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.

Apr 27, 2026

Learning with Embedded Linear Equality Constraints via Variational Bayesian Inference

Machine Learning is becoming more prevalent in science and engineering, but many approaches do not provide meaningful uncertainty estimates and predictions may also violate known physical knowledge. We propose a Bayesian framework to embed linear relationships across inputs and outputs into the learning process, whilst characterizing full predictive uncertainty over both the model parameters and the domain knowledge. We evaluated our method on learning the single particle battery model subject to voltage and energy balances, showing its ability to provide reduced credible intervals and constraint violations compared to standard Bayesian neural networks based on variational inference.

Matthew Marsh, Benoît Chachuat, Antonio del Rio Chanona

Sargent Centre for Process Systems Engineering, Imperial College London

Bayesian Neural Networks Uncertainty Quantification Constrained Optimization Physics-Informed ML

Apr 24, 2026

Control Barrier Functions Solved with Hierarchical Quadratic Programming for Safe Physical Human-Robot Interaction

Physical human-robot interaction offers the potential to leverage human intelligence and robot physical capabilities to enable a range of exciting applications, e.g., collaborative robots for rehabilitation. Safety is critical for the successful deployment of this kind of robotic system. In recent years, Control Barrier Function (CBF) has emerged as an effective approach to enforce safety guarantees, which has been widely applied in various applications, from adaptive cruise control to navigation of legged robots. CBFs can be solved in a Quadratic Programming (QP) problem, which can include many CBF-formulated tasks. To manage a large number of safety tasks, a hierarchical CBF has been used to allow hierarchical relaxation of safety tasks to ensure the feasibility of a solution in the presence of conflicting tasks. In this work, we propose to use a CBF-based Hierarchical Quadratic Programming (HQP) framework in physical human-robot interaction to allow us to design both performance tasks (e.g., preserve the desired behavior at the human-robot interaction point) and safety tasks at any level of a hierarchy to balance the safety and the performance in a more flexible way. Extensive experiments were carried out on a real redundant robot to validate the effectiveness, flexibility, and generality of this approach.

Rui Luo, Jonas Mariager Jakobsen, Wesley Roozing +2

Robotic Control Control Barrier Functions Constrained Optimization Human-Robot Interaction

Apr 24, 2026

Relaxation-Informed Training of Neural Network Surrogate Models

ReLU neural networks trained as surrogate models can be embedded exactly in mixed-integer linear programs (MILPs), enabling global optimization over the learned function. The tractability of the resulting MILP depends on structural properties of the network, i.e., the number of binary variables in associated formulations and the tightness of the continuous LP relaxation. These properties are determined during training, yet standard training objectives (prediction loss with classical weight regularization) offer no mechanism to directly control them. This work studies training regularizers that directly target downstream MILP tractability. Specifically, we propose simple bound-based regularizers that penalize the big-M constants of MILP formulations and/or the number of unstable neurons. Moreover, we introduce an LP relaxation gap regularizer that explicitly penalizes the per-sample gap of the continuous relaxation at training points. We derive its associated gradient and provide an implementation from LP dual variables without custom automatic differentiation tools. We show that combining the above regularizers can approximate the full total derivative of the LP gap with respect to the network parameters, capturing both direct and indirect sensitivities. Experiments on non-convex benchmark functions and a two-stage stochastic programming problem with quantile neural network surrogates demonstrate that the proposed regularizers can reduce MILP solve times by up to four orders of magnitude relative to an unregularized baseline, while maintaining competitive surrogate model accuracy.

Calvin Tsay

Department of Computing, Imperial College London, South Kensington, SW7 2AZ, United Kingdom.

Neural Surrogate Modeling Mixed-Integer Linear Programming Neural Network Optimization Constrained Optimization Integer Linear Programming

Apr 24, 2026

ATRS: Adaptive Trajectory Re-splitting via a Shared Neural Policy for Parallel Optimization

Parallel trajectory optimization via the Alternating Direction Method of Multipliers (ADMM) has emerged as a scalable approach to long-horizon motion planning. However, existing frameworks typically decompose the problem into parallel subproblems based on a predefined fixed structure. Such structural rigidity often causes optimization stagnation in highly constrained regions, where a few lagging subproblems delay global convergence. A natural remedy is to adaptively re-split these stagnating segments online. Yet, deciding when, where, and how to split exceeds the capability of rule-based heuristics. To this end, we propose ATRS, a novel framework that embeds a shared Deep Reinforcement Learning policy into the parallel ADMM loop. We formulate this adaptive adjustment as a Multi-Agent Shared-Policy Markov Decision Process, where all trajectory segments act as homogeneous agents and share a unified neural policy network. This parameter-sharing architecture endows the system with size invariance, enabling it to handle dynamically changing segment counts during re-splitting and generalize to arbitrary trajectory lengths. Furthermore, our formulation inherently supports zero-shot generalization to unseen environments, as our network relies solely on the internal states of the numerical solver rather than on the geometric features of the environment. To ensure solver stability, a Confidence-Based Election mechanism selects only the most stagnating segment for re-splitting at each step. Extensive simulations demonstrate that ATRS accelerates convergence, reducing the number of iterations by up to 26.0% and the computation time by up to 19.1%. Real-world experiments further confirm its applicability to both large-scale offline global planning and real-time onboard replanning within 35 ms per cycle, with no sim-to-real degradation.

Jiajun Yu, Guodong Liu, Li Wang +6

Trajectory Optimization Alternating Direction Method of Multipliers Constrained Optimization

Apr 24, 2026

Fast and Effective Redistricting Optimization via Composite-Move Tabu Search

Spatial redistricting is a practical combinatorial optimization problem that demands high-quality solutions, rapid turnaround, and flexibility to accommodate multi-criteria objectives and interactive refinement. A central challenge is the contiguity constraint: enforcing contiguity in integer-programming or heuristic search can severely shrink the feasible neighborhood, weaken exploration, and trap the search in poor local optima. We introduce a composite-move Tabu search (CM-Tabu) that systematically expands the feasible neighborhood space in Tabu search while preserving contiguity. When a boundary unit cannot be reassigned individually without disconnecting its district, our method identifies a minimal set of units that can move together, or a pair of units (or sets of units) that can be switched, as a contiguity-preserving composite move. Candidate single-unit and composite moves are generated in linear time by analyzing each district's contiguity graph using articulation points and biconnected components. Extensive experiments demonstrate that the proposed approach substantially improves solution quality, run-to-run robustness, and computational efficiency relative to traditional Tabu search and other baselines. For example, in the Philadelphia case, the approach can consistently attain the theoretical global optimum in population-equality and support multi-criteria trade-offs. CM-Tabu delivers optimization performance suitable for real-world practices and decision-support workflows.

Hai Jin, Diansheng Guo

Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

Combinatorial Optimization Constrained Optimization

Apr 21, 2026

Decision-Focused Federated Learning Under Heterogeneous Objectives and Constraints

We consider Decision-Focused Federated Learning (DFFL), a predict-then-optimize setting in which multiple clients collaboratively train predictive models for downstream linear optimization problems without exchanging raw data. Besides the data heterogeneity typical of standard federated learning, clients may also have different objective functions and feasible regions. Building on the SPO+ surrogate loss, we derive heterogeneity bounds that separate objective shift, measured through cost-vector distances, from feasible-set shift, measured through support-function and shape-distance terms. We show that, for general compact feasible sets, small objective perturbations can still induce nonvanishing decision-focused loss discrepancies, while strongly convex feasible regions yield sharper stability-based bounds. We then lift these pointwise bounds to a local-versus-federated excess-risk comparison, showing that federation is beneficial when the statistical advantage of pooling exceeds a client-specific heterogeneity penalty. Computational experiments on polyhedral and strongly convex problems confirm that federation is substantially more robust under strongly convex feasible regions. Finally, we evaluate a simple validation-based interpolation between local and federated DFFL models. This interpolation mitigates the theoretical tradeoff and reduces aggregate regret and worst-client harm in both synthetic experiments and a PJM energy-pricing case study.

Konstantinos Ziliaskopoulos, Alexander Vinel

Auburn University

Predict-Then-Optimize Learning to Optimize Decision-Focused Learning Constrained Optimization Federated Learning Heterogeneous Federated Learning

Apr 21, 2026

HardNet++: Nonlinear Constraint Enforcement in Neural Networks

Enforcing constraint satisfaction in neural network outputs is critical for safety, reliability, and physical fidelity in many control and decision-making applications. While soft-constrained methods penalize constraint violations during training, they do not guarantee constraint adherence during inference. Other approaches guarantee constraint satisfaction via a projection layer, but often rely on the existence of a tractable projection onto the feasible set, limiting their utility in more general problem settings. Many real-world problems of interest are nonlinear and lack the special structure admitting a tractable projection, motivating the development of methods that can enforce general nonlinear constraints. To this end, we introduce HardNet++, a constraint-satisfaction method that enforces linear and nonlinear equality and inequality constraints. Our approach iteratively adjusts the network output via damped local linearizations of the constraints. Each iteration is differentiable, admitting an end-to-end training framework, where the constraint satisfaction layer is active during training. We show that under certain regularity conditions, this procedure enforces nonlinear constraint satisfaction to arbitrary tolerance. Finally, we demonstrate tight constraint adherence without loss of optimality in a learning-for-optimization context, where we apply this method to a nonlinear model predictive control problem.

Andrea Goertzen, Kaveh Alim, Youngjae Min +1

Massachusetts Institute of Technology

Nonlinear MPC Neural Network Optimization Constrained Optimization Constraint Satisfaction

Apr 21, 2026

Policy Gradient Primal-Dual Method for Safe Reinforcement Learning from Human Feedback

Safe Reinforcement Learning from Human Feedback (Safe RLHF) has recently achieved empirical success in developing helpful and harmless large language models by decoupling human preferences regarding helpfulness and harmlessness. Existing approaches typically rely on fitting fixed horizon reward models from human feedback and have only been validated empirically. In this paper, we formulate safe RLHF as an infinite horizon discounted Con- strained Markov Decision Process (CMDP), since humans may interact with the model over a continuing sequence of interactions rather than within a single finite episode. We propose two Safe RLHF algorithms that do not require reward model fitting and, in contrast to prior work assuming fixed-length trajectories, support flexible trajectory lengths for training. Both algo- rithms are based on the primal-dual method and achieve global convergence guarantees with polynomial rates in terms of policy gradient iterations, trajectory sample lengths, and human preference queries. To the best of our knowledge, this is the first work to study infinite horizon discounted CMDP under human feedback and establish global, non-asymptotic convergence.

Qiang Liu, Adrienne Kline, Ermin Wei

Department of Electrical and Computer Engineering, Northwestern University, Evanston, IL, USA · Center for Artificial Intelligence, Bluhm Cardiovascular Institute, Northwestern Medicine, Chicago, IL, USA · Department of Surgery, Northwestern University, Chicago, IL, USA +3

Constrained RL Policy Gradient Methods Constrained Optimization Safe RL RL from Human Feedback

Apr 19, 2026

On The Mathematics of the Natural Physics of Optimization

A number of optimization algorithms have been inspired by the physics of Newtonian motion. Here, we ask the question: do algorithms themselves obey some

natural laws of motion,'' and can they be derived by an application of these laws? We explore this question by positing the theory that optimization algorithms may be considered as some manifestation of hidden algorithm primitives that obey certain universal non-Newtonian dynamics. This natural physics of optimization is developed by equating the terminal transversality conditions of an optimal control problem to the generalized Karush/John-Kuhn-Tucker conditions of an optimization problem. Through this equivalence formulation, the data functions of a given constrained optimization problem generate a natural vector field that permeates an entire hidden space with information on the optimality conditions. An

action-at-a-distance'' operation via a Pontryagin-type minimum principle produces a local action to deliver a globalized result by way of a Hamilton-Jacobi inequality. An inverse-optimal algorithm is generated by performing control jumps that dissipate quantized ``energy'' defined by a search Lyapunov function. Illustrative applications of the proposed theory show that a large number of algorithms can be generated and explained in terms of the new mathematical physics of optimization.

I. M. Ross

Department of Mechanical and Aerospace Engineering, Naval Postgraduate School, Monterey, CA

Constrained Optimization

Apr 19, 2026

Safe and Policy-Compliant Multi-Agent Orchestration for Enterprise AI

Enterprise AI systems increasingly deploy multiple intelligent agents across mission-critical workflows that must satisfy hard policy constraints, bounded risk exposure, and comprehensive auditability (SOX, HIPAA, GDPR). Existing coordination methods - cooperative MARL, consensus protocols, and centralized planners - optimize expected reward while treating constraints implicitly. This paper introduces CAMCO (Constraint-Aware Multi-Agent Cognitive Orchestration), a runtime coordination layer that models multi-agent decision-making as a constrained optimization problem. CAMCO integrates three mechanisms: (i) a constraint projection engine enforcing policy-feasible actions via convex projection, (ii) adaptive risk-weighted Lagrangian utility shaping, and (iii) an iterative negotiation protocol with provably bounded convergence. Unlike training-time constrained RL, CAMCO operates as deployment-time middleware compatible with any agent architecture, with policy predicates designed for direct integration with production engines such as OPA. Evaluation across three enterprise scenarios - including comparison against a constrained Lagrangian MARL baseline - demonstrates zero policy violations, risk exposure below threshold (mean ratio 0.71), 92-97% utility retention, and mean convergence in 2.4 iterations.

Vinil Pasupuleti, Shyalendar Reddy Allala, Siva Rama Krishna Varma Bayyavarapu +1

International Business Machines · International Business Machines (IBM) · South Carolina, United States +6

Multi-Agent System Optimization Multi-Agent Coordination Multi-Agent Orchestration Constrained Optimization AI Agent Governance

Apr 18, 2026

Trajectory-Restricted Optimization Conditions and Geometry-Aware Linear Convergence

Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can be arbitrarily conservative and fail to capture the geometry actually encountered by optimization trajectories. In this paper, we develop a trajectory-restricted framework for linear convergence based on localized geometric regularity. We introduce restricted variants of the Polyak--Łojasiewicz inequality, error bound, and quadratic growth conditions that are required to hold only on subsets of the domain. We show that classical convergence guarantees extend under these localized conditions, and in key cases, we develop new arguments that yield explicit relationships between the corresponding constants. The resulting rates are governed by geometric quantities associated with the regions traversed by the algorithm. For polyhedral composite problems, we prove that convergence is controlled by restricted Hoffman constants corresponding to the active polyhedral faces visited along the trajectory. Once the iterates enter a well-conditioned face, the effective condition number improves accordingly. Our work provides a geometric quantification for fast local convergence after active-set or manifold identification and more broadly suggests that linear convergence is fundamentally governed by the geometry of the subsets explored by the algorithm, rather than by worst-case global conditioning.

Faris Chaudhry, Anthea Monod, Keisuke Yano

Imperial College London, UK · Institute of Statistical Mathematics, Japan

Optimization Convergence Analysis Constrained Optimization

Apr 16, 2026

Adaptive Test-Time Compute Allocation for Reasoning LLMs via Constrained Policy Optimization

Test-time compute scaling, the practice of spending extra computation during inference via repeated sampling, search, or extended reasoning, has become a powerful lever for improving large language model performance. Yet deploying these techniques under finite inference budgets requires a decision that current systems largely ignore: which inputs deserve more compute, and which can be answered cheaply? We formalize this as a constrained optimization problem (maximize expected accuracy subject to an average compute budget) and solve it with a two-stage Solve-then-Learn pipeline. In the solve stage, Lagrangian relaxation decomposes the global constraint into per-instance sub-problems, each admitting a closed-form oracle action that optimally prices accuracy against cost. We prove that the induced cost is monotone in the dual variable, enabling exact budget targeting via binary search. In the learn stage, a lightweight classifier is trained to predict oracle actions from cheap input features, amortizing the allocation rule for real-time deployment. We establish that the task-level regret of the learned policy is bounded by its imitation error times the worst-case per-instance gap, yielding a clean reduction from constrained inference to supervised classification. Experiments on MATH and GSM8K with three LLMs (DeepSeek-V3, GPT-4o-mini, Qwen2.5-7B) show that our method consistently outperforms uniform and heuristic allocation baselines, achieving up to 12.8% relative accuracy improvement on MATH under matched budget constraints, while closely tracking the Lagrangian oracle upper bound with over 91% imitation accuracy.

Zhiyuan Zhai, Bingcong Li, Bingnan Xiao +2

Fudan University · ETH Zurich · Guangming Lab

Test-Time Scaling Constrained Optimization Inference-Time Compute Allocation

Apr 15, 2026

Deployment of AI-Assisted Interventions: Capacity Constraints and Noisy Compliance

AI tools increasingly drive targeted interventions in various service settings, including healthcare, education, and public services. Algorithms score individuals, trigger outreach to those above a threshold (e.g., high-risk or high-value), and encourage them to request service; then providers deliver service to those who request. Much of the work in this area has focused on improving predictive accuracy, implicitly assuming that better predictions lead to better outcomes. We show that predictions are only one component of a larger service system: when service capacity is limited and behavioral responses to outreach are probabilistic, system efficacy depends on operational forces that predictive accuracy does not capture. In such settings, the optimal score threshold must balance two effects: ensuring all capacity is filled (utilization) and, when capacity is constrained, preventing low-value requests from crowding out high-value ones (cannibalization). We characterize the optimal threshold and prove that thresholds based solely on predictive accuracy are generally suboptimal. Further, algorithm selection metrics such as AUC can be misaligned with operational performance: they weight all thresholds equally, while optimal deployment uses a subset of thresholds that depends on both capacity and compliance behavior. We introduce a new metric, Operational AUC (OpAUC), and show that it identifies the efficacy-optimal algorithm. Finally, we conduct a case study on sepsis early warning data that illustrates the magnitude of the improvement available from better threshold selection and shows that a predictor with lower AUC can achieve higher system efficacy under optimal deployment.

Carri W. Chan, Yi Han, Hannah Li +1

Decision, Risk, and Operations, Columbia Business School · Department of Statistics, Columbia University · Division of Pulmonary, Allergy, and Critical Care Medicine, Department of Medicine, Vagelos College of Physicians and Surgeons, Columbia University

AI-Assisted Decision Making Constrained Optimization

Apr 15, 2026

Stochastic Trust-Region Methods for Over-parameterized Models

Under interpolation-type assumptions such as the strong growth condition, stochastic optimization methods can attain convergence rates comparable to full-batch methods, but their performance, particularly for SGD, remains highly sensitive to step-size selection. To address this issue, we propose a unified stochastic trust-region framework that eliminates manual step-size tuning and extends naturally to equality-constrained problems. For unconstrained optimization, we develop a first-order stochastic trust-region algorithm and show that, under the strong growth condition, it achieves an iteration and stochastic first-order oracle complexity of

O(\varepsilon^{-2} \log(1/\varepsilon))

for finding an

\varepsilon

-stationary point. For equality-constrained problems, we introduce a quadratic-penalty-based stochastic trust-region method with penalty parameter

μ

, and establish an iteration and oracle complexity of

O(\varepsilon^{-4} \log(1/\varepsilon))

to reach an

\varepsilon

-stationary point of the penalized problem, corresponding to an

O(\varepsilon)

-approximate KKT point of the original constrained problem. Numerical experiments on deep neural network training and orthogonally constrained subspace fitting demonstrate that the proposed methods achieve performance comparable to well-tuned stochastic baselines, while exhibiting stable optimization behavior and effectively handling hard constraints without manual learning-rate scheduling.

Aike Yang, Hao Wang

ShanghaiTech University, Shanghai, China

Trust-Region Optimization Nonconvex Stochastic Optimization Constrained Optimization

Apr 14, 2026

Optimizing Earth Observation Satellite Schedules under Unknown Operational Constraints: An Active Constraint Acquisition Approach

Earth Observation (EO) satellite scheduling (deciding which imaging tasks to perform and when) is a well-studied combinatorial optimization problem. Existing methods typically assume that the operational constraint model is fully specified in advance. In practice, however, constraints governing separation between observations, power budgets, and thermal limits are often embedded in engineering artefacts or high-fidelity simulators rather than in explicit mathematical models. We study EO scheduling under unknown constraints: the objective is known, but feasibility must be learned interactively from a binary oracle. Working with a simplified model restricted to pairwise separation and global capacity constraints, we introduce Conservative Constraint Acquisition~(CCA), a domain-specific procedure designed to identify justified constraints efficiently in practice while limiting unnecessary tightening of the learned model. Embedded in the Learn&Optimize framework, CCA supports an interactive search process that alternates optimization under a learned constraint model with targeted oracle queries. On synthetic instances with up to 50tasks and dense constraint networks, L&O improves over a no-knowledge greedy baseline and uses far fewer main oracle queries than a two-phase acquire-then-solve baseline (FAO). For $n\leq 30$ , the average gap drops from 65--68% (Priority Greedy) to 17.7--35.8% using L&O. At $n{=}50$ , where the CP-SAT reference is the best feasible solution found in 120s, L&O improves on FAO on average (17.9% vs.\ 20.3%) while using 21.3 main queries instead of 100 and about

5\times

less execution time.

Mohamed-Bachir Belaid

NILU, Climate and environmental research institute, Kjeller, Norway

Constrained Optimization Learning to Optimize Remote Sensing

Apr 14, 2026

HUANet: Hard-Constrained Unrolled ADMM for Constrained Convex Optimization

This paper presents HUANet, a constrained deep neural network architecture that unrolls the Alternating Direction Method of Multipliers (ADMM) into a trainable neural network for accelerating parametric constrained convex optimization. Existing end-to-end learning methods operate as black-box mappings from parameters to solutions, often without explicitly incorporating optimality principles or guaranteeing constraint satisfaction. To address these limitations, HUANet embeds a hard-constrained neural network within each unrolled ADMM iteration, where a differentiable correction stage enforces the affine equalities of the primal subproblem. Furthermore, we incorporate first-order optimality conditions into a self-supervised training loss to promote the convergence of the proposed unrolled algorithm. Extensive numerical experiments for benchmark optimization problems and a control application demonstrate and validate the effectiveness of HUANet in accelerating constrained convex optimization solving.

Trinh Tran, Binh Nguyen, Truong X. Nghiem

Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816, USA

Learning to Optimize Alternating Direction Method of Multipliers Algorithm Unrolling Constrained Optimization

Apr 13, 2026

Complementarity by Construction: A Lie-Group Approach to Solving Quadratic Programs with Linear Complementarity Constraints

Many problems in robotics require reasoning over a mix of continuous dynamics and discrete events, such as making and breaking contact in manipulation and locomotion. These problems are locally well modeled by linear complementarity quadratic programs (LCQPs), an extension to QPs that introduce complementarity constraints. While very expressive, LCQPs are non-convex, and few solvers exist for computing good local solutions for use in planning pipelines. In this work, we observe that complementarity constraints form a Lie group under infinitesimal relaxation, and leverage this structure to perform on-manifold optimization. We introduce a retraction map that is numerically well behaved, and use it to parameterize the constraints so that they are satisfied by construction. The resulting solver avoids many of the classical issues with complementarity constraints. We provide an open-source solver, Marble, that is implemented in C++ with Julia and Python bindings. We demonstrate that Marble is competitive on a suite of benchmark problems, and solves a number of robotics problems where existing approaches fail to converge.

Arun L. Bishop, Micah I. Reich, Zachary Manchester

Authors are with the Robotics Institute, Carnegie Mellon University · Author is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology

Constrained Optimization Quadratic Programming Robotics

Apr 13, 2026

On the Use of Bi-Objective Evolutionary Algorithms for the Stochastic MKP under Dynamic Constraints

The multiple knapsack problem (MKP) generalizes the classical knapsack problem by assigning items to multiple knapsacks subject to capacity constraints. It is used to model many real-world resource allocation and scheduling problems. In practice, these optimization problems often involve stochastic and dynamic components. Evolutionary algorithms provide a flexible framework for addressing such problems under uncertainty and dynamic changes. In this paper, we investigate a stochastic and dynamic variant of MKP with chance constraints, where the item weights are modeled as independent normally distributed random variables and knapsack capacities change during the optimization process. We formulate the problem as a bi-objective optimization formulation that balances profit maximization and probabilistic capacity satisfaction at a given confidence level. We conduct an empirical comparison of four widely used multi-objective evolutionary algorithms (MOEAs), representing both decomposition- and dominance-based search paradigms. The algorithms are evaluated under varying uncertainty levels, confidence thresholds, and dynamic change settings. The results provide comparative insights into the behavior of decomposition-based and dominance-based MOEAs for stochastic MKP under dynamic constraints.

Ishara Hewa Pathiranage, Aneta Neumann

Machine Learning and Optimisation, School of Computer Science and Information Technology, Adelaide University, Adelaide, Australia

Combinatorial Optimization Constrained Optimization Multi-Objective Optimization Multi-Objective Evolutionary Optimization Chance-Constrained Optimization

Apr 12, 2026

RCBSF: A Multi-Agent Framework for Automated Contract Revision via Stackelberg Game

Despite the widespread adoption of Large Language Models (LLMs) in Legal AI, their utility for automated contract revision remains impeded by hallucinated safety and a lack of rigorous behavioral constraints. To address these limitations, we propose the Risk-Constrained Bilevel Stackelberg Framework (RCBSF), which formulates revision as a non-cooperative Stackelberg game. RCBSF establishes a hierarchical Leader Follower structure where a Global Prescriptive Agent (GPA) imposes risk budgets upon a follower system constituted by a Constrained Revision Agent (CRA) and a Local Verification Agent (LVA) to iteratively optimize output. We provide theoretical guarantees that this bilevel formulation converges to an equilibrium yielding strictly superior utility over unguided configurations. Empirical validation on a unified benchmark demonstrates that RCBSF achieves state-of-the-art performance, surpassing iterative baselines with an average Risk Resolution Rate (RRR) of 84.21% while enhancing token efficiency. Our code is available at https://github.com/xjiacs/RCBSF .

Shijia Xu, Yu Wang, Xiaolong Jia +3

Chongqing University, China · Queen Mary University of London, UK · Chongqing Key Laboratory of Big Data Intelligence and Privacy Computing, China +1

Multi-Agent LLMs LLM Agent Safety Constrained Optimization

Apr 8, 2026

LCO: LLM-based Constraint Optimization for Safer Agentic LLMs in Real-world Tasks

Large Language Models (LLMs) are increasingly acting as autonomous agents, but their continuous interaction with the environment can lead to in-context reward hacking (ICRH), a phenomenon where LLMs iteratively optimize their behavior to maximize proxy objectives, inadvertently producing harmful side effects. Existing defense methods are insufficient to address this risk, as ICRH arises not from adversarial inputs but from the model's own over-optimization. To mitigate this issue, we propose LLM-based Constraint Optimization (LCO), a framework that effectively reduces ICRH without model fine-tuning. LCO consists of two modules: self-thought module, which guides the LLM to proactively deliberate and integrate potential safety constraints before execution; and evolutionary sampling module, which employs LLM-based crossover and mutation to constrain the model's actions within a safe solution space while maintaining task performance. Experimental results demonstrate that LCO substantially alleviates ICRH in both output-refine and policy-refine scenarios. In particular, on the tweet engagement optimization task, LCO achieves a 39% reduction in the Toxicity Growth Rate (TGR) on GPT-4, while on the policy optimization benchmark, it reduces the ICRH Occurrence Rate by 15.23%, demonstrating safety improvement without sacrificing task performance.

Jiayong Wan, Jiawei Chen, Zhaoxia Yin +2

East China Normal University, Shanghai, China · Beijing Zhongguancun Academy, Beijing, China · Tsinghua University, Beijing, China

LLM Agent Safety Constrained Optimization

Apr 7, 2026

LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layers

Linear matrix inequalities (LMIs) have played a central role in certifying stability, robustness, and forward invariance of dynamical systems. Despite rapid development in learning-based methods for control design and certificate synthesis, existing approaches often fail to preserve the hard matrix inequality constraints required for formal guarantees. We propose LMI-Net, an efficient and modular differentiable projection layer that enforces LMI constraints by construction. Our approach lifts the set defined by LMI constraints into the intersection of an affine equality constraint and the positive semidefinite cone, performs the forward pass via Douglas-Rachford splitting, and supports efficient backward propagation through implicit differentiation. We establish theoretical guarantees that the projection layer converges to a feasible point, certifying that LMI-Net transforms a generic neural network into a reliable model satisfying LMI constraints. Evaluated on experiments including invariant ellipsoid synthesis and joint controller-and-certificate design for a family of disturbed linear systems, LMI-Net substantially improves feasibility over soft-constrained models under distribution shift while retaining fast inference speed, bridging semidefinite-program-based certification and modern learning techniques.

Sunbochen Tang, Andrea Goertzen, Navid Azizan

Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Constrained Optimization Differentiable Optimization Semidefinite Programming

Apr 5, 2026

Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints

Recent advancements in data science have significantly elevated the importance of orthogonally constrained optimization problems. The Riemannian approach has become a popular technique for addressing these problems due to the advantageous computational and analytical properties of the Stiefel manifold. Nonetheless, the interplay of nonsmoothness alongside orthogonality constraints introduces substantial challenges to current Riemannian methods, including scalability, parallelizability, complicated subproblems, and cumulative numerical errors that threaten feasibility. In this paper, we take a retraction-free primal-dual approach and propose a linearized smoothing augmented Lagrangian method specifically designed for nonsmooth and nonconvex optimization with orthogonality constraints. Our proposed method is single-loop and free of subproblem solving. We establish its iteration complexity of

O(ε^{-3})

for finding

ε

-KKT points, matching the best-known results in the Riemannian optimization literature. Additionally, by invoking the standard Kurdyka-Lojasiewicz (KL) property, we demonstrate asymptotic sequential convergence of the proposed algorithm. Numerical experiments on both smooth and nonsmooth orthogonal constrained problems demonstrate the superior computational efficiency and scalability of the proposed method compared with state-of-the-art algorithms.

Linglingzhi Zhu, Wentao Ding, Shangyuan Liu +1

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA · Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Orthogonal Optimization Constrained Optimization Augmented Lagrangian Method Primal-Dual Optimization

Apr 3, 2026

Improving Feasibility via Fast Autoencoder-Based Projections

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

Maria Chzhen, Priya L. Donti

University of Toronto Toronto, ON, Canada · Massachusetts Institute of Technology Cambridge, MA, USA

Constrained Optimization Constrained RL Autoencoders

Apr 2, 2026

Principled and Scalable Diversity-Aware Retrieval via Cardinality-Constrained Binary Quadratic Programming

Diversity-aware retrieval is essential for Retrieval-Augmented Generation (RAG), yet existing methods lack theoretical guarantees and face scalability issues as the number of retrieved passages

k

increases. We propose a principled formulation of diversity retrieval as a cardinality-constrained binary quadratic programming (CCBQP), which explicitly balances relevance and semantic diversity through an interpretable trade-off parameter. Inspired by recent advances in combinatorial optimization, we develop a non-convex tight continuous relaxation and a Frank--Wolfe based algorithm with landscape analysis and convergence guarantees. Extensive experiments demonstrate that our method consistently dominates baselines on the relevance-diversity Pareto frontier, while achieving significant speedup.

Qiheng Lu, Nicholas D. Sidiropoulos

University of Virginia Charlottesville, VA 22904, USA

Frank-Wolfe Algorithms Constrained Optimization Retrieval-Augmented Generation Mixed-Integer QP

Apr 1, 2026

Tool-Augmented Agent for Closed-loop Optimization,Simulation,and Modeling Orchestration

Iterative industrial design-simulation optimization is bottlenecked by the CAD-CAE semantic gap: translating simulation feedback into valid geometric edits under diverse, coupled constraints. To fill this gap, we propose COSMO-Agent (Closed-loop Optimization, Simulation, and Modeling Orchestration), a tool-augmented reinforcement learning (RL) framework that teaches LLMs to complete the closed-loop CAD-CAE process. Specifically, we cast CAD generation, CAE solving, result parsing, and geometry revision as an interactive RL environment, where an LLM learns to orchestrate external tools and revise parametric geometries until constraints are satisfied. To make this learning stable and industrially usable, we design a multi-constraint reward that jointly encourages feasibility, toolchain robustness, and structured output validity. In addition, we contribute an industry-aligned dataset that covers 25 component categories with executable CAD-CAE tasks to support realistic training and evaluation. Experiments show that COSMO-Agent training substantially improves small open-source LLMs for constraint-driven design, exceeding large open-source and strong closed-source models in feasibility, efficiency, and stability.

Liyuan Deng, Shujian Deng, Yongkang Chen +6

Northwestern Polytechnical University · Shanghai Artificial Intelligence Laboratory

LLM Agent Orchestration Constrained Optimization RL for Tool Use Parametric CAD Modeling Constrained RL

Mar 30, 2026

Transfer Learning in Bayesian Optimization for Aircraft Design

The use of transfer learning within Bayesian optimization addresses the disadvantages of the so-called cold start problem by using source data to aid in the optimization of a target problem. We present a method that leverages an ensemble of surrogate models using transfer learning and integrates it in a constrained Bayesian optimization framework. We identify challenges particular to aircraft design optimization related to heterogeneous design variables and constraints. We propose the use of a partial-least-squares dimension reduction algorithm to address design space heterogeneity, and a meta data surrogate selection method to address constraint heterogeneity. Numerical benchmark problems and an aircraft conceptual design optimization problem are used to demonstrate the proposed methods. Results show significant improvement in convergence in early optimization iterations compared to standard Bayesian optimization, with improved prediction accuracy for both objective and constraint surrogate models.

Ali Tfaily, Youssef Diouane, Nathalie Bartoli +1

1*Department of Mechanical Engineering, McGill University, Canada. · 2Advanced Product Development, Bombardier, Canada. · Department of Mathematical and Industrial Engineering, Polytechnique Montréal, Canada. +4

Bayesian Optimization Surrogate-Assisted Optimization Constrained Optimization

Mar 30, 2026

Rethinking Language Model Scaling under Transferable Hypersphere Optimization

Scaling laws for large language models depend critically on the optimizer and parameterization. Existing hyperparameter transfer laws are mainly developed for first-order optimizers, and they do not structurally prevent training instability at scale. Recent hypersphere optimization methods constrain weight matrices to a fixed-norm hypersphere, offering a promising alternative for more stable scaling. We introduce HyperP (Hypersphere Parameterization), the first framework for transferring optimal learning rates across model width, depth, training tokens, and Mixture-of-Experts (MoE) granularity under the Frobenius-sphere constraint with the Muon optimizer. We prove that weight decay is a first-order no-op on the Frobenius sphere, show that Depth-

μ

P remains necessary, and find that the optimal learning rate follows the same data-scaling power law with the "magic exponent" 0.32 previously observed for AdamW. A single base learning rate tuned at the smallest scale transfers across all compute budgets under HyperP, yielding

1.58\times

compute efficiency over a strong Muon baseline at

6\times10^{21}

FLOPs. Moreover, HyperP delivers transferable stability: all monitored instability indicators, including

Z

-values, output RMS, and activation outliers, remain bounded and non-increasing under training FLOPs scaling. We also propose SqrtGate, an MoE gating mechanism derived from the hypersphere constraint that preserves output RMS across MoE granularities for improved granularity scaling, and show that hypersphere optimization enables substantially larger auxiliary load-balancing weights, yielding both strong performance and good expert balance. We release our training codebase at https://github.com/microsoft/ArchScale.

Liliang Ren, Yang Liu, Yelong Shen +1

Microsoft

Language Model Scaling Laws Hyperparameter Transfer Constrained Optimization Mixture-of-Experts Language Models

Mar 30, 2026

Dogfight Search: A Swarm-Based Optimization Algorithm for Complex Engineering Optimization and Mountainous Terrain Path Planning

Dogfight is a tactical behavior of cooperation between fighters. Inspired by this, this paper proposes a novel metaphor-free metaheuristic algorithm called Dogfight Search (DoS). Unlike traditional algorithms, DoS draws algorithmic framework from the inspiration, but its search mechanism is constructed based on the displacement integration equations in kinematics. Through experimental validation on CEC2017 and CEC2022 benchmark test functions, 10 real-world constrained optimization problems and mountainous terrain path planning tasks, DoS significantly outperforms 7 advanced competitors in overall performance and ranks first in the Friedman ranking. Furthermore, this paper compares the performance of DoS with 3 SOTA algorithms on the CEC2017 and CEC2022 benchmark test functions. The results show that DoS continues to maintain its lead, demonstrating strong competitiveness. The source code of DoS is available at https://ww2.mathworks.cn/matlabcentral/fileexchange/183519-dogfight-search.

Yujing Sun, Jie Cai, Xingguo Xu +4

College of Science, Liaoning Technical University, Fuxin, 123000, China · School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China · College of Computer Science and Technology, Zhejiang University, Hangzhou, 310027, China +1

Metaheuristic Optimization Constrained Optimization

Mar 28, 2026

RDEx-CSOP: Feasibility-Aware Reconstructed Differential Evolution with Adaptive epsilon-Constraint Ranking

Constrained single-objective numerical optimisation requires both feasibility maintenance and strong objective-value convergence under limited evaluation budgets. This report documents RDEx-CSOP, a constrained differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session). RDEx-CSOP combines success-history parameter adaptation with an exploitation-biased hybrid search and an ε-constraint handling mechanism with a time-varying threshold. We evaluate RDEx-CSOP on the official CEC 2025 CSOP benchmark using the U-score framework (Speed, Accuracy, and Constraint categories). The results show that RDEx-CSOP achieves the highest total score and the best average rank among all released comparison algorithms, mainly through strong speed and competitive constraint-handling performance across the 28 benchmark functions.

Sichen Tao, Yifei Yang, Ruihan Zhao +3

Department of Engineering, University of Toyama, Toyama-shi 930-8555, Japan · Cyberscience Center, Tohoku University, Sendai-shi 980-8578, Japan · Faculty of Science and Technology, Hirosaki University, Hirosaki-shi 036-8560, Japan +5

Constrained Optimization Differential Evolution

Mar 28, 2026

RDEx-SOP: Exploitation-Biased Reconstructed Differential Evolution for Fixed-Budget Bound-Constrained Single-Objective Optimization

Bound-constrained single-objective numerical optimisation remains a key benchmark for assessing the robustness and efficiency of evolutionary algorithms. This report documents RDEx-SOP, an exploitation-biased success-history differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session). RDEx-SOP combines success-history parameter adaptation, an exploitation-biased hybrid branch, and lightweight local perturbations to balance fast convergence and final solution quality under a strict evaluation budget. We evaluate RDEx-SOP on the official CEC 2025 SOP benchmark with the U-score framework (Speed and Accuracy categories). Experimental results show that RDEx-SOP achieves strong overall performance and statistically competitive final outcomes across the 29 benchmark functions.

Sichen Tao, Yifei Yang, Ruihan Zhao +3

Department of Engineering, University of Toyama, Toyama-shi 930-8555, Japan · Cyberscience Center, Tohoku University, Sendai-shi 980-8578, Japan · Faculty of Science and Technology, Hirosaki University, Hirosaki-shi 036-8560, Japan +5

Differential Evolution Constrained Optimization