and-pin aiming supersede strict projections in regimes demanding hard feasibility, this approach designed improve feasibility in low-latency settings or in settings where moving predictions closer to the feasible set suffices to guarantee downstream system performance. Our key contributions are follows:
"attachment" to standard neural networks. While not
"Code for the method and all experiments can be found at: https://githiei www.mosstab--aat/.. Aust=Auttoenoder: BBsseddde-----dd------oo

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Published as a conference paper at ICLR 2026
Framework for data-driven feasibility improvement. We pose an approach for learning feasibility improvement mappings that can be appended to neural networks, as an alternative to expensive (albeit exact) constraint enforcement approaches. We propose particular instantiation of this approach, FAB, which employs an autoencoder as the basis of the learned mapping.
Structured latent representation learning. We propose mechanism to learn faithful, feasibility- preserving latent representations of the feasible set, via adversarial training.
Empirical validation. We test our method on range of constrained optimization and reinforcement learning (RL) problems. In our constrained optimization settings, the method consistently provides an efficient approximate projection, learning to map solutions to the feasible set close to 100% of the time in a fraction of millisecond (faster than any other method). For RL settings, the method consistently provides safer actions than methods like proximal policy optimization (PPO) and trust- region policy optimization (TRPO), as well as their constrained variants. Overall, this demonstrates the initial promise of our approach in providing fast, reliable feasibility across a wide range of settings.
2 RELATED WORK
Amortized optimization with constraints. Also known as learning to optimize, amortized optimization is a paradigm designed to accelerate the process of solving optimization problems by using machine learning models as fast function approximators (Amos) (023). Models are trained either via supervised learning on a dataset of known solutions (Chen al., 2022), or through musipervised/ssff-sueerviies methods, where the model's loss function incorporates the optimization problem's is objectives and constraints (Van Henten- ryck 2025). A primary challenge in amortized optimization is ensuring that the model's output satisfies constraints. Several lines of work have emerged to address this, including penalty methods, differentiable projection and/or correction methods, and post-hoc "repair" techniques. Penalty methods turn constrained optimization problems into unconstrained ones by incorporating penalty terms for constraint violations into the optimization objective (Fioretto et al., 2021 Stooke et al. 2020; Raissi et al. 2019): while these methods incentivize feasibility, they do not guarantee it, and may produce highly infeasible solutions in practice. In contrast, differentiablee projection and/or correction methods embed exact solvers directly as layers neural networks (Pham et al. 2018 Chen et al. 2021 Pan et al.) 2022 Nguyen & Donti 2025 Donti et al. 1021). While these methods do provide feasibility guarantees, they are often expensive to run, or highly specialized certain classes of constraints (Min & Azizan 2024 Tordesillas et al. 2023). Likewise, post-hoc "repair"
methods (Boyd et al. 2011 Douglas & Rachford] 1956 Zamzam & Baker 2020) are often either expensive or highly specialized, and the inherent train-test mismatch can further degrade overall performance.
In this work, we propose an alternative, data-driven approach that can learn an inexpensive, non-iterative transformation from a latent convex set to any arbitrary constraint set, thereby speeding up feasibility im- provement in practice (albeit at the expense of provable guarantees). The work closest in the literature to ours is (Liang et al. 2024 2023). proposing a homeomorphic projection approach which learns an invertible mapping from hypersphere to a dure to recover feasibility. However, this procedure is designed for post-hoc feasibility correction rather than end-to-end training, and is limited to holl-homeomorphic constraint sets. Our work presents method that can learn a mapping between a latent hypersphere and any continuous constraint set, and compatible with end-to-end training and inference. In addition, while homeomorphic projections rely bisection procedure, FAB projections are one-shot, leading to significant speed gains in practice.
constraint set, and then uses a bisection proce-
iterative
Safe reinforcement learning. Safe reinforcement learning (RL) is extension of standard RL, where an agent learns to make a sequence of decisions in an environment to maximize a cumulative reward signal while also aiming to minimize the costs associated with constraint violations (Garcıa & Fernández 2015 Gu et al. 2024). While standard RL algorithms, such as Proximal Policy Optimization (PPO, Scchllman et al.; and Trust Region Policy Optimization (TRPO, [Schulman et al. 2013) excel in unconstrained set-
2017
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Critic
Distinguish
Encoder
Decoder
between feasible
Input points *
infeasible
project
outputs solutions
feasible input: encoded into latent set
infeasible project
inference
Figure 1: A schematic of FAB approximate projections. (1) Phase of autoencoder training aims to enable reconstructions of the feasible set. (2) Phase 2 of autoencoder training introduces a discriminator (critic) to enable further structuring and refinement of the latent representation. (3) The trained autoencoder can be utilized as plug-and-play attachment another neural network model.
tings, they are not equipped to handle constraints. Prominent approaches to address this include Lagrangian relaxation (Stooke et al. 2020). safe exploration (Hans et al., 2008), and differentiable projection (Pham et al., 2018; Chen et al. 2021). However, mirroring the discussion on constraint enforcement amortized optimization, these approaches come with distinct trade-offss in terms of reliability and computational cost.
Adversarial training. Adversarial training, widely known for its application Generative Adversarial Net- works (GANs) (Goodfellow et al., 2014) and adversarially robust deep learning (Bai al., 2021), offers a general framework for learning in the face of complex data distributions. In the canonical GAN setup, generator network learns produce realistic data samples from noise vector, while discriminator trained to distinguish between real and generated samples. Minimax training of the generator and discrim- inator drives the generator to produce increasingly plausible outputs. Inspired by this approach, our work uses adversarial training to learn latent representations of feasible sets, where the "discriminator" is trained distinguish between feasible and infeasible outputs produced by another model, and that putput-producing model is thereby forced to learn more robust decision boundaries between feasible and infeasible points.
3 FAST AUTOENCODER-BAN
(FAB) FEASIBILITY IMPROVEMENT
this paper, we consider the task of repeatedly solving parametric optimization problems of the form: y (a) arg min f(;
yEC(2)
where x X' C Rm are the problem parameters, E R" decision variables, f X is the objective function, C(x) S y is a solution. Because the optimal solutions of the problems change the parameters change, this suggests mapping between parameters æ and optimal solutions. In addition to offline optimization, this formulation also captures RI settings, where X' and y are the state and action spaces, respectively (Amos) 2023). Our aim is to use neural networks to approximate mapping from x to y (æ) such that the resultant solution has low objective value, satisfies constraints, and is fast compute inference time.
constraint set, and (x) E C(x) is a
We specifically consider neural networks of the form:
d = ox o No:
where N₂ : + Rd is a standard feedforward network with parameters A and dz Rd + C(x) is a differentiable feasibility improvement procedure that aims to map point in the constraint set. Prior work
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has considered the design of mappings de that can ensure exact feasibility, c.g. by solving a differentiable orthogonal projection or via exact iterative procedures (see Section 2 While such approaches are indeed useful for, e-g., safety-critices settings where provable constraint satisfaction is a must, they tend to be relatively expensive for general classes of constraints. Instead, we propose to learn fast, data-driven, approximate mapping for use in settings where feasiblity is important, but where the benefits of reduced latency outweigh the need for strict constraint satisfaction.
The partieular choice of dr that we propose in this work motivated by the observation that while C(x) may be expensive to enforce directly, we can potentially learn transformation between this set and some other specially-structuree set that is much cheaper to work with. In particular, let E., I + I be an encoder with parameters y, where I := y x X' and C IR*,, and let Ry - y be a decoder ("reconstructor") with parameters 1/, where both Fsy and Rg are standard feedforward neural networks In addition, let S C 2 be a set that by construction simple to map onto (e.g., a simplex or a ball), and let d$ Z + S be a cheap exact mapping to points in S (e.g., a softmax operation or closed-form projection). We then define our feasibility improvement procedure as
Ra Ox : Rup o S a Ey.
(3)
In other words, given output from No alongside its corresponding input, we feed these into encoder, map the encoded latent point into S, and then decode the resultant point. The idea is that once E,y and Rig. are trained, each step of Φh is cheap execute, leading fast overall inference.
The key challenge with this approach is that we must now design the autoencoder parameters (7, v)) such that the output of dh. is actually feasible, i.e., actually lies within C(x). To do this, we propose two-stage autoencoder training procedure aimed at structuring the latent space Z such that, to the extent possible, latent points decode to our feasible set C(x) if and only if they lie within S S Z. A schematic of our overall We now provide additional detail on the two-stage training procedure.
approach is provided in Figure I
3.1 TWO-PHASE AUTOENCODER TRAINING
We train the encoder-decoder pair ( (E,, R2)) in two phases: (1) constraint set reconstruction phase using standard autoencoder training, and (2) a latent space structuring phase involving adversarial training with a discriminator. Together, these aim to enable the autoencoder to faithfully reconstruct the feasible set, and encourage latent points from S to decode to feasible points. Pseudocode for both phases is in Algorithm μ
Phase 1: Constraint set reconstruction. We first train the autoencoder to reconstruct the feasible set. Specifically, we train on a dataset of exclusively feasible points, Trass = (()()) (x(( y( E C(r:())))
As standard in autoencoder training, the encoder E, and the decoder Rg are jointly trained to minimize a standard L2 reconstruction loss to prioritize the fidelity of decoded points:
Lrecon (v. x)) = ||y Ryy. FE (),
(4)
Phase 2: Latent space structuring. The second autoencoder training phase aims to structure the latent space such that any point sampled from S C Z, when decoded, results in a feasible point. To do this, we draw loose inspiration from generative adversarial network (GAN) training, and train autoencoder alter- natingly with a discriminator Dg I > [o, 1] whose role is to distinguish between feasible and infeasible points. Specifically, the discriminator maps from inputs in I to estimated probability that the input is feasible. We let De be standard feedforward neural network.
We focus our discussion in the main text on the case of input-varying constraints C(x), but note that the formulations for
constraints similar. In the latter casc, I := Y; our autoeneoder is simply a standard, rather than conditional, autoencoder; and all mentions of x in Section 3. can be dropped.
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Algorithm 1 TToo-Phase Autoencoder Training for FAB Feasibility Improvement
1: input: Dataset of feasible points Tieas {ly xl0) y E c((z())}
2: input: Discriminator dataset Taise == {(v ) d0) d0 1f y C(z(9), otherwise }
3:
4: procedure PHASEI(Tieax)
// Constraint set reconstruction
inít encoder E,, decoder Rgg
6: while not converged do for batch of (y,x) E Treas
7: compute Crecon(1) 2) via Eq.
8: update 7, using V, Frecon + Creeon
9: end while
10: return E,, R
11: end procedure
12:
13: procedure PHASE2((Tais, En Rep)
// Latent set structuring
init discriminator De
15:
while not converged do for batch of ((,,z),)) E Taise
// Update discriminator De compute Laisc c) via Eq. update& using VgLaiic // Update cutoencoder (Ee, Rss))
17:
18:
19:
20:
compute Ly, T), Lhinge (y, x, c) via Eq. sample batch of 2 ~ compute Liate((2) (z), Lycum( (batch of 2) via Eq. compute Limnee from Lrecon; Lhinges Llatent; Lgrom
21:
22:
23:
Eq.
24:
update 7/ using V, Catmie Larruc end while
25:
26:
return Ey. Rs
27:
end procedure
We leverage datasets during this training phase: (a) labeled dataset of feasible and infeasible points, Taisc (()(()) (0)), clf) ( if γ E C(x(+)), 3, 0 otherwise ). and (b) samples z01 S from latent subset. Unlike in GANs, where the generator and discriminator are trained via minimax loss, here, the autoencoder and discriminator are trained using distinct loss functions.
Specifically, the discriminator Dg is trained to distinguish between feasible and infeasible points, by mini- mizing the negative log likelihood Taisc:
Laise (y, æ, c) = (clog (e(y, x) + (1 e) log (1 De x)))
(5)
The autoencoder is trained to minimize composite loss function using both Taise and 2(1) ~ S, defined as:
Extruc
1
Arecon C Crecun (1) am(0))
+
Aringe Cringe (v (1()) c(.))
N
N
(6)
Matem Liaten ( (2()))
Agenom Eyeomm ((x(1) (,))),,((M))))
+
M
1=1
where Arecon; Atatent Abinge Agoo
IR; Lrecon defined above; and the remaining loss terms defined
The hinge loss Lainge: Structures the latent space by mapping feasible points to the interior hy- persphere and infeasible points to its exterior. Here, write the hinge loss for the specific choice of
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8 {z ||=112 S r for some radius
Lninge (y. = e RRLLU (1, -) + (1 ReLU (r - (v, 12).
(7)
The latent loss Cistent: Encourages outputs decoded from latent points S to be feasible, via super- vision from the discriminator:
Dlaten (2) log (
(8)
The Jacobian regularization term Cesom: Encourages uniform coverage of the feasible set by the decoder (Nazari al. 2023). and computed over a of latent points $ _ S:
Egoom (S) Variance [log dct(J_J] =Ix)
where J= = V (z) is the Jacobian of the decoder with respect to It. is the k x k identity matrix, and & is small scalar value. This loss term measures how the decoder's output changes under small changes in the latent code (=/z). Specifically, it calculates the determinant of the Gram matrix J and measures how much the decoder locally stretches or shrinks the space.
3.2 LEVERAGING THE TRAINED FEASIBILITY MAPPING
After autoencoder training, we fix the weights of the encoder and decoder and use them to construct the feasibility mapping Φ. = Ryg. di Ery, as given by Equation 3 We then append the feasibility mapping our base neural network as ÿø Öz Ng. given by Equation 2 The resultant neural network ğe (with learnable parameters 0) can then be trained end-to-end as usual to address the amortized optimization problem at hand, i.e., aim to learn optimal solutions Problem [ In particular, No learns to adapt the autoencoder's mapping procedure as part end-to-end training. 1.e., can account for this mapping in picking neural network parameters aimed improving the optimality of the end-to-end solution. (While the encoder/decoder parameters v could also be further trained end-to-end alongside the parameters 0, rather than being fixed, we do consider that case here.) Overall, our learned mapping serves as plug-and play attachment to standard deep learning models that can be used during training and inference to improve model feasibility, while also being fast run. In the following sections, we demonstrate the performance of in learning feasible amortized optimization solutions in both offline optimization and RI. settings.
4 EXPERIMENTS ON CONSTRAINED OPTIMIZATION PROBLEMS
Problem classes. We test our approach on constrained optimization problems with three objective types: linear, quadratic, and distance minimization. We consider four classes of 2-dimensional nonconvex con- straint families (Figure as well as 3-, 5-, and 10-dimensional hyperspherical shell constraints == {x E Rd 1 S ||>1 S 2}. We note that these are input-independent constraint sets C y (we will consider input-dependent constraint sets in Section The problems are formulated as:
Linear:
Quadratic:
Distance minimization:
min a T
miny TQ++aa aay
min ||y - ||
s.t. y E
s.t. y E
s.t. y E C,
where y E ) = TR is the decision variable, and problem parameters a E R Q E Raxa E T vary between instances. Our goal is to learn neural network approximator jo for each problem class that minimizes the objective while satisfying constraints.
Baselines. We compare FAB feasibility improvement against:
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Blob with Bite
Concentric Circles
Star Shaped
Two Moons
*
O
4
4
Figure 2: The nonconvex constraint sets tested in constrained optimization settings, termed (from left to
right): Blob with Bite, Concentric Circles, Star-Shaped, and Two Moons.
Projected Gradient Descent. A classical optimization algorithm that iteratively takes gradient descent step and then projects the current iterate onto the feasible set to enforce constraints.
Penalty. A classical optimization approach that solves an unconstrained version of the problem, minimize, (v) = f(v + uP(y), where μ is large penalty coefficient.
A classical optimization method that combines penalty terms and La- grange multipliers to handle constraints. The objective is to minimize J(y, A) f(y) + X c(v) + {1Ic(y)]1 We alternate between updating the primal variables y using estimated gradients and updating the dual variables A.
Augmented-Lagrangee
Interior Point (a.k.a barrier method). A classical optimization algorithm that ensures feasibility by strictly penalizing the optimizer it approaches constraint boundaries. Iterates are initialized within the feasible set, and logarithmic barrier function - - Σ (d(y)) is added the objective, where d(y) a smoothed proximity score (derived from local sampling). The barrier coefficient l is annealed over time to guide the solution toward the optimum.
Penalty Neural Network. A neural network that learns to minimize the unconstrained version of the problem, minimize, (y) = f(y) + >d(l)), where is a large penalty coefficient and d(y) the distance between the constraint set boundary and an infeasible predicted point.
FSNet (Nguyen & Donti 2025). A learning-base based method that combines NNs with an exact solu- tion of an iterative constraint violation- violation-minimizatio minimmzaiiinnn optimization problem.
Homeomorphic Projection (Liang & Chen 2025). A method that learns an invertible mapping be- tween a hypersphere and ball-homeemorphic constraint using a post-hoc bisection procedure to recover feasibility neural network outputs.
Ablations. We also test the following ablations and variants of our method:
Different numbers decoder networks: We examine the effect of training the autoencoder with multiple decoder networks, in principle allowing different decoder networks to "specialize" in different parts of the latent space. Specifically, we consider collection of p decoder networks
(())) ..... (((e) each mapping from Z + V. In addition, we introduce a weighting network M_ 2 + TRP that maps from the latent space to a set of logits (one per decoder). For any 2, we compute mixture weights ax2) softmax(M,, (z)) R. and define the final decoded output as the convex combination (z) = Ei= ox(2) (()) The weighting network Ma, and all decoder networks p(() are trained jointly and-to-end using the same loss functions described in Section 3.1 with R, (2) the decoded output. In our ablations, we test the settings where p = 2,3.
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Quadratic
Linear
Dist. Min.
Method
Feas (%)
Time (ms)
Gap
(%)
Time (ms)
J. Gap
Feas (%)
4 Time (ms)
Gap
80.5
79.07
1.18
76.3
48.23
0.90
79.2
69.79
4.65
Projected Gradient
o= 39.6
o= 91.63
1,58
o= 42.5
o= 52.86
o= 0.99
o= 40.6
25.09
5.87
20.4
78.39
1.36
10.4
61.58
1.52
14.6
38.67
6.39
Clussical
Penalty Method
o= 40.3
o= 71.27
o= 1.67
o= 30.5
o= 111.21
o= 1.49
o= 35.3
o= 10.02
o= 7.07
20.9
85.71
1.33
10.9
56,98
1.56
15.7
42.58
6.43
Augmented Lagrangian
o= 40.7
o= 179.17
o= 1.59
o= 31.1
o= 62.66
o= 1.56
o= 36.3
o= 6.69
o= 6.70
78.3
82.80
1.64
77.3
46.37
1,03
79.9
42.00
6.12
Interior Point
o= 41.2
o= 110.99
o= 2.02
o= 41.9
o= 33.68
o= 1.03
o= 40.0
o= 12.03
o= 6.41
91.1
0.20
0.61
80.0
0.14
0.85
40.7
0.24
4.34
Penalty NN
<= 28.4
o= 1.02
o= 1.06
o= 38.4
o= 0.00
o=
o= 49.1
o= 1.30
o= 6.30
98.7
4,42
1.13
98.3
3.40
0.96
99.3
4.03
5,30
FSNet
d= 11.5
d= 2.77
d= 1.45
o= 12.8
o= 2.08
0= 1.01
d= 8,5
0= 1.67
0= 5.51
73.9
247
1.74
76.1
240
1.67
76.1
8.12
Homeomorphic Projection
T= 43,9
d= 105
d= 1,90
o= 42.6
0= 104
0= 1.48
d= 42.6
d= 105
0= 7.80
96.8
0.55
0.85
83.5
0.43
0.73
88.3
0.41
4.22
FAB
o= 17.6
o= 0.40
o= 1.16
o= 37.1
0.27
o= 0,73
o= 32.2
o= 0.20
o= 4.15
ILL-Based
100.0
0.59
1.31
100.0
0.55
0.94
100.0
0.50
5.64
FAB: 2 Decoder Nets
o= 0.0
o= 0.35
o= 1,36
o= 0.0
o= 0.37
o= 0.89
0.0
o= 0.25
o= 0.73
100.0
1.02
100.0
0.62
100.0
5.44
FAB: 3 Decoder Nets
o= 0.0
o= 1.03
d= 1.15
o= 0.0
d= 0.36
0= 0.84
o= 0.0
o= 0.32
o= 4.84
82.5
0.64
0.86
1.07
64.1
0.54
3.83
FAB: Phase 1 Only
o= 38.0
o= 0.14
o= 1,21
o= 45.9
4.12
o= 0.69
o= 48.0
o= 0.09
o= 4.07
28.4
0.63
1.13
18.8
0.76
28.1
5.14
FAB: No Hinge Loss
o= 45.1
o= 0.16
o= 1,23
o= 39.1
o= 0.84
o= 0,80
o= 44.9
oz 0,16
o= 4.96
75.1
0.67
FAB: No Dg
de 43.2
aw 0.22
aw 1.21
ow 49.8
0.31
ow 0.69
dm 34.3
ow 0.19
d 4.00
Table 1:
Mean and std. dev. for feasibility (%), time (ms), and optimality gap for each method: Two Moons.
Phase Only: The autoencoder was trained using only the reconstruction loss and no Phase 2.
No Hinge Loss: The autoencoder was trained without the hinge loss.
No discriminator De: The adversarial discriminator was removed from Phase training entirely, relying only on the other losses, to isolate the discriminator's contribution.
Appendix C we also provide experiments testing the effect of (a) incomplete coverage of the constraint set in the autoencoder training data, and (b) using different decoder depth and width configurations.
Implementation details. We run experiments over 5 seeds with 300 problems per seed (trained for 500 epochs with batch size of 32, tested on 1,500 problems), totaling 18,000 optimization problems across the entire testing suite. All methods were run on a workstation equipped with one NVIDIA RTX 4090 GPU, We choose S to be 0.5-radius hypersphere. Hyperparameters given in Appendix A.
Evaluation metrics. We evaluate the methods by their feasibility rates, optimality gaps, and inference times over the 4 constraint types and 3 objective types.
Results. As presented Table I and Appendix B our methods achieve a substantial reduction com- putation time, with sub-millisecond inference times (1-2 orders of magnitude faster than exact feasibility methods), while consistently achieving high feasibility rates across the feasible sets. Across most experi- ments, the best-pergorming method was found to be FAB with one decoder; FAB with multiple decoders
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emerges as the strongest method for the Two Moons disjoint constraint set. Notably, we see that FAB does well at finding feasible solutions even in difficult cases with disjoint or non-ball-homeomorphe sets.
5 EXPERIMENTS ON SAFETY GYM (SAFE RL)
Safe RL problem formulation. Safe RL can be framed within parametric optimization setup from Equation We consider discrete-time dynamical system where the state evolves according to:
8k+1 = [skk uk);
(10)
where S6 E X' C R is the current state and uy E y S I²' is the control action at timestep k. this context, the actions u& the decision variables, and the states 84 correspond to the problem parameters. The constraint set ((sg) is state-dependent (i.e., wubblen-prrmetee-ddeeeneee safe, i.c. uk C(sx), if executing from state 8₁ does not lead an immediate constraint violation (e.g..
An action ug is considered
collision). The objective is to learn policy X - V, that maximizes the expected cumulative reward while also satisfying constraints.
Environment. We evaluate our approach using the SafetyPointGoal2-00 and Safety PiiettyPointushh-vee envi- ronments from the Safety Gymnasium benchmark suite (Ji al.) 2023). In the Goal task, an agent must navigate to series of goal locations while avoiding randomly-placed hazards. In the Push task, an agent must navigate to and manipulate a movable box, pushing toward a target location while avoiding randomly placed hazards. The state vector comprises simulated sensor readings (e.g.. accelerometers, gyroscope, and a LiDAR-like sensor for hazard detection), while the action vector controls the agent's forward/backward movement, well its rotational velocities.
Baselines. We evaluate the following baselines:
PPO (Schulman et al. 2017). A state-of-the-arr unconstrained RL algorithm.
TRPO (Schulman al., 2015). Another robust unconstrained RL algorithm.
Lagrangian PPO (PPO-LAG) and Lagrangian TRPO (TRPO-LAG) (Stooke et al. 2020 Ray
et al. 2019). Constrained versions of PPO and TRPO that use Lagrangian relaxation to penalize constraint violations during training.
We compare these against the approach of using PPO with FAB projections (PPO-FAB).
Implementation details. For each enviromment, the FAB autoencoder was trained on a dataset of 100,000 state-action pairs, approximately evenly balanced between safe and unsafe examples, which were collected by running a random policy in the environment. The performance of all algorithms was evaluated over 50 independent seeds. All methods were run on workstation equipped with one NVIDIA RTX 4090 GPU. We choose S to be 0.5-radius hypersphere. Hyperparameters are given in Appendix A.2
Evaluation metries. We assess performance based on several metrics, averaged over the evaluation episodes: episode rewards, episode costs, and mean inference time. Episode reward is the cumulative rewards obtained by the agent in one episode (which is 1000 steps). Episode cost is the total number of constraint violations (i.e., entering hazard zones).
Results. As presented Tables and PPO-FAB achieves the lowest constraint violation costs among the evaluated methods, substantially outperforming standard baselines like PPO and TRPO on this metric, and even somewhat outperforming the safe variants PPO-LAG and TRPO-LAG. Furthermore, it also exhibits the highest levels of reliability, in the sense of having the lowest standard deviations in both cost and reward across all methods compared. However, this focus on safety also corresponds to a more conservative reward- seeking policy, resulting in lower reward means (sometimes much lower) compared to the other algorithms.

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Published as a conference paper ICLR 2026
Algorithm
Reward Mean f
Reward Std +
Cost Mean +
Cost Std +
Time Mean (s) }
PPO-FAB
-1.12
1.11
24.32
37.84
2.75
PPO
13.26
14.05
167.46
87.06
2.47
PPO-LAG
2.24
5.10
54.10
64.50
2.40
TRPO
15.58
10.31
164.14
88.43
2.59
TRPO-LAG
2.37
8.46
89.04
187.67
2.78
Table 2: Results for SafetyPointGoal2-000 environment.
Algorithm
Reward Mean f
Reward Std +
Cost Mean |
Cost Std J
Time Mean (s) |
PPO-FAB
-0,07
0.52
7.70
37.51
3.71
PPO
0.41
3.11
59.86
120.18
3.78
PPO-LAG
0.60
1.58
31.34
58.17
4.09
TRPO
0.20
2.50
106.88
216.19
4.03
TRPO-LAG
-0.77
5.51
28.22
67.21
4.05
Table 3: Results for SafetyPoinnPush2-vO0 environment.
6 CONCLUSION
In this work, introduced novel-data driven method, Fast Autoencoder-Basee (FAB) projections. address the challenge of efficiently enforcing nonconvex constraints learning-based systems. Our method specially trains autoencoders to learn approximate projections onto general (e.g., potentially nonconvex disjoint) feasible sets. The trained autoencoders can be leveraged as plug-and-play attachments to standard neural networks part of broader constrained learning pipeline. By structuring the autoencoder's latent space to correspond to a simple convex shape that is easy to map into, we can perform a fast projection within this latent space inference and decode the result back into a point that highly likely to be feasible.
On our constrained optimization benchmarks, we show FAB consistently achieves near-perfect feasibility rates (often approaching 100%) across a diverse set of challenging, nonconvex feasible sets. In the Safety- Gym benchmark, it consistently achieves low constraint violation costs. Crucially, it achieves both of these with inference times that are significantly faster than those of other methods. While methods with formal guarantees can ensure exact feasibility, their computational overhead can be prohibitive for certain real-time applications. Therefore, FAB presents a compelling alternative for latency-sensitive domains where rapid, high-fidelity approximate projections may be sufficient.
Limitations FAB include lack of hard guarantees, and its reliance on having a representative dataset of feasible and infeasible points in order for autoencoder training perform well. We also observe that the adversarial training process can be somewhat unstable, requiring, c.g., extensive hyperparameter tuning. We also note that the experiments in this work, while spanning diverse settings, relatively small-scale. Future work includes both larger-scale evaluations and identifying ways to improve sample efficiency, distributional performance, and interpretability of the data-driven mapping e.g., through the use of specialized neural network structures such as input-convex neural networks (Amos et al.) 2017) or operator learning-based networks (Kovachki al. 2024), even simpler parameterized functions as well as exploring toolkits such as formal verification to better understand and assess the behavior of the learned data-driven mapping after it is trained. Another future direction includes exploring adaptive co-training of the data-driven feasi- bility improvement mapping with the neural network to which it will be attached (rather than training the mapping fully separately), motivated by prior results on the benefits of end-to-end learning (Cameron et al.) un-
2022). Overall, we believe that the paradigm of data-driven feasibility mapping offers compelling derexplored middle ground between penalty methods and exact feasibility enforcement, and look forward to future work that investigates alternative approaches within this space beyond simply the one presented here.
10

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Published as conference paper ICLR 2026
ACKNOWLEDGMENTS
We thank Hoang Nguyen and Khai Nguyen for their input and feedback this work. This work was supported by the U.S. National Science Foundation under award #2325956.
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Published as conference paper ICLR 2026
A HYPERPARAMETERS
A.1 CONSTRAINED OPTIMIZATION HYPERPARAMETERS
Hyperparameters for autoencoder training in the constrained optimization experiments (Section 4). The optimal loss weights were found using grid search over the following values:
Aecon options = [1.5, 2.0]
Aeeus options = [1.0, 1.5, 2.0]
Alatent options [1.0, 1.5]
Agomm options [0.025]
Abinge options [0.5, 1.0]
The remaining hyperparameters for both our methods and the baselines were manually chosen via informal experimentation.
Phase 1 Training (phasel training.py)
Batch size: 256
Epochs: 500
Learning rate: 0.001
Optimizer: Adam
Training/validation split: 80% / 20%
Hidden dimension: 64
Number decoders:
Number of samples: 60000
Phase 2 Training (phase2 training.py)
Batch size: 256
Epochs: 150
Learning rates: AE=0.0005, Discriminato==0.001
Loss weights: A100 Ateasibility =1.00 Atatent 1.0, Aninge 0.1, Agcometric 0.1
Optimizer: Adam
Critic steps
Training/validation split: 80% / 20%
Normalization: enabled
Hidden dimension: 64
Number of decoders:
Hyperparameters for the ablation and decoder configuration experiments described in Appendix C are in Table A.1
'This hyperparameter deseribes the number of discriminator gradient descent steps per pass through the autoen-
coder.
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Published as conference paper ICLR 2026
Shape
Exp. Type
Config
recon
feas
Atatent
Agcom *
Aninge
Throughput
blb_wwth_bbte
coverage
Cov_10
1.5
1.0
1.0
0.025
0.5
457.44
blob bith..iitttt
coverage
Cov 25
1.5
1.0
1.0
0.025
0.5
453.23
blob_with with_bite
coverage
Cov_50
1.5
1.0
1.0
0.025
0.5
446.47
blob_we with_bit
coverage
Cov_75
1.5
1.0
1.0
0.025
0.5
446.20
blob_with. bith
capacity
W32_D2
1.5
1.0
1.0
0.025
0.5
404.14
blob_with. with_bite
capacity
W32 D4
1.0
0.025
0.5
384.20
blob_with bith
capacity
W32_D6
1.5
1.0
1.0
0.025
0.5
368.88
blob_with.
capacity
W64_D2
1.5
1.0
1.0
0.025
0.5
399.35
Blob_with.bid
capacity
W64_D4
1.5
1.0
1.0
0.025
0.5
379.72
blob_with. with_bite blob with. bite Blob_with_bes
capacity
W64_D6
1.5
1.0
1.0
0.025
0.5
366.03
capacity
W128 D2
1.0
0.025
0.5
394.16
capacity
W128 D4
1.5
1.0
1.0
0.025
0.5
379.71
blob with bite blob_with.Nd with
capacity
W128 D6
1.5
1.0
1.0
0.025
0.5
367.23
num_dee dee
2_decoeers
1.5
1.0
1.0
0.025
0.5
349.43
star shaped coverage
coverage
Cov_10
1.5
1.0
1.0
0.025
0.5
448.20
star shaped
Cov .25
1.5
1.0
1.0
0.025
0.5
444.26
shaped star shaped
coverage
Cov_50
1.5
1.0
1.0
0.025
0.5
441.18
coverage
Cov 75
1.5
1.0
1.0
0.025
0.5
436.32
star shaped
capacity
W32_D2
1.5
1.0
1.0
0.025
0.5
399.33
star shaped
capacity
W32 D4
1.5
1.0
1.0
0.025
0.5
379.74
star shaped
capacity
W32_D6
1.5
1.0
1.0
0.025
0.5
368.52
shaped
capacity
W64 D2
1.5
1.0
1.0
0.025
0.5
394.05
star shaped
capacity
W64_D4
1.5
1.0
1.5
0.025
0.5
385.30
star shhpeed
cupacity
W64_D6
1.5
1.0
1.0
0.025
0.5
366.63
star shaped
capacity
W128 D2
1.5
1.0
1.5
0.025
0.5
401.35
star_shage sha_es
capacity
W128. DD
1.5
1.0
1.0
0.025
0.5
382.05
star shaped
capacity
W128 D6
1.5
1.0
1.0
0.025
1.0
369.05
star_shaped
num_dec
2_decoders
1.5
1.0
1.0
0.025
1.0
352.57
two moons
coverage
Cov
1.0
1.0
0.025
0.5
805.99
two_moons
coverage
Cov.25
1.5
1.0
1.0
0.025
0.5
809.56
two_moons
coverage
Cov_50
1.5
1.0
1.0
0.025
0.5
802.55
two moons
coverage
Cov 75
1.5
1.0
1.0
0.025
0.5
810.32
Iwo_moons
capacity
W32_D2
2.0
2.0
1.5
0.025
0.5
650.05
IWO moons
capacity
W32 D4
1.5
1.5
0.025
1.0
469.24 24
two_moons
capacity
W32_D6
1.5
1.0
1.5
0.025
0.5
449 40
capacity
W64 D2
1.5
1.0
1.5
0.025
0.5
503.61
two_moons
capacity
W64_D4
1.5
1.0
1.0
0.025
0.5
532.68
two_moons
capacity
W64_D6
1.5
1.0
1.0
0.025
0.5
447.05
two moons
capacity
W128 D2
2.0
1.5
1.5
0.025
0.5
604.86
two_moons
capacity
W128 D4
1.5
2.0
1.5
0.025
0.5
553.15
moons
capacity
W128 D6
1.5
1.0
1.5
0.025
0.5
490.69
two_moons
num_dec
2_decoders
1.5
1.0
1.5
0.025
0.5
433.12
concentric circles concentrie.circles
coverage
Cov.
1.0
0.025
0.5
834,29
concentric circles
coverage
Cov 25
1.5
1.0
1.0
0.025
0.5
829.25
concentric_cine cirrcls
coverage
Cov_50
1.5
1.0
1.0
0.025
0.5
736.49
concentric circles
coverage
Cov.75
1.5
1.0
1.0
0.025
0.5
830.27
concentric _circle
capacity
W32_D2
1.5
1.0
1.0
0.025
0.5
862.89
concentric circles
capacity
W32.D4
1.0
1.0
0.025
0.5
703,99
concentric_circles
capacity
W32.D6
1.5
1.0
1.0
0.025
0.5
637.08
concentric circles
capacity
W64
1.5
1.0
1.5
0.025
1.0
764.57
concentric. cirrels
capacity
W64
1.5
1.0
1.0
0.025
1.0
791.68
concentric _circles
capacity
W64_D6
1.5
1.0
1.0
0.025
1.0
708.25
concentric.ciree
capacity
W128 D2
15.55
1.0
1.0
0.025
1.0
895.01
concentric Aiccentrccccrcless
capacity
W128 D4
1.5
1.0
1.0
0.025
1.0
788.96
concentric.cim circles
capacity
W128.DD
1.5
1.0
1.0
0.025
1.0
737.83
concentric.cime cirrelss
num_des dec
2_decoders
1.5
1.0
1.0
0.025
1.0
634.10 10
Table A.1: Optimal configurations for ablations and training throughput comparisons. The default decoder configuration W64_D4.

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Published as conference paper ICLR 2026
A.2 SAFETY GYM HYPERPARAMETERS
Hyperparameters for autoencoder training in the Safety Gym experiments (Section 5) Most of the hyperpa-
rameters were borrowed from the Safety Gymnasium benchmark suite implementations (Ji ct al. 2023). as well as manually chosen via informal experimentation.
Phase Training (safety-gynn Saaeetyyyymmphaeel...o http:/pphase_ttaiinnnp..p
Batch size: 256
Epochs: 500
Learning rate: 0.001
Optimizer: Adam
Training set: 80%
Validation set: 20%
A 1.0
Hidden dimension: 64
Number of decoders:
Number of samples: 100000
State dimension: 60 (safety-gym)
Action dimension: (safety -ym)
Latent dimension: equal to action dimension
Phase 2 Training htnpe//ggym/hhaaeeettainngge
Batch size: 256
Epochs: 100
Learning rates: AE=0.0001, Diseriminator=0.0002
Loss weights: Arecnn 1.0, Aresibility 0.5, Ataent
0.5, Abinge 0.5, Ageometric 0.1
Optimizer: Adam
Validation split: 0.2
Hidden dimension: 64
Number decoders:
16

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Published as conference paper ICLR 2026
B ADDITIONAL EXPERIMENTAL RESULTS
Tables B.2 B.4
provide experimental results for the Blob with Bite, Concentric Circles, and Star-Shaped
constraint families described Section 4 Figure 2
Results for the Two Moons constraint family are pro-
vided in the main body (Table
Table _
B,5
provides results for the 3-, 5-, and 10-dimensional hyperspherical shell constraints : {or 2}.
r
Quadratic
Lincar
Dist. Min.
Method
Feas (%)
Time (ms)
J Gap
Feas (%)
Time (ms)
J. Gap
Feas (%)
Time (ms)
U Gap
100.0
38.73
0.60
100.0
19.80
1.04
100.0
32.34
0.95
Projected Gradient
o= 0.0
o= 28.68
oz 0.88
o= 0.0
o= 31.73
o= 1.11
o= 0.0
o= 49.68
o= 2.42
59.7
64.76
0.97
43.7
38.72
1.29
55.1
55.01
5.37
Classica
Penalty Method
o= 49.0
o= 47.86
o= 1.39
o= 49.6
a= 52.87
o= 1.28
o= 49.7
a= 156.36
o= 6.47
58.5
71.64
0.98
44.5
42.68
1.33
56.5
45,49
5.54
Augmented Lagrangian
d= 49.3
o= 64.04
o= 1.42
o= 49.7
o= 49.49
o= 1.29
o= 49.6
d= 33.47
d= 6.73
95.3
61.82
2.50
92.6
45.66
1.63
94.5
40.50
11.89
Interior Point
d= 21.2
d= 35.51
d= 2.61
o= 26.2
o= 56.51
o= 1.66
d= 22.7
o= 41.82
o= 11.35
97.1
0.19
0.50
90.4
0.14
1.04
66.0
0.14
1.24
Penalty NN
d= 16.9
d= 0.06
d= 0.76
o= 29.5
d= 0.03
d= 1.14
O= 47.4
d= 0.02
a= 2.41
99.5
2.66
1.15
98.7
2,49
2.09
99.8
2.31
13.63
FSNet
O= 7.3
dm 11.09
dm 1.67
d= 11.2
d= 1.43
de 1,98
du 4,5
de 1.33
de 13.30
100.0
196.631
4.37
100.0
196.247
2.58
100.0
196.021
13.36
Homeomorphic Projection
n= 0.0
dm 193.4
aw 4.28
aw 0.0
om 195.8
ow 2.30
am 0.0
dw 195.5
aw 12.73
100.0
0.69
0.53
100.0
0.50
1.14
100.0
0.39
2.89
FAB
o= 0.0
o= 1.44
o= 0.71
o= 0.0
a= 0.48
o= 0.96
a= 0.0
n= 0.17
n= 4.26
MI -Based
62.0
1.14
0.52
55.9
0.59
1.09
86.0
0.45
2.59
FAB: Decoder Nets
n= 48.5
o= 4.08
o= 0.69
<= 49.7
o= 0.43
o= 0.95
o= 34.7
o= 0.15
o= 4.23
95.6
1.15
0.53
95.3
0.60
1.11
94.6
0.55
3.59
FAB: 3 Decoder Nets
o= 20.5
o= 3.18
o= 0.70
o= 21.2
o= 0.35
o= 0.92
o= 22.6
o= 0.42
o= 4.49
66.7
0.77
0.51
64.1
1.16
1.06
76.7
0.56
1.32
FAB: Phuse Only
47.1
0.34
a= 0.62
o= 48.0
o= 2.88
o= 0.83
o= 42.3
o= 0.13
o= 1.48
100.0
0.94
0.63
100.0
1.15
1.29
100.0
0.61
6.47
FAB: No Hinge Loss
o=
1.79
dm 0.79
d= 1.58
de
d=
0= 5.28
80.8
0.74
0.52
80.6
0.82
1.11
77.9
0.73
4.79
FAB: No Dg
Gw 0.69
ow
39.5
ow 0.47
dw 0.92
ar 41.5
de
Table B.2: Mean and std.dev. for feasibility (%), time (ms), and optimality gap for each method: Blob with Bite.
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Published as conference paper ICLR 2026
Quadratic
Min.
Method
Feas (%)
Time (ms)
J Gap
Feas (%)
Time (ms)
J. GGap
Feas (%)
Time (ms)
Gap
100.0
31.24
0.96
100.0
14.83
1.16
100.0
24.36
Projected Gradient
o= 0.0
d= 13.55
d= 1.27
d= 0.0
o= 14.92
o= 1.16
0.0
17.34
d= 0.88
56.68
1.12
42.5
39.15
Clussica
Penalty Method
47.8
d= 44.47
d= 1.41
d= 49.4
on 63.11
d= 1.30
o= 50,0
31.26
d= 6,60
37.2
57.59
42.1
52.09
Augmented Lagrangian
48.3
dw 43.84
dw 1.43
du 49.4
dm 85,00
dw 1,30
aw 50,0
46.
de 6.91
54.53
2.67
89.9
39.76
Interior Point
24.8
o= 34.56
o= 2.71
o= 30.2
o= 29.55
o= 1.79
o= 26.2
o= 38.55
o= 11.28
0.24
0.14
Penalty NN
26.7
0.51
o= 1,19
o= 19.1
o= 0,00
d= 1.17
o= 50.0
a= 0.02
o= 2,08
1.75
FSNet
o= 2.6
1.01
o= 1.72
o= 12.0
o= 6,11
a= 2.01
o= 9.3
0= 0.97
14.52
166
4.83
168
Homeomorphic Projection
24.4
86
o= 4.31
o= 24.8
o= 87
a= 2.32
o= 24.8
o= 90
13.51
100.0
0.45
100.0
0.38
1.48
0.48
5.72
FAB
0= 0.0
d= 0.21
d= 1.67
o= 0.0
o= 0.13
d= 1.36
d= 2.6
d= 0.36
a= 7.72
ML -l--Based
99.8
0.50
1.49
92.7
0.49
1.52
0.52
1.54
FAB: Decoder Nets
Om 4.5
dw 0.17
dm 1.44
ow 27.5
dm 0.30
dm 1.33
Oe 38.9
de 0.30
ae 1.53
99.3
0.86
1.50
96.9
0.61
1.60
0.71
1.25
FAB: Decoder Nets
Ow 8.1
ow 1.94
dw 1.43
ow 17.4
aw 0.44
dw 1.47
aw 38.0
dm 1.21
aw 1.25
99.1
0.77
0.89
98.0
0.54
1.12
25.9
0.51
4.61
FAB: Phase 1 Only
n=
2.51
n= 0.82
o= 14.0
o= 0.11
n= 0.81
o= 43.8
a= 0.04
o= 3.63
99.1
0.74
98.1
0.58
1.48
68.7
7.08
FAB: No Hinge Loss
o= 9.9
0.57
o= 1.29
o= 13.5
o= 0.21
o= 1.29
o= 46.4
o= 0.03
o= 7.85
98.3
0.74
98.5
0.72
1.23
82.7
1.58
FAB: No Dg
12.8
o=
0.67
o= 0.92
o= 12.3
o=
2.67
o=
0.96
o= 37.9
o= 0.03
o= 2.15
Table B.3: Mean and std. dev. for feasibility (%), time (ms), and optimality gap for each method: Concentric Circles constraint family. 5 seeds, 300 problems each per problem per objective type.
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Published as conference paper ICLR 2026
Quadratic
Min.
Method
Feas (%)
Time (ms)
Gap
Feas (%)
Time (ms)
Gap
Feas (%)
Time (ms)
Gap
100.0
38.41
100.0
18.99
100.0
31.18
Projected Gradient
0= 0.0
d= 39.50
d= 0.41
d= 0.0
a= 10.97
o= 0.69
0.0
d= 27.47
0= 1.86
61.37
0.87
38.0
40.03
39.41
Clussica
Penalty Method
42.8
d= 33.75
d= 1,32
d= 48.5
dm 33.05
d= 1,15
o= 49.5
13,16
d= 5.99
75.0
70.70
40.4
39,49
4.99
Augmented Lagrangian
43.3
dw 74.02
dw 1,38
du 49.1
dm 20.97
dm 1.10
dw 49.3
32,70
0 6, 17
70.65
92.9
40.25
Interior Point
18.1
de 73.77
dw 1.99
oa 25.7
d 21.66
aw 1.46
de 21.8
62,60
ne 8,70
0.20
83.5
Penalty NN
13.3
0.07
o= 0.32
o= 37.1
o= 0,03
a= 0.71
o= 48.0
o= 0.00
02 2.71
3.13
3.76
FSNet
o= 3.6
o= 3.00
o= 1.91
o= 5.2
a= 2.06
0= 1.04
o= 5.2
11.90
0= 7.61
100.0
128.653
3.70
100.0
126.350
126.455
Homeomorphic Projection
o= 0.0
48.4
o= 3.35
o= 0.0
o= 48.9
a= 2.07
o= 0.0
o= 48,9
11.84
0.56
0.34
99.9
0.60
0.77
0.64
2.01
FAB
o= 2.6
o= 0.45
o= 0.40
o= 2.6
o= 1.61
o= 0.65
oz 20.9
o= 0.93
oz 1.96
ML-Based -Bosed
100.0
0.60
0.37
99.5
0.49
0.73
94.7
0.71
2.30
FAB: Decoder
o= 0.0
o= 0.32
o= 0.43
or 7.3
o= 0.26
o= 0.59
o= 22.3
o= 0.83
o= 2.28
99.9
0.65
0.34
99.9
0.55
0.75
94.5
2.02
FAB: Decoder
o= 2.6
o= 0.29
o= 0.39
o= 3.6
o= 0.24
a= 0.62
o= 22.9
d= 2.39
a= 2.00
100.0
0.61
0.34
100.0
0.54
0.76
100.0
0.53
3.05
FAB: Phase 1 Only
d= 0.0
o= 0.08
a= 0.41
o= 0.0
o= 0.08
a= 0.65
o= 0.0
a= 0.08
d= 3.04
100.0
0.60
0.38
100.0
0.54
0.83
100.0
0.52
5,00
FAB: No Hinge Loss
0= 0.0
o= 0.07
a= 0.49
o= 0.0
o= 0.07
a= 0.75
o= 0.0
a= 0.04
0= 4.27
100.0
0.86
0.35
100.0
0.54
0.78
96.8
0.52
1.62
FAB: No D3
0= 0.0
o= 2.05
d=
0.40
o= 0.0
o= 0.07
d=
0.65
o= 17.6
a= 0.04
a= 1.64
Table B.4: Mean and std. dev. for feasibility (%), time (ms), and optimality gap for each method: Star Shaped constraint family. 5 seeds, 300 problems each per problem per objective type.
Quadratic
Linear
Dist. Min.
Constraint Dimension
Feas (%) L Time (ms)
Gap
Feas (%)
Time (ms)
J GGp
Feus (%)
Time (ms)
J Gap
86.6
0.12
1.6316
77.8
0.08
1.5220
55.0
0.09
6.8719
Penalty NN: 3D
a= 34.1
o= 0.06
o= 1.6897
o= 41.6
o= 0.02
a= 1.4711
o= 49.7
o= 9.6887
88.3
0.12
1,8296
85.8
0.08
1.7951
28.4
0,08
12,5372
Penalty NN: 5D
d= 32.1
d= 0.03
d= 1.7225
o= 34.9
o= 0.02
o= 1.1600
o= 45.1
o= 0.02
a= 12.5638
64.3
0.13
3,1529
80.4
0.09
3.6322
0.5
0,08
49,9375
Penalty NN: 10D)
1.6177
7.3
27.8050
100.00
0.81
2.08
99.47
0.77
1.66
97.67
0.80
6.28
FAB: 3D
d= 0.00
d= 0.01
1.74
o=
o= 1.40
d= 0.15
d= 0.04
G= 6.98
90.50
0.83
2.08
95,20
0.74
1.94
96.47
0.77
4,30
FAB: 5D
0.08
1.20
0.18
4.77
100.00
0.81
10.97
100.00
0.74
1.94
100.00
0.74
12.69
FAB:
0.00
6.45
1.48
0.00
o=0.01
8.96
Table B.5: Mean and std. dev. for feasibility (%), time (ms), and optimality gap for Penalty NN and FAB run on the 3-, 5-, and 10-dimensional hyperspherical shell constraint sets.
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Published as conference paper ICLR 2026
C ADDITIONAL ABLATIONS AND CONFIGURATIONS
we provide results for two additional types of variants of our FAB method for the 2- dimensional constrained optimization settings:
In Tables C.6 C.6.C.7
Incomplete coverage ablations test the performance of method when the training data only consists of part of the constraint set (10%, 25%, 50%, and 75%).
Decoder capacity experiments test the performance multiple decoder depth and width configu- rations. W indicates decoder width, while D indicates decoder depth.
For the incomplete coverage ablations, we find that FAB is generally able to maintain feasibility even when the feasible set not fully covered during training, achieving relatively high feasibility rates across all ablations (with the lowest being 80.1 + 0.0% for Concentric Circles Cov-75 and 76.5 + 24.7% for Two Moons Cov-25, and with other scores all above 90% and largely at 100%). Notably, the 10% coverage ablation achieves 100% feasibility across all feasibility sets, likely because the feasibility sets may become topologically simpler if the coverage is low enough, which means they are easier for the autoencoder to learn. Optimality gaps vary across the experiments, depending on whether the optimal solution happens lie within the part of the constraint set that sampled. (The reported optimality gaps also vary based on whether feasibility is maintained, find that while perfect coverage of the constraint set is not necessary, better coverage of the constraint set during training unsurprisingly enables better representation of the full constraint set and thereby better end- to-end learning outcomes in general.
infeasible solutions
achieve artificially low "optimality gaps.") Overall,
For the decoder capacity experiments, we find that performance varies depending on the problem type. Therefore, while we did not systematically tune the decoder width and depth during our main experiments, tuning these parameters may indeed be beneficial in general for overall performance.
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Published as conference paper ICLR 2026
Quadratic
Linear
Dist. Min.
Method
Feas (%)
J. Time (ms)
Gap
Feas (%)
Time (ms)
Gap
Feas (%)
1. Time (ms)
Gap
100.0
1.12
0.82
100.0
0.87
1.53
100.0
0.85
10.25
Incomplete Coverage
Cov-10
o= 0.0
o= 1.10
o= 1.02
o= 0.0
o= 0.04
o= 1.39
o= 0.0
o= 0.04
o= 7.33
100.0
1.08
0.81
100.0
0.86
1.52
100.0
0.85
10.15
Cov-25
o= 0.0
o= 0.09
o= 1.01
o=
o= 0.007
o= 1.39
o=
o= 0..03
o= 7.20
100.0
1.08
0.80
100.0
0.86
1.47
100.0
0.85
9.15
Cov-50
G=
0.09
o= 0.98
d=
0.01
1.30
o= 0.03
o= 6.97
100.0
1.08
0.81
100.0
0.86
1.52
100.0
0.85
9.46
Cov-75
G=
o= 0.09
o= 1.01
d=
0.03
1.38
o=
d=
o= 6.86
99.5
1.09
0.62
98.5
0.79
1.21
0.80
6.36
W32-DD2
o= 6.8
1.48
0.80
o=
0.03
1.03
o=
0.03
5.14
98.6
1.11
0.67
0.86
1.35
0.88
9.52
W32-D4
11.8
0.08
0.88
12.3
0.03
1.23
41.9
0.03
7.37
99.4
1.17
0.72
98.7
0.93
1.20
0.95
6.54
W32-D6
Decoder capacity
7.7
0.08
0.81
11.5
0.03
1.06
17.6
0.03
5.17
100.0
1.03
0.58
100.0
0.79
1.17
100.0
0.80
6.19
W64-D2
0.0
0.07
0.77
0.0
0.03
0.99
0.0
0.02
5.09
67.3
1.16
0.76
68.6
0.93
1.42
52.2
0.94
7.10
W64-D6
46.9
0.08
0.93
46.4
0.03
1.28
50.0
0.02
5.50
99.9
1.04
0.57
99.7
0.79
1.22
95.3
0.80
6.13
W128-D2
3.7
0.07
01.76
5.2
0.03
1.03
0= 21.2
d= 0.02
5.08
100.0
1.10
0.82
100.0
0.86
1.51
100.0
0.87
9.70
W128-D4
G= 0.0
0.07
0.98
0.0
0.03
1.36
0.0
0.02
7.01
100.0
1.16
0.82
100.0
0.93
1.55
100.0
0.94
10.26
W128-D6
0= 0.0
0.08
o= 1.02
o= 0.0
o=
0.03
o= 1.41
o= 0.0
o= 0.06
o= 7.38
Table C.6: Mean and std. dev. for feasibility (%), time (ms), and optimality gap for each ablation: Blob with Bite.
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Published as conference paper ICLR 2026
Quadratic
Linear
Dist. Min.
Method
Feas (%)
Time (ms)
J Gap
Feas (%)
Time (ms)
Gap
1 Feas (%)
4 Time (ms)
Gap
100.0
1.37
1.62
100.0
0.52
1.70
100.0
0.53
11.54
Incomplete Coverage
Cov-10
o= 0.0
d= 0.33
d= 1.54
d= 0.0
o=
d= 1.61
o=
o= 0.03
o= 9.31
100.0
1.36
2.75
100.0
0.52
1.69
0.54
10.71
Cov-25
0.0
2.42
1.60
0.47
o= 9.97
100.0
1.36
2.43
100.0
0.52
1.60
100.0
0.53
9,69
Cov-50
0.0
0.13
2.16
d= 0.0
0.02
1.51
0.0
0.02
11.02
80.1
1.34
1.39
84.3
0.52
1.16
35.5
0.53
4.29
Cov-75
0.0
0.14
1.38
0.0
0.02
1,02
0.0
0.02
o= 5.27
99.5
0.94
0.95
94.5
0.48
1.18
14.7
0.47
1.82
W32-D2
om 0.0
om 0.37
a 0.80
ore 0.0
orm 0.02
0.89
0.0
0.02
or 1.65
99.0
0.97
1.36
95.8
0.52
1.32
23.7
0.51
3.75
W32-D4
o= 0.0
o= 0.19
o= 1.31
o= 0.0
o= 0,01
o= 1.11
o= 0.0
o= 0.01
o= 6,07
99.1
1.02
1.33
91.0
0.55
1.27
23.0
0.55
3.17
W32-D6
o= 0.0
o= 0.19
o= 1.27
o= 0.0
o= 0.01
o= 1.08
o= 0.0
o= 0.01
o= 5.37
Decoder capacity
W64-D2
98.7
0.94
0.96
95.3
0.47
1.19
14.7
0.47
1.44
o= 0.0
d= 0.19
(= 0.83
d= 0.0
o=
d= 0).86
o=
o= 0.01
o= 1.00
98.7
0.98
1.25
98.1
0.51
3.08
W64-D4
o= 0.0
o=
o= 1.16
0.0
o=
0.01
4.28
98.9
1.01
1.38
94.1
0.55
1.30
0.55
2.89
W64-D6
0.0
0.18
1.30
0.0
0.01
1.10
0.0
0.01
3.90
99.9
0.93
1.40
98.3
0.48
1.43
27.3
0.48
3.91
W128-D2
o= 0.0
0.19
1.32
0.0
o= 0.01
1.25
0.0
0.01
5.96
98.8
0.99
1.32
94.1
0.52
1.36
21.7
0.52
2.95
W128-D4
o= 0.0
d= 0.20
om 1.22
dm 0.0
d= 0.01
1.18
a= 0.0
o= 0.01
o= 3.96
99.5
1.03
1.35
96.9
0.56
1,40
16.4
0.56
3,22
W128-D6
o= 0.0
o=
0,21
o=
1.30
o= 0.0
o= 0.01
o=
1.22
o= 0,0
o= 0,01
o= 5,07
Table C.7: Mean and std. dev. for feasibility (%), time (ms), and optimality gap for each ablation: Concentric Circles.
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Published as conference paper ICLR 2026
Quadratic
Linear
Dist. Min.
Method
Feas (%)
J. Time (ms)
Gap
Feas (%)
Time (ms)
Gap
Feas (%)
4. Time (ms)
Gap
100.0
1.06
1.30
100.0
0.87
1.43
100.0
0.88
9.57
Incomplete Coverage
Cov-10
o= 0.0
o= 0.11
o= 1.25
o= 0.0
o= 0.02
o= 1.40
o= 0.0
o= 0.03
o= 7.80
100.0
1.04
1.21
100.0
0.86
1.00
100.0
0.87
5.31
Cov-25
o= 0.0
o= 0.09
o= 1.36
o=
o= 0.01
o= 0.99
o=
d= 0.02
o= 4.25
100.0
1.05
0.84
100.0
0.86
0,92
100.0
0.87
5.93
Cov-50
o=
0.09
o= 0.93
0.01
0.91
d=
d= 5.43
100.0
1.05
0.75
100.0
0.86
0,82
100.0
0.87
4.72
Cov-75
o=
0.09
o= 0.78
d==
0.01
0.75
d=
d= 4.39
1.01
0.41
100.0
0.81
0.76
3.66
W32-DD2
0.11
0.48
0.03
0.66
3.48
1.07
0.37
100.0
0.88
0.79
3.55
W32-D4
0.09
0).4
0.05
0.68
0.04
3.46
100.0
1.13
0.39
100.0
0.95
0.77
100.0
0.95
3.77
W32-D6
Decoder capacity
0.0
0.09
0.46
0.0
0.03
0.66
0.0
0.02
3.58
100.0
1.00
0.39
100.0
0.80
0.76
100.0
0.81
3.74
W64-D2
0.0
0.09
0.46
0.0
0.02
0.65
0.0
0.02
3.54
100.0
1.14
0.87
100.0
0.95
1.06
100.0
0.95
7.90
W64-D6
0.0
0.09
1.10
0.0
0.03
1.11
0.0
0.02
7.51
100.0
0.99
0.38
100.0
0.81
0.77
100.0
0.81
3.62
W128-D2
0.0
0.09
0).44
0.0
0.02
0.67
0.0
d= 0.02
d= 3.48
100.0
1.05
0.38
100.0
0.88
0.83
100.0
0.88
3.51
W128-D4
o= 0.0
0.09
0.44
0.0
0.01
0.73
0.0
0.02
T= 3.34
100.0
1.14
0..7
100.0
0.96
0.74
100.0
0.95
3,40
W128-D6
o= 0.0
0.09
o= 0.44
o= 0.0
0.01
o=
0.63
o= 0.0
o= 0.02
o= 3.39
Table C.8: Mean and std. dev. for feasibility (%), time (ms), and optimality gap for each ablation: Star Shaped.
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Published as conference paper ICLR 2026
Quadratic
Linear
Dist. Min.
Method
Feas (%)
Time (ms)
J Gap
Feas (%)
Time (ms)
Gap
1 Feas (%)
4 Time (ms)
Gap
100.0
1.48
1.64
100.0
0.52
1.70
100.0
0.53
11.54
Incomplete Coverage
Cov-10
o= 0.0
d= 1.05
o= 1.50
d= 0.0
o=
d= 1.61
o=
o= 0.03
o= 9.31
76.5
1,46
2.15
81.1
0.52
1.69
0.54
10.71
Cov-25
24.7
1.92
0.0
1.60
0.47
o= 9.97
92.7
1,45
0.82
100.0
0.52
0.53
9,69
Cov-50
26.5
0.11
0.90
0.0
0.02
1.51
0.0
0.02
11.02
91.2
1.46
0.90
84.3
0.52
1.16
23.1
0.53
4.29
Cov-75
28.3
0.13
1.02
0.0
0.02
1,02
0.0
0.02
o= 5.27
99.5
0,94
0.95
94.5
0.48
1.18
14.7
0.47
1.82
W32-D2
om 0.0
or 0.37
a 0.80
or 0.0
orm 0.02
0.89
0.0
0.02
or 1.65
99.0
0.97
1.36
95.8
0.52
1.32
23.7
0.51
3.75
W32-D4
o= 0.0
o= 0.19
o= 1.31
o= 0.0
o= 0,01
o= 1.11
o= 0.0
o= 0.01
o= 6,07
99.1
1.02
1.33
91.0
0.55
1.27
23.0
0.55
3.17
W32-D6
o= 0.0
o= 0.19
o= 1.27
o= 0.0
o= 0.01
o= 1.08
o= 0.0
o= 0.01
o= 5.37
Decoder capacity
W64-D2
98.7
0.94
0.96
95.3
0.47
1.19
14.7
0.47
1.44
o= 0.0
o= 0.19
(= 0.83
d= 0.0
o=
d= 0).86
o=
o= 0.01
o= 1.00
98.7
0.98
1.25
98.1
0.51
3.08
W64-D4
o= 0.0
o=
o= 1.16
0.0
o=
0.01
4.28
98.9
1.01
1.38
94.1
0.55
1.30
0.55
2.89
W64-D6
0.0
0.18
1.30
0.0
0.01
1.10
0.0
0.01
3.90
99.9
0.93
1.40
98.3
0.48
1.43
27.3
0.48
3.91
W128-D2
0.0
0.19
1.32
0.0
o= 0.01
1.25
0.0
0.01
5.96
98.8
0.99
1.32
94.1
0.52
1.36
21.7
0.52
2.95
W128-D4
o= 0.0
dr 0.20
om 1.22
dre 0.0
d= 0.01
1.18
a= 0.0
0.01
o= 3.96
99.5
1.03
1.35
96.9
0.56
1,40
16.4
0.56
3,22
W128-D6
o= 0.0
o= 0,21
o=
1.30
o= 0.0
o= 0.01
o=
1.22
o= 0,0
o= 0,01
o= 5,07
Table C.9: Mean and std. dev. for feasibility (%), time (ms), and optimality gap for each ablation: Two Moons.
D STATEMENT ON USE OF LARGE LANGUAGE MODELS (LLMS)
LLMs did not play significant role in research ideation and/or writing for this paper. Perplexity was used to facilitate literature discovery as part of the research process, alongside other (non-LLM-based)) tools. LLMs were also used to aid in sentence-level rewording for small number of sentences during the preparation of this manuscript.
24


Reviews:
[{"review_id": "VaxC74AM0Q", "reviewer": "Reviewer_3AaQ", "summary": "This paper introduces the Feasibility Autoencoder-Based (FAB) method, which uses an offline-trained autoencoder to project infeasible DNN outputs into a structured latent space for fast feasibility enforcement in constrained optimization. Key contributions include a two-phase training algorithm for the autoencoder, empirical speedups, and demonstrations of high feasibility with near-optimal performance on benchmarks like portfolio optimization.", "questions": "1. How does the autoencoder ensure optimality conservation during projection, and can you provide bounds or discuss failure modes?\n\n2. How does training handle multi-lable T_feas (could have multiple y per x)? Will different y harms the model performance as it could be confused on the targeted y.\n\n3. What are FAB's specific advantages over homeomorphic projections (e.g., in non-convexity or generalization), and how could the pretrained encoder-decoder be extended to new domains?\n\n4. The rationale of optimality consideration is questionable, like the decoder only see feaible/infeasible points, given an infeasible but close to optimal point, how could the model reconstruct a good solution with good optimality performance?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The method's originality lies in its pretrained encoder-decoder as a reusable feasibility projector, combining autoencoders with adversarial structuring for non-convex constraints and showing potential for generalization across tasks.\n\n2. Quality is good, with well-designed algorithms, and good presentation.\n\n3. The studied problem is important, as FAB enables real-time constraint handling in practical ML applications, potentially impacting fields like operations research and safe AI.", "weaknesses": "1. The lack of theoretical guarantees, especially on optimality preservation post-projection, weakens reliability, as the method relies on empirical data without bounds on gaps or failure modes. The feasibility and optimality guarantee is not presented and the method itself seems only rely on the performance of the decoder\n\n2. Contributions are vague compared to baselines like homeomorphic projections, which offer low-complexity feasibility guarantees; clearer differentiation (e.g., quantitative edges in non-convex settings) is needed.\n\n3. The rationale of optimality consideration is questionable, please see below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8jw3OQJstg", "reviewer": "Reviewer_nF3p", "summary": "The paper proposes FAB, a fast “approximate projector” built from a conditional autoencoder. Training has two phases: (1) reconstruct feasible points; (2) use a discriminator + simple penalties to make latent samples decode to feasible outputs. Experiments on several synthetic constraint families and one safe-RL task look promising.", "questions": "1. Could you report throughput vs. dimension and memory for different decoder widths and number of decoders, and a cost breakdown?\n\n2. Does a FAB trained on one hazard layout transfer to unseen layouts? Any results under domain shift?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Easy to attach after an existing predictor to improve feasibility with negligible latency.\n\n2. The two-phase training architecture is clear and easy to follow\n\n3. Tables report near-perfect feasibility and big time gains vs. homeomorphic projection and other baselines", "weaknesses": "It would further strengthen the paper if some form of theoretical guarantee—even a weak or probabilistic feasibility bound—could be established.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TTEmU7LBUp", "reviewer": "Reviewer_WGVG", "summary": "The paper proposes a FAB projection method, which encodes a prediction and context into a latent space, projects onto a simple convex set 𝑆, and decodes back to the original space. A two-phase training is used. Phase 1 reconstructs the feasible set from feasible samples, while Phase 2 structures the latent via a discriminator with a hinge loss, plus a latent loss and a Jacobian regularizer. FAB is then attached as a plug-and-play mapping 𝜙 around a base network.", "questions": "Two biggest concerns with L2O methods remain (1) the expensive training cost, and (2) the lack of feasibility guarantees. Can the authors discuss more about these two aspects?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The main contribution is an amortized, non-iterative feasibility mapper that is fast at inference and empirically effective on diverse nonconvex toy sets. \n2. Its plug-and-play design is attractive for deployment, since 𝜙 can be mounted onto arbitrary predictors without rewriting training loops.\n3. The Safe RL experiment demonstrates the safety and latency advantages.", "weaknesses": "1. FAB explicitly does not provide hard feasibility guarantees, making it unsuitable for safety-critical regimes, and no robustness evidence is given for distribution shift or dataset coverage failures that the authors acknowledge as limitations. \n2. Comparisons emphasize feasibility, but optimality gaps vary and many classical baselines (e.g., Projected Gradient, FSNet) can reach 100% feasibility as well.\n3. The experiments focus on toy nonconvex sets rather than realistic, parameter-dependent instances, so the path to real deployments remains unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)