Constrained Optimization

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The field studies how to optimize an objective over a feasible set, from convex programs and combinatorial problems to nonconvex models with complex or conflicting constraints. Classical approaches include projection and Frank–Wolfe methods, interior-point and augmented-Lagrangian algorithms, and evolutionary search; newer work also learns feasible solutions or uses neural projections and solver warm starts. Applications range from robotic trajectory planning and power-grid operation to fair machine learning, resource allocation, and scheduling. A central challenge is balancing scalable computation with reliable feasibility and convergence guarantees, especially for nonconvex or infeasible problems and settings without standard regularity conditions.

Latest papers

337 papers40 papers in the last four weeks, up 29% on the four weeks before. 0.2% of all new papers.

Jun 11, 2026

Neural Slack Variables for Shape Constraints

Enforcing functional inequality constraints such as monotonicity and convexity in neural networks is a fundamental challenge in many industrial and scientific applications. Classical one-sided penalty methods, along with primal-dual methods gated by complementary slackness, provide constraint gradients only at violated locations, resulting in fragile satisfaction. Architectures that guarantee feasibility by construction, on the other hand, remain largely limited to elementary cases and impose additional inductive biases. We introduce neural slack variables, a deep learning native primal-side approach that converts constraint enforcement into a regression problem by coupling the primary network with a jointly learned auxiliary network. The auxiliary network serves as a valid target for the primary network's constraint quantities, inducing feasibility and regularity. Neural slack variables achieve zero measured violations on dense-grid monotonicity and convexity test cases, where penalty and primal-dual baselines leave residual violations, and enable arbitrage-free learning of volatility surfaces, an open industrial challenge in quantitative finance.

Ruben Wiedemann, Antoine Jacquier, Lukas Gonon

Imperial College London London, UK · University of St. Gallen St. Gallen, Switzerland

Monotonicity Constraints Constrained Optimization

Jun 11, 2026

Different Layers, Different Manifolds: Module-Wise Weight-Space Geometry in Transformer Optimization

Weight-space geometry plays a central role in neural network optimization, yet manifold constraints are often applied uniformly across all weight matrices. In this work, we ask whether different transformer modules prefer different manifold geometries. We study Manifold Muon for GPT-2 pretraining and compare layer-wise assignments of Stiefel and DGram constraints across attention and MLP blocks. Our results show a clear asymmetry: constraining attention layers with Stiefel geometry while assigning DGram geometry to MLP layers gives the best performance among the tested configurations, whereas the inverted assignment and all-DGram configuration become unstable under the shared hyperparameter setting. We trace this failure to singular value growth in DGram-constrained attention weights, which can amplify attention logits and induce softmax saturation. These findings suggest that symmetry-aware and geometry-aware optimization for transformers should be module-specific rather than uniform.

Kirato Yoshihara

School of Engineering Science, The University of Osaka, Osaka, Japan.

Stiefel Manifold Optimization Neural Network Optimization Weight-Space Analysis Constrained Optimization Neural Network Training Dynamics

Jun 10, 2026

Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen--Cahn reaction identification, and unsteady Navier--Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at substantially reduced cost.

Zhen Zhang, Alessandro Alla, George Em Karniadakis

Division of Applied Mathematic, Brown University, Providence, RI, USA · Department of Mathematics, Sapienza Università di Roma, Rome, Italy

Constrained Optimization Physics-Informed ML PDE Inverse Problems

Jun 10, 2026

Range-Aware Bayesian Optimization for Discovering Diverse Designs within Target Property Windows

In many materials and product design problems, desirable candidates exhibit properties that fall within an acceptable range rather than achieve a single optimum. Recovering multiple, distinct solutions that satisfy such specifications is also practically valuable, as some candidates may be preferred for reasons of cost, processability, or robustness that are difficult to encode directly in an objective function. Here, we develop a range-aware Bayesian optimization (BO) framework in which the acquisition function directly scores the posterior probability that a candidate satisfies a target range. The framework naturally extends to parallel pursuit of multiple distinct specifications over a shared candidate space. Across benchmark tasks, range-aware acquisition consistently recovers larger and more diverse sets of valid designs than standard BO baselines and recent goal-seeking methods. Its utility is further demonstrated in two practically motivated design case studies involving optimizing reaction conditions for polymer synthesis and sequence-defined oligomer discovery for prescribed optical absorption bands, supported by quantum chemical calculations. These results suggest that range-aware BO can provide a practical and sample-efficient foundation for specification-driven design, particularly when design flexibility and solution diversity are important considerations.

Shengli Jiang, Jason Wu, Charles M. Schroeder +1

Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08540, USA

Molecular Optimization Bayesian Optimization Safe Bayesian Optimization Constrained Optimization Quality-Diversity Optimization

Jun 9, 2026

PL-KKT-hPINN: Enforcing Nonlinear Equality Constraints on Neural Networks via Piecewise-Linear Projection

While physics-informed neural networks (PINNs) have shown strong potential for process modeling, physical equations are only enforced as soft constraints during training, and thus, they do not guarantee constraint satisfaction at inference. We propose a framework, called piecewise-linear Karush--Kuhn--Tucker hard-constrained PINNs (PL-KKT-hPINNs), that strictly enforces nonlinear equality constraints through piecewise-linear projection. This extends the KKT-hPINN framewor, which exactly enforces linear equalities through the Karush--Kuhn--Tucker (KKT) conditions associated with orthogonally projecting neural network outputs onto the constraint feasible region. The method is demonstrated on a continuous stirred-tank reactor (CSTR) case study for both one and two inputs. Results show that PL-KKT-hPINN preserves predictive accuracy comparable to that of a standard neural network while achieving substantially lower constraint violations. In addition, the proposed model shows improved robustness in low-data regimes, yielding lower RMSE than the unconstrained neural network for limited training sample sizes. These results demonstrate that PL-KKT-hPINN provides a computationally efficient and physically consistent framework for surrogate modeling of nonlinear chemical engineering systems.

Fateme Mohammad Mohammadi, Hector Budman, Joshua L. Pulsipher

Department of Chemical Engineering University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada

Neural Surrogate Modeling Constrained Optimization Physics-Informed ML

Jun 9, 2026

LieIPM: Lie Group Interior Point Method for Direct Trajectory Optimization of Rigid Bodies

Designing dynamically feasible trajectories for rigid bodies is a fundamental problem in robotics. While direct methods are widely used, the existing constrained optimizers typically operate in Euclidean space and ignore the manifold structure of rigid body motions. This mismatch may introduce singularities or lead to poorly conditioned optimization problems. To bridge this gap, we develop a structure-aware framework for constrained trajectory optimization directly on matrix Lie groups. Our approach is based on the second-order rigid body models utilizing Lie group structures, which enables efficient Newton-type updates while preserving the underlying geometry. Building on this model, we propose a line-search Lie Group Interior Point Method (LieIPM) to handle constraints on the manifolds. We instantiate the framework for rigid body motion planning using Lie group variational integrators and derive closed-form intrinsic derivatives that exploit group symmetries. The LieIPM preserves the topology of rotation motions by construction and avoids singularities. Numerical results demonstrate superior robustness and faster convergence compared to general-purpose solvers and structure-exploiting optimal control methods.

Sangli Teng, Ruiqi Zhang, Tzu-Yuan Lin +5

University of California, Berkeley · MIT · Ohio University +1

Trajectory Optimization Riemannian Optimization Lie Group Methods Constrained Optimization Robot Motion Planning

Jun 8, 2026

Duality for Optimal Multi-Item, Multi-Bidder Auction Design: Revenue Certificates through Deep Learning

Characterizing revenue-optimal auctions for multi-item, multi-bidder settings remains a fundamental open problem, with no known closed-form solution existing beyond restrictive binary-type instances. This has motivated interest in computational approaches to optimal auction design. In this paper, we introduce the first computational framework that directly tackles the dual problem for multi-item, multi-bidder auctions and dominant-strategy incentive compatibility (DSIC), generating certified revenue upper bounds. Our approach parametrizes Lagrange multipliers with a structurally guaranteed strict flow-conservation property using neural networks, enabling efficient optimization over feasible dual solutions via gradient descent. To bridge the gap between discrete computational methods and theoretical guarantees for continuous types, we develop a novel lifting technique that maps dual certificates from coarse discretizations to fine refinements. We prove that lifting gives valid revenue upper bounds for multi-item, multi-bidder auctions with continuous uniform valuations. Furthermore, we give a generalized lifting construction for arbitrary continuous distributions and demonstrate that these lifted duals converge to the revenue of the original continuous problem in the discrete limit. We validate this computational framework for the dual auction design problem by recovering known analytical mechanisms for canonical instances. For multi-item multi-bidder problems, our framework establishes a small gap between the optimal revenue and best-known DSIC mechanisms, providing computational certificates of near-optimality.

Yanchen Jiang, David C. Parkes, Tonghan Wang

Harvard University, USA · College of AI, Tsinghua University, China

Primal-Dual Optimization Game Theory Constrained Optimization

Jun 7, 2026

Parameter Tuning with Generalization Guarantees for GPU-Accelerated Linear Programming

Recent research has developed practical, parallelizable first-order methods for large scale linear programming, but performance is highly dependent on hyperparameter selection. We derive generalization guarantees for hyperparameter tuning within (cu)PDLP, a state-of-the-art first-order LP solver designed for modern hardware. First, we pin down the behavior of PDHG, the primal-dual hybrid gradient algorithm that underlies PDLP, as a function of its step size and primal weight, leading to linear sample complexity guarantees for learning those parameters. We then conduct a structural analysis of PDLP, which augments PDHG with several specialized techniques like preconditioning, adaptive step sizes, averaging, adaptive restarts, and smoothed primal weight updates. Our analysis captures the behavior of the solution trajectory as a function of the hyperparameters and leverages recent advances in data-driven algorithm design to obtain polynomial sample complexity guarantees for learning those hyperparameters. Finally, we conduct proof-of-concept experiments that demonstrate the need for data-driven PDLP parameter tuning. Our results showcase the versatility of the data-driven algorithm design toolkit for principled hyperparameter tuning within solver-grade implementations of complex modern optimization algorithms.

Siddharth Prasad, Dravyansh Sharma

Toyota Technological Institute at Chicago

Primal-Dual Optimization Constrained Optimization Linear Programming Hyperparameter Optimization

Jun 7, 2026

Bayesian Optimization of a Multi-Product Chemical Reactor Using Composite Models and Partial Physics Knowledge

We study data-driven real-time economic optimization of a multi-product chemical reactor when no reliable first-principles model is available beyond a steady-state energy balance. Instead of learning the economic objective directly as a black-box function, we use a composite formulation in which Gaussian process (GP) models predict physically meaningful outputs, including product concentrations and reactor temperature, while profit is computed analytically from these predictions together with raw-material, product, and utility prices. This preserves the structure of the economic objective, makes it parametric in changing prices without needing retraining, and allows candidate operating points to be checked against the available energy balance through a physics residual. The GPs also provide predictive uncertainty, which is exploited in a Bayesian optimization (BO) framework both for data-efficient exploration and for conservative enforcement of the reactor temperature constraint through an upper confidence bound. The acquisition function additionally penalizes large energy-balance mismatch obtained by substituting the GP-predicted outputs and candidate inputs into the available steady-state energy balance. The approach is demonstrated on a benchmark simulation of a non-isothermal multi-product reactor. Relative to a trust-region safe BO implementation, the proposed method achieves better simulated economic performance within the available iteration budget. Relative to a purely data-driven BO approach that does not use the available physics information, it avoids reactor temperature constraint violations.

Liqiu Dong, Marta Zagórowska, Mehmet Mercangöz

Department of Chemical Engineering, Imperial College London, London, UK · DCSC, Delft University of Technology, Delft, Netherlands

Bayesian Optimization Safe Bayesian Optimization Constrained Optimization

Jun 5, 2026

Residual-Controlled Multiplier Learning for Stochastic Constrained Decision-Making

Stochastic constrained decision-making requires optimizing performance objectives while enforcing statistical requirements such as safety or fairness. However, standard primal--dual methods struggle to update multipliers robustly under stochastic mini-batch feedback, as the noise of mini-batch gradients and constraint estimates can be directly accumulated into the multiplier memory. To address this issue, we propose Residual-Controlled Multiplier Learning (RCML), which reformulates multiplier updating as projected-pressure feedback. The central idea is to decompose the projected multiplier into an effective pressure signal for primal descent and a pressure-memory residual for finite-gain multiplier tracking. To handle heterogeneous and noisy observations, we further augment this residual-integral backbone with modular stochastic stabilization components. For the convex-affine backbone, we establish finite-gain convergence, derive a stochastic residual bound under mini-batch feedback, and show that the residual feedback law admits a local KKT-residual interpretation near regular KKT points of nonconvex problems. Experiments across optimization, allocation, and fair-ranking tasks show that RCML improves feasibility control and multiplier stability while maintaining competitive objective performance. Code is released at https://anonymous.4open.science/r/RCML-3114/.

Kang Liu, Jianchen Hu, Ziyu Qu +3

School of Future Technology Xi’an Jiaotong University Xi’an, China, 710049 · School of Automation Science and Engineering Xi’an Jiaotong University Xi’an, China, 710049 · School of Mathematics and Physics China University of Geosciences Wuhan, China, 430074 +2

Stochastic Optimization Primal-Dual Optimization Constrained Optimization

Jun 4, 2026

Multi-ResNets for Subspace Preconditioning in Constrained Optimization

We propose MResOpt, a staged residual neural network architecture for constrained optimization problems. Our architecture fits within predict-complete-correct pipelines and decomposes constraint satisfaction by priority through intermediate re-completion and stage-aware losses. The framework enables domain-informed ordered constraint satisfaction which allows the network to utilize ordinal structure when present. Under an idealized infinite-width regime, we show that our design behaves as sequential Gaussian Process regression. On synthetic QP, QCQP, and SOCP benchmarks, the staged architecture improves high-priority constraint satisfaction across convex and non-convex settings. On line-flow-constrained AC optimal power flow, we introduce a physics-motivated constraint ordering and show that MResOpt supports a learned division of labor that keeps iterates on the equality manifold, achieving substantially lower high-priority violation than reprojected baselines while remaining computationally efficient.

Merve Karakas, Christopher J. Williams, Emmanuel O. Balogun +3

UCLA · University of Oxford · Sunrun† +1

Residual Learning Learning to Optimize Optimal Power Flow Constrained Optimization Constraint Satisfaction

Jun 4, 2026

Finding Most Influential Sets

Identifying most influential sets (MIS) - size-

k

subsets whose removal maximally changes a target estimand - is typically infeasible because it requires searching over

\binom{n}{k}

subsets. For estimands with linear-fractional leave-set-out effects, we show that MIS selection reduces to a one-parameter sequence of top-

k

problems. Dinkelbach's method yields an algorithm with

\mathcal{O}(n)

cost per iteration and finite termination. For fixed residualized inputs, the algorithm returns a globally optimal set for the univariate ratio objective, including the oracle-residualized partial linear model. With estimated nuisance functions, uniform denominator and generated-score stability imply approximation to the first-order oracle orthogonal-score objective; exact set recovery follows under a separation condition. Simulations and applications show that the method recovers exact MIS that were previously computationally inaccessible.

Lucas D. Konrad, Nikolas Kuschnig

Vienna University of Economics and Business, Austria · Monash University, Australia

Constrained Optimization

Jun 3, 2026

R-APS: Compositional Reasoning and In-Context Meta-Learning for Constrained Design via Reflective Adversarial Pareto Search

Large language models (LLMs) are fluent on open-ended tasks, yet in agentic settings, where a system must plan, use tools, and act over extended horizons, fluency does not ensure reliable delivery. We trace this gap to three coupled structural failures: errors propagate without localization, worst-case perturbations go unevaluated, and accumulated knowledge is never invalidated. We argue these share a root cause: abductive, counterfactual, meta-inductive, corrective, and inductive reasoning pull a shared context in incompatible directions. We introduce Reflective Adversarial Pareto Search (R-APS), to our knowledge the first method addressing all three failures jointly via reasoning-mode decomposition, allocating each reasoning mode its own context and orchestrating interaction across three timescales: staged compositional reasoning with a typed validation critic (failure localization), sensitivity-guided counterfactual stress-testing as a first-class Pareto objective (robustness), and meta-inductive rule extraction with explicit invalidation (persistent memory). R-APS requires no fine-tuning and operates on a frozen LLM purely via structured protocol design. We evaluate on planar mechanism synthesis (robotics, prosthetics, mechanical design), with every candidate checked by a kinematic solver. On 32 target trajectories, R-APS delivers robustness certificates 3.5x tighter than uniform-perturbation baselines, 46% faster iterations-to-first-admission, and 2.1x Chamfer-distance reduction over Enum+GA while jointly controlling bar-count and worst-case robustness. Small 4B reasoning-specialized models prove competitive with general-purpose 70B backbones inside the protocol, suggesting structured protocols can partially offset model scale.

João Pedro Gandarela, Thiago Rios, Stefan Menzel +1

Idiap Research Institute, Switzerland · École Polytechnique Fédérale de Lausanne (EPFL), Switzerland · Honda Research Institute Europe, Germany +2

Mechanism Design Meta-Learning Compositional Reasoning Constrained Optimization

Jun 3, 2026

DiffSlack: Learning under Nonlinear Inequality Constraints via Learnable Slack Variables

Enforcing nonlinear inequality constraints in neural networks remains challenging, especially when the output is subject to many coupled constraints. Existing hard constraint methods often impose structural restrictions on the constraint set or introduce substantial computational overhead for large-scale nonlinear problems. Here, we propose DiffSlack, a differentiable projection layer for nonlinear inequality-constrained neural prediction. DiffSlack reformulates inequalities as equalities with learnable slack variables, which are predicted as part of the augmented network output and provide a data-driven warm start for damped Gauss-Newton projection. The projection layer maps raw predictions onto the augmented feasible manifold while preserving end-to-end differentiability. A two-stage curriculum further stabilizes training and improves constraint satisfaction. We evaluate DiffSlack on vehicle path planning with 200 nonlinear inequality constraints from collision avoidance, curvature limits, and waypoint spacing. Compared with existing learning-based baselines, DiffSlack achieves a higher planning success rate and stronger geometric constraint satisfaction under a comparable inference budget. Ablation studies further show that the hard projection layer reduces sensitivity to supervision quality. Closed-loop tracking in CARLA and real-world vehicle experiments confirms the executability of the generated trajectories. These results demonstrate that DiffSlack provides a practical and scalable approach to embedding hard inequality constraints into neural networks for engineering applications.

Ziqian Wang, Chenxi Fang, Zhen Zhang

State Key Laboratory of Tribology in Advanced Equipment, Tsinghua University, Beijing 100084, China · Beijing Key Laboratory of Transformative High-end Manufacturing Equipment and Technology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China · Automotive Electronics Business Unit at Hirain Inc., China

Constrained Optimization Differentiable Optimization

Jun 3, 2026

Optuna Constrained Tree-Structured Parzen Estimator Is a Joint Density Generalization of c-TPE

Constrained hyperparameter optimization (HPO) is common in practice, yet Optuna's widely used constrained TPE lacks algorithmic analysis. While c-TPE proposes an expected constrained improvement (ECI) approach assuming independence between the objective and constraints, Optuna uses a single joint density over both. We show that Optuna's constrained TPE is joint c-TPE -- the same ECI acquisition function using a joint likelihood. We demonstrate joint c-TPE is invariant to constraint duplication whereas independent c-TPE degrades as the product accumulates duplicated factors. We outline practical tradeoffs between the formulations and directions for future study.

Shuhei Watanabe, Kaichi Irie

Independent Researcher · Kyoto University

Bayesian Optimization Constrained Optimization Hyperparameter Optimization

Jun 2, 2026

Rethinking Neural Width for Alternating Current Optimal Power Flow Proxies

Deep learning proxies for Alternating Current Optimal Power Flow (ACOPF) lack systematic methods for determining architectural size. This paper conducts a constructive thought experiment to answer a fundamental inquiry: how wide must a neural network be to almost accurately approximate the ACOPF manifold? We introduce a Loss-Guided Neural Densification (LG-ND) algorithm that incrementally discovers necessary capacity by expanding only when the current deep neural network topology fails to improve further. Empirical results across various IEEE systems show that LG-ND achieves performance parity with literature baselines using up to ten times fewer neurons per layer. Such architectural minimalism is critical for the formal verification required in safety-critical grid operations.

Dhruvi Khandelwal, Anurag Basistha, Ayushi Jolotia +1

Department of Electrical Engineering, National Institute of Technology Kurukshetra, India. · Indraprastha Institute of Information Technology Delhi, India. · Department of Electrical Engineering, Indian Institute of Technology Roorkee, India.

Neural Surrogate Modeling Optimal Power Flow Constrained Optimization Neural Network Approximation Theory Neural Architecture Search Neural Network Expressivity

Jun 2, 2026

The Shadow Price of Reasoning: Economic Perspective on Optimal Budget Allocation for LLMs

Inference-time scaling has emerged as a critical avenue for enhancing Large Language Models' performance, yet real-world deployment is constrained by strict computational budgets. In this work, we formulate inference budget allocation as a global constrained optimization problem governed by economic principles. By modeling per-query reasoning utility with a shifted-surge function, we derive an optimal allocation policy based on a global shadow price that equilibrates marginal utility under resource scarcity. Based on this theory, we propose Constrained Latent-utility Equilibrium Allocation for Reasoning (CLEAR). It performs rational abandonment and reallocates resources from insolvent queries to solvable queries near their emergence thresholds. Extensive experiments on several reasoning tasks with different traffic streams demonstrate that CLEAR significantly improves the Pareto frontier of total token cost versus mean accuracy. In resource-scarce regimes, CLEAR achieves up to a 3x improvement in global accuracy compared to uniform allocation.

Xu Wan, Speed Zhu, Jianwei Cai +4

1Zhejiang University · 2Tecent HY Team · 3Peking University

Cost-Aware Inference Inference-Time Scaling Constrained Optimization Inference-Time Compute Allocation

Jun 1, 2026

Coordination Graphs for Constrained Multi-Agent Reinforcement Learning

Constrained Multi-agent reinforcement learning (CMARL) faces two intertwined challenges: the joint action space grows exponentially with the number of agents, and additional requirements couple agents in ways that reward structure alone does not capture. We introduce Coordination Graphs for Constrained Multi-Agent Reinforcement Learning (CG-CMARL), a framework that addresses both challenges by combining coordination graphs with Lagrangian duality. The system decomposes the joint problem into pairwise regions, each served by a set of shared Q-functions, one for the primary objective and one for each of the constraints, so that the number of learned models is independent of the number of agents. At execution time, Max-Sum message passing coordinates actions across the factor graph, while a Lagrangian multiplier controls the objective--constraint tradeoff, allowing a single trained model to trace a Pareto front without retraining. We provide convergence guarantees under mild conditions, together with a compositional error bound that decomposes into separate interpretable sources, each traceable to a specific design choice and independently controllable. Experiments on cooperative navigation tasks (where teams of up to 10 agents must coordinate to reach target positions while satisfying pairwise constraints) show that our method produces Pareto fronts dominating established baselines trained at fixed reward-shaping ratios, while scaling to team sizes where centralized approaches become intractable.

Santiago Amaya-Corredor, Miguel Calvo-Fullana, Anders Jonsson

Department of Engineering, Universitat Pompeu Fabra, Barcelona, Spain

Multi-Objective Reinforcement Learning Multi-Agent System Optimization Constrained RL Constrained Optimization Cooperative MARL Constrained MARL

Jun 1, 2026

Everywhere Learning: Artificial Intelligence with Pointwise Constraints

Everywhere learning is a new paradigm whereby Artificial Intelligence (AI) systems are trained to satisfy loss constraints with probability one over the data distribution. This is in contrast to the standard paradigm of training AI systems to minimize average losses. We develop an approximate duality theory to substantiate a generalization analysis that establishes the proximity between solutions of empirical and statistical everywhere learning problems. Our results show that dual variables reweigh the data distribution towards points in which loss constraints are more difficult to satisfy and that generalization is controlled by the mismatch between the concentration of mass of the data distribution and the concentration of mass on points where constraints are more difficult to satisfy. We further show that we can control generalization with a sparse L1 penalty on constraint relaxations. We illustrate the merits of everywhere learning with an experiment in agentic classification for language model tasks.

Ignacio Boero, Ignacio Hounie, Luiz Chamon +1

Department of Electrical and Systems Engineering, University of Pennsylvania · École polytechnique, Institut Polytechnique de Paris

Constrained Optimization Statistical Learning Theory

May 31, 2026

Global Convergence of a Line-Search Filter Differential Dynamic Programming Method

In this article, we establish the global convergence properties of the FilterDDP algorithm, which extends the discrete-time differential dynamic programming (DDP) algorithm of Mayne and Jacobson [International Journal of Control, 3, (1966), pp. 85-95] to handle nonlinear constraints over states and controls, in addition to the dynamics. FilterDDP adopts a line-search filter procedure for step acceptance. However, instead of a damped Newton step applied in the general nonlinear programming setting, the computation of a trial point involves applying a backward recursion and a forward simulation. We establish the global convergence of FilterDDP by showing that for a subset of constrained optimal control problems, the this backward-forward procedure satisfies the same properties as a Newton step for the purpose of establishing global convergence of a line-search filter method, following the analysis of Wächter and Biegler [SIAM Journal on Optimization, 16 (2005), pp. 1-31].

Ming Xu, Iman Shames

School of Computer and Communication Sciences EPFL 1015 Lasanne, Switzerland · Department of Electrical and Electronic Engineering University of Melbourne Parkville, VIC, 3010, Australia

Optimization Convergence Analysis Constrained Optimization Optimal Control

May 30, 2026

Infeasible optimization problems and the hierarchical augmented Lagrangian method in imitation learning

Imitation learning (IL) is an effective approach to train complex robotics policies. Recent works have introduced hard constraints into imitation-learning optimization problems to ensure safety, stability, and robustness of the learned policy. However, we argue that these constraints are sometimes infeasible, which can lead to unstable or difficult training dynamics. We study a simple remedy for such situations based on recent theoretical results on the augmented Lagrangian method in infeasible settings. We show that our approach drives the learned policy toward the solution of a closest-feasible constrained IL problem with desirable properties. The method is illustrated on a toy driving example with a total-acceleration constraint and pedestrian-safety constraints, a setting in which infeasibility can naturally arise while still allowing a safe learned policy.

Roland Andrews, Justin Carpentier, Ajay Sathya

Imitation Learning Augmented Lagrangian Method Constrained Optimization

May 29, 2026

Unlearning in Diffusion Models: A Unified Framework with KL Divergence and Likelihood Constraints

Unlearning in diffusion models aims to remove undesirable data or concepts while preserving the utility of pretrained models -- two fundamentally conflicting objectives. We propose a principled constrained optimization framework that formulates unlearning as minimizing the deviation from a pretrained model, subject to explicit separation constraints from the unlearning distributions. Specifically, we formulate three constrained optimization problems based on reverse and forward KL divergences, and likelihood constraints. The first two generalize existing approaches for concept and data unlearning, while the third offers a novel and natural formulation for unlearning. Despite the nonconvexity of the KL constraints, we establish strong duality for all three problems, enabling us to explicitly characterize their optimal solutions as unlearning targets and develop primal-dual algorithms for each formulation. Experimental results demonstrate that our KL-constrained approach achieves superior retention-unlearning tradeoffs compared to weight-based baselines for concept and data unlearning, and that our likelihood-based approach matches unlearning effectiveness while better preserving retained concepts compared to baselines.

Shervin Khalafi, Alejandro Ribeiro, Dongsheng Ding

University of Pennsylvania · University of Tennessee, Knoxville

Primal-Dual Optimization Diffusion Model Unlearning Constrained Optimization Machine Unlearning Diffusion Models

May 28, 2026

Exploiting Chordal Sparsity for Globally Optimal Estimation with Factor Graphs

Robust and efficient state estimation is crucial for perception, navigation, and control in robotics. State estimation problems are conveniently modeled using the factor-graph framework as enabled by modern software packages such as GTSAM or g2o. However, the standard solvers included in such frameworks are local and may converge to poor local minima, posing significant safety concerns. Conversely, techniques based on convex relaxations have been shown to provide a means of globally solving or certifying many state estimation problems. However, these relaxations 1) often require substantial effort to formulate, and 2) may incur significantly higher cost compared to efficient local solvers, as they require solving a large semidefinite program (SDP). In this work, we address both shortcomings by 1) creating a new procedure within the GTSAM framework for automatically constructing convex SDP relaxations for any factor graphs with common factor and variable types, and by 2) exploiting the Bayes tree constructions native to GTSAM to decompose the SDP problem, leading to significant speedup in solver time for chordally sparse problems. We demonstrate the favorable scaling of this structure-exploiting global estimator compared to standard local solvers for two case studies: A 3D pose-graph SLAM problem with a ring factor graph and a 2D localization problem with a chain factor graph. The software framework is available at https://github.com/borglab/gtsam.

Avinash Subramanian, Connor Holmes, Timothy D. Barfoot +2

College of Computing, Georgia Institute of Technology, Atlanta, GA, USA · Robotics Institute, University of Toronto, Toronto, ON, Canada · Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, USA

Semidefinite Programming Factor Graph Optimization Constrained Optimization

May 27, 2026

Bound-Constrained Sparse Representation for Electrical Impedance Tomography

This study proposes a bound-constrained sparse representation (BC-SR) framework for electrical impedance tomography (EIT), aimed at improving conductivity estimation without explicit regularization. BC-SR adopts a representation-driven strategy, generating conductivity from low-dimensional latent variables via an implicit composite parameterization. Structural priors are embedded using a truncated graph-Laplacian basis, while a bound-preserving nonlinear mapping enforces admissible conductivity ranges and improves conditioning through implicit gradient modulation. The approach ensures robust convergence, even under noisy or incomplete data. Extensive validation on 2D/3D simulations, tank experiments, and in-vivo lung data shows that BC-SR improves physical consistency and structural fidelity, offering enhanced robustness compared to traditional methods. Additionally, BC-SR enables 3D time-difference EIT reconstruction, offering improved spatial resolution and a more coherent representation of 3D conductivity distributions, particularly for in-vivo lung data. This suggests potential for improved performance in EIT, particularly in clinical applications for respiratory monitoring.

Chun Zhang, Dong Liu

School of Biomedical Engineering and Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215123, China · Laboratory of Spin Magnetic Resonance and Anhui Province Key Laboratory of Scientific Instrument Development and Application, University of Science and Technology of China, Hefei 230026, China · Jiangsu Provincial Key Laboratory of Multimodal Digital Twin Technology, Suzhou 215123, China

Sparse Recovery Image Reconstruction Constrained Optimization Image Inverse Problems

May 27, 2026

Constrained Auto-Bidding via Generative Response Modeling

Auto-bidding systems aim to maximize advertiser value over long horizons under budget constraints and ratio targets such as cost-per-acquisition, yet future traffic and auction dynamics are non-stationary and uncertain. Existing approaches face distinct limitations: control-based pacing reacts to deviations but cannot anticipate future conditions, while RL and generative methods fold constraints into reward signals, obscuring violations and degrading under distribution shift. We shift the learning target from actions to responses with the Generative Response Model (GRM), a history-conditioned sequence model that jointly predicts future traffic volume and horizon-aggregate cost/value curves as functions of a single bid multiplier. We show that under mild monotonicity conditions, the optimality gap relative to full per-tick control is bounded by the dispersion of per-tick marginal value-per-cost. Given predicted responses, a lightweight analytic controller enforces each active constraint via a 1D root-finding step. We prove this controller is exact for the single-multiplier problem and bound constraint violations under receding-horizon replanning in terms of prediction error. Experiments on AuctionNet show that GRM improves constraint stability and overall score compared to existing baselines.

Eunseok Yang, Xingdong Zuo, Kyung-Min Kim

NAVER Corporation Seongnam-si, Republic of Korea

Automated Bidding Online Advertising Constrained Optimization

May 26, 2026

Developing a Totally Unimodular Linear Program for Optimal Conformance Checking: When and Why It Complements A*

Alignment-based conformance checking is the state-of-the-art approach for comparing observed process executions with normative process models. The standard exact solution relies on an A*-based heuristic search, which can exhibit exponential runtime in the presence of long traces or substantial deviations. This paper introduces a reformulation of alignment-based conformance checking as a totally unimodular linear program (LP) defined on the reachability graph of the synchronous product. By exploiting the underlying network-flow structure, the proposed formulation guarantees the existence of an integral optimal extreme-point solution through LP relaxation, thereby avoiding the combinatorial overhead associated with integer variables and branch-and-bound search. We conduct an extensive empirical evaluation on more than 2.1 million conformance checking instances derived from real-world and synthetic benchmark datasets. The results show that A* and the LP approach exhibit complementary performance characteristics: the former performs best on short, well-conforming traces, while the LP formulation provides substantial speedups for longer traces with deviations, precisely where conformance checking is most informative. Based on these findings, we derive simple algorithm-selection guidelines that combine both approaches, achieving average runtime savings of 38.6% with 96% selection accuracy compared to always using A*.

Izack Cohen

aFaculty of Engineering, Bar-Ilan University, Ramat Gan, Israel

Algorithm Selection Process Mining Combinatorial Optimization Constrained Optimization Linear Programming

May 24, 2026

Blocked Gibbs meets Diffusion Transformers: Unsupervised Learning for Constraint Optimization

Diffusion models have shown promise in learning to solve constraint optimization problems. However, they are mostly restricted to problems with binary variables and rely on graph neural networks, hindering their application to a broader range of problems such as those with general discrete variables or constraint structures that necessitate global rather than local reasoning. We investigate the use of Diffusion Transformers to address the aforementioned limitations. A naive implementation performs poorly due to a fundamental mismatch between the standard diffusion process and constraint solving: while the former applies small, incremental denoising across all variables, the latter requires substantially altering specific subsets of variables to attain feasibility or optimality. Our method, Blocked Gibbs Diffusion Transformer (BloGDiT), is the first to address this limitation by replacing standard joint Gaussian denoising with blocked Gaussian denoising. BloGDiT uses iterative block resampling and anneals the block size over time to facilitate large, targeted edits within a block of variables. Across Sudoku, Graph Coloring, Maximum Independent Set, and MaxCut, BloGDiT matches or outperforms existing methods, demonstrating that blocked Gibbs-style diffusion provides a highly effective inductive bias for Transformer-based constraint satisfaction and optimization.

Yudong W. Xu, Wenhao Li, Xiaoyu Wang +2

University of Toronto · Vector Institute

Diffusion Transformer Constrained Optimization Constraint Satisfaction Neural Combinatorial Optimization

May 24, 2026

Mitigating Gradient Pathology in PINNs through Aligned Constraint

While Physics-Informed Neural Networks (PINNs) are powerful for solving Partial Differential Equations (PDEs), their training is often paralyzed by gradient pathology. The gradients from the PDE residuals and boundary constraints oppose each other, trapping the model in local minima. Current solutions, such as adaptive weighting or hard constraints, either fail to fundamentally resolve this ill-conditioning or are limited to simple geometries. In this study, we systematically analyze the possible causes of this gradient pathology from the perspectives of loss landscapes and optimization dynamics. Based on the obtained conclusion, we propose Constraint-Aligned loss with Manifold Lifting (CAML). By reformulating all zeroth-order terms into aligned constraints, our method effectively mitigates gradient conflicts. In addition, we introduce a delay factor to help the optimizer skip the high-curvature area. Experiments demonstrate that our CAML significantly enhances numerical stability and efficiency in highly complex PINN problems. Our code is open-sourced on https://github.com/YichenLuo-0/CAML.

Yichen Luo, Peiyu Zhu, Dongxiao Hu +5

Department of Information Science and Engineering, KTH Royal Institute of Technology, Stockholm, Sweden · School of Advanced Manufacturing and Robotics, Peking University, Beijing, China · School of Advanced Technology, Xi’an Jiaotong-Liverpool University, Suzhou, China +2

Neural PDE Solvers Gradient Interference Constrained Optimization Physics-Informed ML Neural Network Training Dynamics

May 24, 2026

Manifold-Constrained MPPI: Real-Time Sampling-Based Control for Nonlinear Equality-Constrained Robotic Systems

Sampling-based model predictive control methods, such as Model Predictive Path Integral (MPPI), offer derivative-free optimization and robustness in complex robotic systems. However, standard MPPI relies on cost-based soft penalties that cannot guarantee hard-constraint satisfaction, severely limiting its applicability to highly constrained tasks such as closed-chain manipulation. To address this, we propose Manifold-Constrained MPPI (MC-MPPI), a real-time sampling-based control framework that regulates nonlinear equality constraints while preserving the computational advantages of MPPI. The key idea is to decouple the constrained optimal control problem into latent-space planning and execution-level correction. At the planning stage, a Variational Autoencoder (VAE) learns a low-dimensional latent representation of the constraint manifold, enabling MPPI to efficiently generate near-feasible candidate trajectories without per-sample modification. Since this reference enables accurate linearization of the equality constraints, an execution-level Quadratic Programming (QP) controller resolves the residual manifold mismatch in a single solve rather than through iterative projection. Experiments on a 14-DoF closed-chain dual-arm system in both simulation and real-world settings demonstrate 100 Hz planning and 500 Hz execution, equality-residual regulation in static and dynamic environments, and a 95% success rate over 40 randomized hardware trials. Supplementary videos and implementation details are available at https://rcilab.github.io/mcmppi.

Seulchan Lee, Sanghyun Kim

School of Mechanical Engineering, Kyung Hee University, Yongin, South Korea

Robotic Control Constrained Motion Planning Constrained Optimization MPPI Control

May 22, 2026

Scalable Heterogeneous Graph Foundation Models for Data-Driven Optimal Power Flow in Smart Grids

Fast and reliable optimal power flow (OPF) approximation is important for power system operation, yet heterogeneous OPF graph models are often evaluated either with architecture-specific implementations or on limited training corpora. This paper presents a large-scale heterogeneous graph-learning workflow, built on HydraGNN, for data-driven OPF surrogate modeling and graph foundation-model (GFM) development. The workflow preserves buses, generators, loads, shunts, alternating-current (AC) lines, transformers, and device-to-bus relations and provides a common implementation for distributed preprocessing, multi-graphics-processing-unit (GPU) training, hyperparameter optimization (HPO), and downstream adaptation. Using approximately three million heterogeneous graphs from ten Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms (PGLib-OPF) cases spanning 14 to 13,659 buses, we compare six heterogeneous graph neural network (GNN) families under a common HPO campaign on Frontier, resulting in two compact HeteroSAGE and HeteroHEAT models (~1.6-1.7 million parameters) that achieve the lowest observed validation MSE. Strong and weak scaling using up to 1,024 Frontier nodes show that 256 nodes (corresponding to 1,024 AMD MI250X GPUs) provides the best time-to-solution balance for the measured workload. Two downstream tasks evaluate task adaptation on the IEEE 118-bus case: a classification task to distinguish nominal operating conditions from artificially generated extreme-overload conditions and N-1 single-component-outage regression to predict bus voltage magnitude and phase angle. Partial fine-tuning provides the strongest balance between predictive performance and computational requirements.

Massimiliano Lupo Pasini, Yijiang Li, Kibaek Kim +1

Computational Sciences and Engineering Division, Oak Ridge National Laboratory, 1 Bethel Valley Road, Oak Ridge, TN 37831, USA

Neural Surrogate Modeling Heterogeneous GNNs Optimal Power Flow Graph Foundation Models Constrained Optimization Power Systems