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.

May 9, 2026

Broximal Gradient Descent: A Projection-Free Sister of Projected Gradient Descent

We propose Broximal Gradient Descent (BroxGD), a projection-free sister method to projected gradient descent for constrained optimization. Its forward--backward construction replaces the proximal backward operation by the broximal operation of Gruntkowska et al. (2025). Instead of projecting, each step minimizes a linear function over the intersection of the constraint set

\mathcal{X}

and a ball

\mathbb{B}(x_k,t_k)

centered at the current iterate

x_k

, of suitable radius

t_k>0

:

x_{k+1}\in\arg\min_{z\in\mathcal{X}\cap\mathbb{B}(x_k,t_k)}\langle\nabla f(x_k),z\rangle.

We develop a comprehensive convergence theory spanning a wide range of optimization regimes and radius rules. We expect BroxGD to find many applications and inspire numerous extensions, much like projected gradient descent. Our contribution is theoretical; potential applications and toy experiments illustrate the method and suggest directions for future work.

Peter Richtárik, Kaja Gruntkowska, Hanmin Li

King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

Gradient Descent Constrained Optimization

May 9, 2026

FlowADMM: Plug-and-play ADMM with Flow-based Renoise-Denoise Priors

Plug-and-play (PnP) methods for solving inverse problems have recently achieved strong performance by leveraging denoising priors based on powerful generative diffusion and flow models. However, existing diffusion- and flow-based PnP methods typically rely on stochastic renoise-denoise operations, which complicate the analysis of their convergence behavior. In this work, we identify and formalize the deterministic renoise-denoise operator underlying flow-based plug-and-play methods. This perspective reveals that these methods implicitly define a deterministic operator given by the expectation of a denoiser over the latent noise distribution. Building on this insight, we propose FlowADMM, a PnP algorithm that integrates the renoise-denoise operator into the classical alternating direction method of multiplier (ADMM) framework. We establish convergence guarantees for FlowADMM under weak Lipschitz conditions on the underlying flow network, and extend the analysis to non-stationary time schedules. Empirically, FlowADMM achieves state-of-the-art performance among flow-based PnP methods on a range of inverse problems, including denoising, deblurring, super-resolution, and inpainting, while requiring fewer data consistency evaluations than prior approaches.

Hendrik Sommerhoff, Michael Moeller

Computer Vision Group, University of Siegen Hölderlinstraße 3, 57076 Siegen, Germany

Alternating Direction Method of Multipliers Image Restoration Constrained Optimization Inverse Problems

May 8, 2026

AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Training

Physics-informed neural networks (PINNs) provide a flexible framework for solving forward and inverse problems governed by partial differential equations (PDEs), but standard PINN training typically relies on soft penalty formulations that combine PDE residuals, data mismatch, and initial/boundary conditions using manually chosen weights. This often leads to ill-conditioning, sensitivity to loss weights, and poor constraint satisfaction. In this work, we reformulate PINN training as an equality-constrained optimization problem and propose a novel Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN (AdamFLIP). The key idea is to view the constraint residuals as the output of a controlled dynamical system and to compute the Lagrange multiplier as a feedback input that locally drives these residuals toward stable linear contraction dynamics. AdamFLIP then applies Adam-style first- and second-moment adaptation to the resulting feedback-linearized Lagrangian gradient, combining principled constraint handling with the scalability and robustness of adaptive neural-network optimization. We test AdamFLIP on a range of benchmark forward and inverse PDE problem, and it consistently outperforms both the standard soft-constrained PINN and state-of-the-art constrained optimizers. Specifically, on the Navier--Stokes equations benchmark, AdamFLIP \textbf{reduces relative

L_2

error by more than two thirds} for the predicted solution compared to the next best method. Our AdamFLIP framework provides an effective and computationally scalable hard constraint optimization method for PINN training.

Binghang Lu, Runyu Zhang, Changhong Mou +2

Purdue University · Massachusetts Institute of Technology · Utah State University +1

Neural PDE Solvers Constrained Optimization Physics-Informed ML Adaptive Gradient Methods Feedback Linearization Optimization

May 8, 2026

Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems

We study a class of bilevel optimization problems in which both the upper- and lower-level problems have minimax structures. This setting captures a broad range of emerging applications. Despite the extensive literature on bilevel optimization and minimax optimization separately, existing methods mainly focus on bilevel optimization with lower-level minimization problems, often under strong convexity assumptions, and are not directly applicable to the minimax lower-level setting considered here. To address this gap, we develop penalty-based first-order methods for bilevel minimax optimization without requiring strong convexity of the lower-level problem. In the deterministic setting, we establish that the proposed method finds an

ε

-KKT point with

\tilde{O}(ε^{-4})

oracle complexity. We further show that bilevel problems with convex constrained lower-level minimization can be reformulated as special cases of our framework via Lagrangian duality, leading to an

\tilde{O}(ε^{-4})

complexity bound that improves upon the existing

\tilde{O}(ε^{-7})

result. Finally, we extend our approach to the stochastic setting, where only stochastic gradient oracles are available, and prove that the proposed stochastic method finds a nearly

ε

-KKT point with

\tilde{O}(ε^{-9})

oracle complexity.

Yiyang Shen, Yutian He, Weiran Wang +1

University of Iowa

Min-Max Optimization Bilevel Optimization Constrained Optimization

May 8, 2026

Sensitivity-Based Robust NMPC for Close-Proximity Offshore Wind Turbine Inspection with a Tilted Multirotor

Close-proximity offshore wind turbine inspection requires strict clearance control around large cylindrical structures under wind and model mismatch. Nominal Nonlinear Model Predictive Control (NMPC) may violate safety constraints when mass, inertia, thrust effectiveness, drag, or wind conditions differ from nominal assumptions. We propose a sensitivity-based robust NMPC for a tilted multirotor that robustifies the tower-clearance constraint via online constraint tightening. First-order parametric state sensitivities provide a structured-uncertainty margin, while bounded gusts are handled by a stage-dependent additive margin. The formulation augments the nominal NMPC with sensitivity propagation and margin evaluation only, leaving the receding-horizon optimization structure unchanged. Monte-Carlo evaluation over 500 uncertainty realizations on a boundary-critical helical inspection trajectory shows that the proposed controller eliminates the clearance violations observed under nominal NMPC at the cost of a moderate increase in solve time.

Giuseppe Silano, Martin Saska

Ricerca sul Sistema Energetico S.p.A., Milan, Italy, and the Czech Technical University in Prague, Czechia · Czech Technical University in Prague, Czechia

Industrial Inspection Robust Control Model Predictive Control Constrained Optimization Quadrotor Control

May 8, 2026

SkillSelect-Serve: QoS-Aware Budgeted Skill Service Recommendation for LLM Agents

Reusable agent skills are emerging as a service-oriented capability layer for Large Language Model (LLM) agents. Unlike plain retrieval items, a skill exposes functional capabilities, input-output assumptions, tool dependencies, context cost, and risk metadata. Selecting skills is particularly challenging for small LLM agents, which can load only a few capability units under restricted context, tool availability, and risk tolerance. Existing fixed Top-k methods rank skills by textual relevance and overlook requirement satisfaction, deliverability, and operational constraints. We present SkillSelect-Serve, a QoS-aware, budget-constrained Skill Service recommendation framework. Raw skills are profiled as structured Skill Services, the task is converted into a structured requirement object, and candidates discovered from a large-scale registry are ranked by a calibrated task-conditioned suitability estimator and packed by a constrained projection enforcing token-budget, aggregated-risk, and tool-availability constraints, using only deployment-observable features. On a registry of 35,353 skills with pooled multi-positive relevance judgments verified by two independent assessors, the unconstrained top-5 recommendation fits a realistic 4,000-token context for only 9.1% of tasks; the constrained projection restores 100% deliverability at a cost of only 1.14 points of hit rate, outperforming retrieve-and-rerank, budget truncation, and diversity-based selection under identical budgets. The same mechanism halves delivered risk exposure and eliminates the 44-81% tool-violation rates of tool-agnostic recommendation. At an identical three-service budget, hit rate improves from 0.8864 to 0.9091 over fixed Top-3 retrieval. The results support managing reusable agent skills as discoverable, comparable, and constraint-aware service units instead of plain retrievable documents.

Jingyuan Zheng, Dongjing Wang, Xin Zhang +7

School of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou, 310018, China · Sanya Traditional Chinese Medicine Hospital, Sanya, 572000, China · HDU-ITMO Joint Institute, Hangzhou Dianzi University +3

LLM Agent Skill Retrieval LLM Agents Constrained Optimization Recommender Systems Large Language Model-Based Recommendation

May 8, 2026

Convex Optimization with Nested Evolving Feasible Sets

Convex Optimization with Nested Evolving Feasible Sets (CONES)} is considered where the objective function

f

remains fixed but the feasible region evolves over time as a nested sequence

S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T

. The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known nested convex body chasing problem. When the loss function is convex, we propose a lazy-algorithm and show that it achieves

O(T^{1-β}), O(T^β)

simultaneous regret and movement cost for any

β\in (0,1]

, over a time horizon of

T

. When the loss function is strongly convex or

α

-sharp, we propose an algorithm Frugal that simultaneously achieves zero regret and a movement cost of

O(\log T)

. To complement this, we show that any online algorithm with

o(T)

regret has a movement cost of

Ω(\log{T})

for both cases, proving optimality of Frugal.

Karthick Krishna M., Haricharan Balasundaram, Rahul Vaze

Online Convex Optimization Constrained Optimization

May 8, 2026

Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks

Exact solution of hard combinatorial optimization problems often relies on strong convex relaxations, but solving these relaxations repeatedly inside a branch-and-bound algorithm can be prohibitively expensive. Hence, we consider this challenge for Max-Cut, where branch and bound commonly uses semidefinite programming (SDP) relaxations to bound subproblems. We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for these SDP solvers and can be plugged directly into an exact branch-and-bound framework. The proposed architecture has update steps of complexity

\mathcal{O}(n^2 + ne)

, and predicts both primal- and dual-feasible SDP solutions. The primal SDP solutions yield feasible Max-Cut solutions via the Goemans--Williamson algorithm. In addition, it is trained in a self-supervised fashion without requiring solved SDP relaxations as labels. Empirically, we show that our architecture can substantially reduce the cost of bounding in exact Max-Cut solving by up to

10.6 \times

compared with using the state-of-the-art SDP solver Mosek. Our work highlights the potential of learned, validity-preserving surrogates for accelerating exact optimization over structured convex relaxations.

Hao Chen, Chendi Qian, Christopher Morris +2

1Davidson School of Chemical Engineering, Purdue University · Faculty of Computer Science, RWTH Aachen University · 3Jacobs Technion-Cornell Institute, Cornell Tech

Max-Cut Semidefinite Programming Graph Neural Networks Branch and Bound Combinatorial Optimization Global Optimization+2

May 7, 2026

Matrix-Valued Optimism is Matrix-Valued Augmentation: Additive Hybrid Designs for Constrained Optimization

Augmented Lagrangian and optimistic primal--dual methods stabilize equality-constrained optimization through seemingly different mechanisms: the former adds constraint-dependent primal curvature, while the latter adds dual memory. Recent work has shown that these mechanisms are equivalent for scalar parameters. We extend this equivalence to matrix-valued correction. We prove an additivity principle: for symmetric matrix parameters, the ideal primal trajectory depends only on the summed correction matrix, not on how it is split between augmented and optimistic channels. This exposes a design freedom: algebraically equivalent decompositions can have different finite-step feasibility because augmented correction affects primal curvature, whereas optimistic correction affects the scale of the dual memory correction. We formulate the resulting step-size-limited design problem and derive a closed-form hybrid rule that selects a matrix correction, splits it between the two channels, and chooses primal and dual steps using local spectral weights. Experiments on nonlinear equality-constrained problems with controlled constraint-Jacobian conditioning show that the hybrid design improves over pure augmented and pure optimistic endpoints, closely tracks a grid-search hybrid oracle, and is competitive with first-order primal--dual baselines under mild-to-moderate ill-conditioning. The experiments also identify the expected limitation: exact cancellation requires increasingly large matrix corrections as the constraint Jacobian becomes ill-conditioned.

Jiayi Zhao

University of Washington Seattle, WA, USA

Primal-Dual Optimization Augmented Lagrangian Method Constrained Optimization

May 7, 2026

WARP: A Benchmark for Primal-Dual Warm-Starting of Interior-Point Solvers

Solving AC Optimal Power Flow (AC-OPF) is of central importance in electricity market operations, where interior-point methods (IPMs) such as IPOPT are the standard solvers. A growing body of work uses machine learning to predict primal warm-start iterates, reporting iteration reductions of 30-46%. We show that these reported gains rest on an inappropriate evaluation baseline: prior methods benchmark against the flat start

V_m = 1, V_a = 0

, whereas the solver's actual default - the variable-bound midpoint

(l+u)/2

- is near-optimal for log-barrier centrality. Against this corrected baseline, no primal-only warm-start method reduces solver iterations. We trace the failure to a geometric property of interior-point methods: primal prediction accuracy is anticorrelated with convergence speed, and providing the ground-truth optimal solution

x^*

without dual variables causes the solver to diverge. Oracle experiments establish that the complete primal-dual-barrier state

(x^*, λ^*, z^*, μ^*)

reduces IPOPT iterations from 23 to 3 - an 85% reduction that is structurally inaccessible to primal-only methods. To enable rigorous evaluation of warm-start methods on this task, we release a benchmark suite comprising dual-labeled AC-OPF datasets with IPOPT-extracted solutions, a corrected evaluation protocol, and WARP - a topology-conditioned encode-process-decode interaction network that predicts the full interior-point state

(\hat{x}, \hatλ, \hat{z}, \hatμ)

on the heterogeneous constraint graph. WARP achieves a 76% reduction in IPOPT iterations while natively accommodating N-1 contingency topology variations without retraining.

Dhruv Suri, Helgi Hilmarsson, Shourya Bose

Pravah

Primal-Dual Optimization Graph Neural Networks Optimal Power Flow Constrained Optimization Power Systems

May 7, 2026

Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature

Kernel quadrature can exploit RKHS spectral structure and outperform Monte Carlo on smooth integrands, but optimized quadrature weights are generally signed and may be numerically unstable. We study whether spectral acceleration remains possible when the weights are constrained to be positive, i.e., simplex weights. In the exact-target fixed-pool setting, an evaluated i.i.d. candidate pool of size

N

is already available and the task is to reweight it so as to approximate the kernel mean embedding. We show that this positive reweighting problem is governed not by the equal-weight empirical average, but by the random convex hull generated by the pool. Our main geometric result shows that the mean of a bounded

d

-dimensional random vector can be approximated by a convex combination of

N

i.i.d. samples at accuracy

O(d/N)

with high probability, sharper than equal-weight averaging in the fixed-dimensional regime. We transfer this

d

-dimensional convex-hull approximation to full RKHS worst-case error through an augmented Mercer-truncation argument. The resulting positive-weight KQ bounds consist of a spectral tail term and a finite-sample convex-hull term, yielding Monte-Carlo-beating rates in favorable spectral regimes, including near-

O(1/N)

rates up to logarithmic factors under exponential spectral decay. We also provide a constructive Frank--Wolfe algorithm that operates directly on the pool atoms, maintains simplex weights, and admits an explicit optimization-error bound.

Satoshi Hayakawa

The University of Tokyo

Constrained Optimization Kernel Methods Approximation Algorithms Frank-Wolfe Algorithms

May 6, 2026

Differentiable Parameter Optimization for DAEs with State-Dependent Events

Differential-algebraic equations (DAEs) with state-dependent events arise in systems whose continuous dynamics are constrained by algebraic equations and interrupted by mode changes, switching logic, impacts, or state reinitializations. Gradient-based parameter learning for such systems is challenging because algebraic variables are implicitly defined, event times depend on the parameters, and reset maps introduce discontinuities. This paper studies differentiable parameter optimization for semi-explicit DAEs with events. We formulate the learning problem as a constrained least-squares problem with DAE dynamics, algebraic constraints, guard equations, and reset maps. We then develop two complementary gradient-computation strategies. The first is an automatic-differentiation-through-simulation method that solves algebraic variables inside the vector field, differentiates the algebraic solve using the implicit function theorem, and handles events through segmented differentiable integration. The second is an explicit discrete-adjoint method that represents the forward simulation as an event-split residual system and computes gradients by solving for the Lagrange multipliers of smooth-segment and event residuals. The formulation clarifies that residual terms in the adjoint method are equality constraints, not heuristic penalties. We compare the two approaches in terms of gradient interpretation, event-time handling, implementation complexity, and local validity. Both methods provide gradients for the event path selected by the forward simulation and are valid under fixed event ordering and transversal guard crossings.

Ion Matei, Maksym Zhenirovskyy, Anthony Wong

Fujitsu Research of America

Dynamical Systems Automatic Differentiation Constrained Optimization Differentiable Physics Differentiable Optimization

May 6, 2026

Exact Dual Geometry of SOC-ICNN Value Functions

Input Convex Neural Networks (ICNNs) are commonly used in a two-stage manner: one first trains a convex network and then minimizes it over its input in a downstream inference problem. Recent second-order-cone ICNNs (SOC-ICNNs) enrich ReLU-based ICNNs with quadratic and conic modules and admit an exact representation as value functions of second-order cone programs (SOCPs). This value-function structure enables an explicit convex-analytic treatment of SOC-ICNN inference. In this paper, we study the exact first-order and local second-order geometry of SOC-ICNNs from the dual viewpoint. We show that supporting slopes, subdifferentials, directional derivatives, and local Hessians can be recovered directly from optimal dual variables. These results provide the geometric primitives for white-box SOC-ICNN inference, going beyond black-box automatic differentiation. Numerical experiments validate the exact multiplier readout, the local Hessian formula, and the set-valued behavior at structurally degenerate inputs. We also provide a step-by-step tutorial showing how the readout mechanism instantiates a complete white-box inference loop. The code is available at https://anonymous.4open.science/r/SOC-ICNN-Theory-BEFC/.

Kang Liu, Jianchen Hu, Wei Peng

School of Future Technology Xi’an Jiaotong University Xi’an, China · School of Future Technology School of Automation Science and Engineering Xi’an Jiaotong University Xi’an, China · School of Automation Science and Engineering Xi’an Jiaotong University Xi’an, China

Second-Order Cone Programming Neural Network Optimization Input Convex Neural Networks Constrained Optimization

May 6, 2026

Geometry-Aware Neural Optimizer for Shape Optimization and Inversion

Geometry is central to PDE-governed systems, motivating shape optimization and inversion. Classical pipelines conduct costly forward simulation with geometry processing, requiring substantial expert effort. Neural surrogates accelerate forward analysis but do not close the loop because gradients from objectives to geometry are often unavailable. Existing differentiable methods either rely on restrictive parameterizations or unstable latent optimization driven by scalar objectives, limiting interpretability and part-wise control. To address these challenges, we propose Geometry-Aware Neural Optimizer (GANO), an end-to-end differentiable framework that unifies geometry representation, field-level prediction, and automated optimization/inversion in a single latent-space loop. GANO encodes shapes with an auto-decoder and stabilizes latent updates via a denoising mechanism, and a geometry-informed surrogate provides a reliable gradient pathway for geometry updates. Moreover, GANO supports part-wise control through null-space projection and uses remeshing-free projection to accelerate geometry processing. We further prove that denoising induces an implicit Jacobian regularization that reduces decoder sensitivity, yielding controlled deformations. Experiments on three benchmarks spanning 2D Helmholtz, 2D airfoil, and 3D vehicles show state-of-the-art accuracy and stable, controllable updates, achieving up to +55.9% lift-to-drag improvement for airfoils and ~7% drag reduction for vehicles.

Guoze Sun, Tianya Miao, Haoyang Huang +4

Gaoling School of Artificial Intelligence, Renmin University of China

Neural Surrogate Modeling Geometric Representation Learning Constrained Optimization Differentiable Optimization

May 6, 2026

Demystifying Manifold Constraints in LLM Pre-training

The recent success of matrix optimizers (e.g., Muon) suggests that specific normalization of momentum, such as orthogonalization and row-wise normalization, benefits both the stability and acceleration of LLM training. Consequently, several recent studies have suggested that weights should also be normalized, leading to a Riemannian optimization problem. While such constrained training frameworks demonstrate superior performance, the effects of explicitly constraining weights, and their interaction with existing stabilization mechanisms, remain less understood. To bridge this gap, we study manifold constrained training dynamics through activation scales, rotational dynamics, and the update-to-weight ratio. We propose a Riemannian spectral steepest descent optimizer called MACRO, alongside a radius selection principle to serve as our testbed. Our analysis and numerical experiments reveal that RMSNorm and manifold constraints serve overlapping roles, and that weight decay can be completely eliminated when manifold constraints are applied. By controlling the update-to-weight ratio, constrained training significantly alleviates update cancellation, empirically demonstrating that MACRO is robust to low-precision computation and competitive with existing algorithms for standard LLM pre-training.

Kang An, Jiaxiang Li, Donald Goldfarb +1

Rice University · Independent Researcher · Columbia University

Language Model Pretraining Riemannian Optimization Constrained Optimization Neural Network Training Dynamics

May 5, 2026

Adaptive Diagonal Loading for Norm Constrained Beamforming

Reliable adaptive beamforming is critical for large microphone arrays operating in highly dynamic acoustic environments. In scenarios characterized by fast-moving talkers and interferers, the available sample support for estimating the spatial correlation matrix is often snapshot-deficient. This deficiency, coupled with array imperfections, degrades the White Noise Gain (WNG), leading to severe target signal cancellation. To ensure stable and robust beamforming, we propose a novel adaptive diagonal loading method that guarantees the WNG remains strictly within specified bounds. By leveraging the Kantorovich inequality, we map the desired WNG to a strict upper bound on the condition number of the correlation matrix. Furthermore, we present three estimation techniques for the adaptive loading level, ranging from trace-based bounding to exact eigenvalue decomposition, offering scalable computational complexities of

\mathcal{O}(M)

,

\mathcal{O}(M^2)

, and

\mathcal{O}(M^3)

. Our approach demonstrates highly stable beamforming under fast-changing interference.

Manan Mittal, Ryan M. Corey, John R. Buck +1

Stony Brook University1 · University of Illinois Chicago2 · University of Massachusetts Dartmouth3

Constrained Optimization

May 5, 2026

A Provably Convergent and Practical Algorithm for Gromov--Wasserstein Optimal Transport

Gromov--Wasserstein optimal transport (GWOT) aligns metric measure spaces by matching their within-domain relational structures, but large-scale GWOT remains challenging because its objective is nonconvex and projection onto the transport polytope is often solved only approximately in practice. This leads to a gap between practical projected-gradient implementations and convergence theory, which typically assumes exact projections. For squared-loss GWOT, we propose an inexact projected-gradient framework with a verifiable feasibility-residual-based inexact condition for the projection subproblem. This condition is directly computable and avoids unknown quantities such as the exact projection point. Under this implementable condition, we prove subsequential convergence to stationary points and, with a mild tolerance-decay condition, convergence of the whole sequence. The resulting method retains the simplicity and sparsity of projected-gradient schemes while providing rigorous convergence guarantees, turning projected-gradient methods into a principled and scalable approach for GWOT with provable reliability.

Ling Liang, Lei Yang

Department of Mathematics, The University of Tennessee, Knoxville, TN 37916, USA · School of Computer Science and Engineering, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, Guangdong 510275, China

Gradient Descent Nonconvex Optimization Constrained Optimization Optimal Transport

May 4, 2026

A second-order method landing on the Stiefel manifold via Newton $\unicode{x2013}$ Schulz iteration

Retraction-free approaches offer attractive low-cost alternatives to Riemannian methods on the Stiefel manifold, but they are often first-order, which may limit the efficiency under high-accuracy requirements. To this end, we propose a second-order method landing on the Stiefel manifold without invoking retractions, which is proved to enjoy local quadratic (or superlinear for its inexact variant) convergence. The update consists of the sum of (i) a component tangent to the level set of the constraint-defining function that aims to reduce the objective and (ii) a component normal to the same level set that reduces the infeasibility. Specifically, we construct the normal component via Newton

\unicode{x2013}

Schulz, a fixed-point iteration for orthogonalization. Moreover, we establish a geometric connection between the Newton

\unicode{x2013}

Schulz iteration and Stiefel manifolds, in which Newton

\unicode{x2013}

Schulz moves along the normal space. For the tangent component, we formulate a modified Newton equation that incorporates Newton

\unicode{x2013}

Schulz. Numerical experiments on the orthogonal Procrustes problem, principal component analysis, and real-data independent component analysis illustrate that the proposed method performs better than the existing methods.

Xinhui Xiong, Bin Gao, P. -A. Absil

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China · State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China · ICTEAM Institute, UCLouvain, Belgium

Newton-Schulz Iteration Stiefel Manifold Optimization Second-Order Optimization Constrained Optimization

May 4, 2026

Strategy-Aware Optimization Modeling with Reasoning LLMs

Large language models (LLMs) can generate syntactically valid optimization programs, yet often struggle to reliably choose an effective modeling strategy, leading to incorrect formulations and inefficient solver behavior. We propose SAGE, a strategy-aware framework that makes Modeling Strategy explicit in both data construction and post-training. SAGE builds a solver-verified multi-strategy dataset and trains a student model with supervised fine-tuning followed by Segment-Weighted GRPO using a composite reward over format compliance, correctness, and solver efficiency. Across eight benchmarks spanning synthetic and real-world settings, SAGE improves average pass@1 from 72.7 to 80.3 over the strongest open-source baseline. With multiple generations, SAGE discovers more distinct correct formulations and improves component-level diversity at pass@16 by 19-29%. At the largest scale, SAGE produces more compact constraint systems with 14.2% fewer constraints than the baseline, consistent with solver-efficient modeling. Overall, these results show that making Modeling Strategy explicit improves automated optimization modeling. Code is available at https://github.com/rachhhhing/SAGE.

Ruiqing Zhao, Fengzhi Li, Yuan Zuo +5

School of Computer Science and Engineering, Beihang University, Beijing, China · JIUTIAN Research, Beijing, China · MIIT Key Laboratory of Data and Decision Intelligence, Beihang University, Beijing, China

RL for Code Generation LLM-Based Program Synthesis Constrained Optimization Natural Language to Optimization Modeling

May 4, 2026

LUMINA: A Grid Foundation Model for Benchmarking AC Optimal Power Flow Surrogate Learning

AC optimal power flow (ACOPF) is foundational yet computationally expensive in power grid operations, driving learning-based surrogates for large-scale grid analysis. These surrogates, however, often fail to generalize across network topologies, a critical gap for deployment on grids not seen during training and for routine operational what-if studies. We introduce LUMINA-Bench, a comprehensive benchmark suite for ACOPF surrogate learning covering multi-topology pretraining, transfer, and adaptation. The benchmark evaluates homogeneous and heterogeneous architectures under single- and multi-topology learning settings using unified metrics that capture both predictive accuracy and physics-informed constraint violations. We additionally compare constraint-aware training objectives, including MSE, augmented Lagrangian, and violation-based Lagrangian losses, to characterize accuracy-robustness trade-offs across settings. Data processing, training, and evaluation frameworks are open-sourced as the LUMINA suite to support reproducibility and accelerate future research on feasibility-aware OPF surrogates.

Hongwei Jin, Keunju Song, Zeeshan Memon +5

Mathematics and Computer Science Division, Argonne National Laboratory · Department of Electronic Engineering, Sogang University · Department of Computer Science, Emory University +1

Neural Surrogate Modeling Benchmark Design Optimal Power Flow Constrained Optimization Power Systems

May 3, 2026

Optimization of CV-QKD Under Practical Constraints

Using reinforcement learning, we optimize for practical hardware constraints, including limited FIR filter taps at the transmitter and receiver, mean photon number and finite DAC/ADC resolution. Under these realistic conditions, the proposed approach achieves significant performance improvements.

Svitlana Matsenko, Amirhossein Ghazisaeidi, Marcin Jarzyna +2

DTU Electro, Technical University of Denmark, DK-2800, Kgs. Lyngby, Denmark · Nokia Bell Labs, 91300 Massy, France · Centre for Quantum Optical Technologies, CeNT, University of Warsaw, 02-097 Warszawa, Poland +1

Constrained Optimization

May 3, 2026

Towards Systematic Generalization for Power Grid Optimization Problems

AC Optimal Power Flow (ACOPF) and Security-Constrained Unit Commitment (SCUC) are fundamental optimization problems in power system operations. ACOPF serves as the physical backbone of grid simulation and real-time operation, enforcing nonlinear power flow feasibility and network limits, while SCUC represents a core market-level decision process that schedules generation under operational and security constraints. Although these problems share the same underlying transmission network and physical laws, they differ in decision variables and temporal coupling, and prior learning-based approaches address them in isolation, resulting in disjoint models and representations.We propose a learning framework that jointly models ACOPF and SCUC through a shared graph-based backbone that captures grid topology and physical interactions, coupled with task-specific decoders for static and temporal decision-making. Training includes solver supervision with physics-informed objectives to enforce AC feasibility and inter-temporal operational constraints. To evaluate generalization, we assess cross-case transfer on unseen grid topologies for ACOPF and SCUC without retraining, and systematic generalization on the UC-ACOPF problem using unsupervised, physics-based objectives and a power-dispatch consensus mechanism. Experiments across multiple grid scales demonstrate improved performance and transferability relative to existing learning-based baselines, indicating that the model can support learning across heterogeneous power system optimization problems.

Zeeshan Memon, Yijiang Li, Hongwei Jin +2

Emory University Atlanta, GA, USA · Argonne National Laboratory Lemont, IL, USA

Optimal Power Flow Constrained Optimization Compositional Generalization Power Systems

May 1, 2026

Augmented Lagrangian Multiplier Network for State-wise Safety in Reinforcement Learning

Safety is a primary challenge in real-world reinforcement learning (RL). Formulating safety requirements as state-wise constraints has become a prominent paradigm. Handling state-wise constraints with the Lagrangian method requires a distinct multiplier for every state, necessitating neural networks to approximate them as a multiplier network. However, applying standard dual gradient ascent to multiplier networks induces severe training oscillations. This is because the inherent instability of dual ascent is exacerbated by network generalization -- local overshoots and delayed updates propagate to adjacent states, further amplifying policy fluctuations. Existing stabilization techniques are designed for scalar multipliers, which are inadequate for state-dependent multiplier networks. To address this challenge, we propose an augmented Lagrangian multiplier network (ALaM) framework for stable learning of state-wise multipliers. ALaM consists of two key components. First, a quadratic penalty is introduced into the augmented Lagrangian to compensate for delayed multiplier updates and establish the local convexity near the optimum, thereby mitigating policy oscillations. Second, the multiplier network is trained via supervised regression toward a dual target, which stabilizes training and promotes convergence. Theoretically, we show that ALaM guarantees multiplier convergence and thus recovers the optimal policy of the constrained problem. Building on this framework, we integrate soft actor-critic (SAC) with ALaM to develop the SAC-ALaM algorithm. Experiments demonstrate that SAC-ALaM outperforms state-of-the-art safe RL baselines in both safety and return, while also stabilizing training dynamics and learning well-calibrated multipliers for risk identification.

Jiaming Zhang, Yujie Yang, Yao Lyu +2

School of Vehicle and Mobility & College of AI, Tsinghua University, Beijing, 100084, China

Constrained RL Augmented Lagrangian Method Constrained Optimization Safe RL

May 1, 2026

Model Compression with Exact Budget Constraints via Riemannian Manifolds

Assigning one of K options to each of N groups under a total cost budget is a recurring problem in efficient AI, arising in mixed-precision quantization, non-uniform pruning, and expert selection. The objective (model loss) depends on all assignments jointly and does not decompose across groups, so combinatorial solvers can only optimize proxy objectives. Evolutionary search evaluates the actual loss but lacks gradients, while penalty-based methods enforce the budget only approximately and often require heavy hyperparameter tuning. We show that under softmax relaxation, the budget constraint defines a smooth Riemannian manifold in logit space with unusually clean geometry: the normal vector is available in closed form, shifting logits along the cost vector changes expected cost monotonically, and vector transport reduces to a single inner product. Building on this, we propose Riemannian Constrained Optimization (RCO), which wraps tangent projection, binary-search retraction, and momentum transport around a standard Adam step. Combined with Gumbel straight-through estimation and budget-constrained dynamic programming for discrete feasibility, RCO optimizes the actual loss with first-order methods, enforces the expected budget exactly at every iterate, and introduces no constraint-related hyperparameters. The same construction handles multiple simultaneous budgets with no additional coefficients. RCO matches or exceeds state-of-the-art methods on synthetic problems and realistic LLM compression settings, often at considerably lower wall-clock cost. Source code is available at https://github.com/IST-DASLab/RCO.

Michael Helcig, Dan Alistarh

ETH Zürich · IST Austria

Model Compression Riemannian Optimization LLM Compression Constrained Optimization

Apr 30, 2026

NLPOpt-Net: A Learning Method for Nonlinear Optimization with Feasibility Guarantees

Nonlinear Parametric Optimization Network (NLPOpt-Net) is an unsupervised learning architecture to solve constrained nonlinear programs (NLP). Given the structure of an NLP, it learns the parametric solution maps with guaranteed constraint satisfaction. The architecture consists of a backbone neural network (NN) followed by a multilayer (

k

-layered) projection. While the NN drives toward optimality through a loss function consisting of a modified Lagrangian augmented with a consistency loss, the projection ensures feasibility by projecting the NN predictions in the original constraint manifold. Instead of typical distance minimization, our projection exploits local quadratic approximations of the original NLP. Under certain conditions (such as convexity), the projection has a descent property, which improves the NN predictions further. NLPOpt-Net deploys an inversion-free, modified Chambolle-Pock algorithm to solve the constrained quadratic projections during the forward pass and uses the implicit function theorem for efficient backpropagation. The fixed structure of the projection further allows decoupling of the NN and the projection once the training is complete. NLPOpt-Net solves large-scale convex QP, QCQP, NLP, and nonconvex problems with near zero optimality gap and constraint violations reduced to machine precision. Additionally, it provides near accurate prediction of the active sets and corresponding dual variables, thereby enabling a scalable approach for multiparametric programming. Compiling the projection in C provides order of magnitude improvement in inference time compared to JAX. We provide the codes and NLPOpt-Net as a ready to use package that includes GPU support.

Bimol Nath Roy, Rahul Golder, MM Faruque Hasan

Texas A&M University

Neural Network Optimization Learning to Optimize Constrained Optimization Differentiable Optimization

Apr 30, 2026

Global Optimality for Constrained Exploration via Penalty Regularization

Efficient exploration is a central problem in reinforcement learning and is often formalized as maximizing the entropy of the state-action occupancy measure. While unconstrained maximum-entropy exploration is relatively well understood, real-world exploration is often constrained by safety, resource, or imitation requirements. This constrained setting is particularly challenging because entropy maximization lacks additive structure, rendering Bellman-equation-based methods inapplicable. Moreover, scalable approaches require policy parameterization, inducing non-convexity in both the objective and the constraints. To our knowledge, the only prior model-free policy-gradient approach for this setting under general policy parameterization is due to Ying et al. (2025). Unfortunately, their guarantees are limited to weak regret and ergodic averages, which do not imply that the final output is a single deployable policy that is near-optimal and nearly feasible. In this work we take a different approach to this problem, and propose Policy Gradient Penalty (PGP) method, a single-loop policy-space method that enforces general convex occupancy-measure constraints via quadratic-penalty regularization. PGP constructs pseudo-rewards that yield gradient estimates of the penalized objective, subsequently exploiting the classical Policy Gradient Theorem. We further establish the regularity of the penalized objective, providing the smoothness properties needed to justify the convergence of PGP. Leveraging hidden convexity and strong duality, we then establish global last-iterate convergence guarantees, attaining an

ε

-optimal constrained entropy value with

ε

bounded constraint violation despite policy-induced non-convexity. We validate PGP through ablations on a grid-world benchmark and further demonstrate scalability on two challenging continuous-control tasks.

Florian Wolf, Ilyas Fatkhullin, Niao He

The Computing & Mathematical Sciences Department, California Institute of Technology, Pasadena, CA. · Department of Computer Science, ETH Zurich, Switzerland. · 3ETH AI Center, ETH Zurich, Switzerland.

Constrained RL Maximum Entropy RL Policy Gradient Methods Constrained Optimization

Apr 30, 2026

On the Expressive Power of GNNs to Solve Linear SDPs

Semidefinite programs (SDPs) are a powerful framework for convex optimization and for constructing strong relaxations of hard combinatorial problems. However, solving large SDPs can be computationally expensive, motivating the use of machine learning models as fast computational surrogates. Graph neural networks (GNNs) are a natural candidate in this setting due to their sparsity-awareness and ability to model variable-constraint interactions. In this work, we study what expressive power is sufficient to recover optimal SDP solutions. We first prove negative results showing that standard GNN architectures fail on recovering linear SDP solutions. We then identify a more expressive architecture that captures the key structure of SDPs and can, in particular, emulate the updates of a standard first-order solver. Empirically, on both synthetic and SdpLib benchmarks of various classes of SDPs, this more expressive architecture achieves consistently lower prediction error and objective gap than theoretically weaker baselines. Finally, using the learned high-quality predictions to warm-start the first-order solver yields practical speedups of up to 80%.

Chendi Qian, Christopher Morris

Faculty of Computer Science, RWTH Aachen University, Germany

Semidefinite Programming Graph Neural Networks GNN Expressivity Constrained Optimization

Apr 29, 2026

Learning Over-Relaxation Policies for ADMM with Convergence Guarantees

The Alternating Direction Method of Multipliers (ADMM) is a widely used method for structured convex optimization, and its practical performance depends strongly on the choice of penalty and relaxation parameters. Motivated by settings such as Model Predictive Control (MPC), where one repeatedly solves related optimization problems with fixed structure and changing parameter values, we propose learning online updates of the relaxation parameter to improve average performance on problem classes of interest, while guaranteeing that asymptotic convergence is not compromised for the worst-case realization of such problems. This choice is computationally attractive in the Operator Splitting Quadratic Program (OSQP)-like architectures, since adapting relaxation does not trigger the matrix refactorizations associated with penalty updates. We establish convergence guarantees for ADMM with time-varying penalty and relaxation parameters under mild assumptions, and show on benchmark quadratic programs that the resulting learned policies improve both iteration count and wall-clock time on average over baseline OSQP.

Junan Lin, Paul J. Goulart, Luca Furieri

Department of Engineering Science, University of Oxford, OX1 3PJ Oxford, UK

Learning to Optimize Alternating Direction Method of Multipliers Constrained Optimization

Apr 28, 2026

Deflation-Free Optimal Scoring

Sparse Optimal Scoring (SOS) reformulates linear discriminant analysis to enable feature selection through elastic net regularization, making it well-suited for high-dimensional settings where the number of features exceeds observations. Most existing SOS methods use deflation-based strategies that compute discriminant vectors sequentially, which can propagate errors and produce suboptimal solutions. We propose a novel approach that estimates all discriminant vectors simultaneously under an explicit global orthogonality constraint, which we call Deflation-Free Sparse Optimal Scoring (DFSOS). DFSOS combines Bregman iteration with orthogonality-constrained optimization, decomposing the problem into tractable subproblems for scoring vectors, discriminant vectors, and orthogonality enforcement. We establish convergence to stationary points of the augmented Lagrangian under mild conditions. Extensive experiments using synthetic data and real-world time series data demonstrate that DFSOS achieves classification accuracy comparable to or better than existing deflation-based methods. These results indicate that deflation-free approaches offer a robust and effective framework for sparse discriminant analysis in high-dimensional problems.

Sharmin Afroz, Brendan Ames

Department of Computational & Informational Sciences, Stillman College, Tuscaloosa, AL, USA · School of Mathematical Sciences, University of Southampton, Southampton, UK

Orthogonal Optimization Constrained Optimization Dimensionality Reduction

Apr 27, 2026

Feasible-First Exploration for Constrained ML Deployment Optimization in Crash-Prone Hierarchical Search Spaces

Deploying machine learning models under production constraints requires joint optimization over model family, quantization scheme, runtime backend, and serving configuration. This induces a hierarchical mixed-variable search space in which many configurations are invalid: evaluations may crash, exceed memory limits, or violate latency constraints. Standard black-box optimizers such as Tree-structured Parzen Estimators (TPE) and constrained Bayesian optimization are effective when valid configurations are common, but they can spend a large fraction of a small evaluation budget on invalid or uninformative trials in hostile deployment spaces. This paper studies that regime and asks whether optimization should be decomposed into an explicit exploration stage followed by model-guided exploitation. We propose Thermal Budget Annealing (TBA), a feasible-first exploration procedure that maps valid and feasible regions before warm-starting TPE. The method includes two robustness mechanisms for hostile hardware: trial timeouts that abort clearly infeasible evaluations early, and subspace blacklisting that temporarily suppresses categorical subspaces after repeated failures. We also introduce DeployBench, a benchmark suite for deployment optimization with hierarchical structure, hidden crash zones, hard constraints, and unequal evaluation costs. On synthetic benchmarks and real GPU deployment with five pre-trained vision models across five GPU targets (NVIDIA H100, A100, RTX 5080, L4, and T4), the proposed hybrid improves model-family discovery under tight constraints while reducing wasted budget relative to cold-start TPE.

Christian Lysenstøen

Department of Electrical Engineering and Computer Sciences University of California, Berkeley

Bayesian Optimization Black-Box Optimization Constrained Optimization