In federated learning (FL), the assumption that datasets from different devices are independent and identically distributed (i.i.d.) often does not hold due to user differences, and the presence of various data modalities across clients makes using a single model impractical. Personalizing certain parts of the model can effectively address these issues by allowing those parts to differ across clients, while the remaining parts serve as a shared model. However, we found that partial model personalization may exacerbate client drift (each client's local model diverges from the shared model), thereby reducing the effectiveness and efficiency of FL algorithms. We propose an FL framework based on the alternating direction method of multipliers (ADMM), referred to as FedAPM, to mitigate client drift. We construct the augmented Lagrangian function by incorporating first-order and second-order proximal terms into the objective, with the second-order term providing fixed correction and the first-order term offering compensatory correction between the local and shared models. Our analysis demonstrates that FedAPM, by using explicit estimates of the Lagrange multiplier, is more stable and efficient in terms of convergence compared to other FL frameworks. We establish the global convergence of FedAPM training from arbitrary initial points to a stationary point, achieving three types of rates: constant, linear, and sublinear, under mild assumptions. We conduct experiments using four heterogeneous and multimodal datasets with different metrics to validate the performance of FedAPM. Specifically, FedAPM achieves faster and more accurate convergence, outperforming the SOTA methods with average improvements of 12.3% in test accuracy, 16.4% in F1 score, and 18.0% in AUC while requiring fewer communication rounds.

With the proliferation of Internet of Things (IoT) devices, the demand for addressing complex optimization challenges has intensified. The Lyapunov Drift-Plus-Penalty algorithm is a widely adopted approach for ensuring queue stability, and some research has preliminarily explored its integration with reinforcement learning (RL). In this paper, we investigate the adaptation of the Lyapunov Drift-Plus-Penalty algorithm for RL applications, deriving an effective method for combining Lyapunov Drift-Plus-Penalty with RL under a set of common and reasonable conditions through rigorous theoretical analysis. Unlike existing approaches that directly merge the two frameworks, our proposed algorithm, termed Lyapunov drift-plus-penalty method tailored for reinforcement learning with queue stability (LDPTRLQ) algorithm, offers theoretical superiority by effectively balancing the greedy optimization of Lyapunov Drift-Plus-Penalty with the long-term perspective of RL. Simulation results for multiple problems demonstrate that LDPTRLQ outperforms the baseline methods using the Lyapunov drift-plus-penalty method and RL, corroborating the validity of our theoretical derivations. The results also demonstrate that our proposed algorithm outperforms other benchmarks in terms of compatibility and stability.

With the widespread application of Unmanned Aerial Vehicles (UAVs) in domains like military reconnaissance, emergency rescue, and logistics delivery, efficiently planning the shortest flight path has become a critical challenge. Traditional heuristic-based methods often suffer from the inability to escape from local optima, which limits their effectiveness in finding the shortest path. To address these issues, a novel Improved Grey Wolf Optimizer (IGWO) is presented in this study. The proposed IGWO incorporates an Advanced Cooperative Predation (ACP) and a Lens Opposition-based Learning Strategy (LOBL) in order to improve the optimization capability of the method. Simulation results show that IGWO ranks first in optimization performance on benchmark functions F1-F5, F7, and F9-F12, outperforming all other compared algorithms. Subsequently, IGWO is applied to UAV shortest path planning in various obstacle-laden environments. Simulation results show that the paths planned by IGWO are, on average, shorter than those planned by GWO, PSO, and WOA by 1.70m, 1.68m, and 2.00m, respectively, across four different maps.

Combinatorial Optimization (CO) consists of finding the best solution from a discrete set of possibilities by optimizing a given objective function subject to constraints. It has widespread applications across various domains, including vehicle routing (Veres and Moussa, 2019), production planning (Dolgui et al., 2019), and drug discovery (Liu et al., 2017). However, its NP-hard nature and the complexity of many problem variants make solving CO problems highly challenging. Traditional heuristic methods (e.g., (Kirkpatrick et al., 1983; Glover, 1989; Mladenovi c and Hansen, 1997)) rely on hand-crafted rules to guide the search, providing near-optimal solutions with significantly lower computational costs. Inspired by the success of deep learning in computer vision (Krizhevsky et al., 2012; He et al., 2016) and natural language processing (Vaswani et al., 2017; Devlin, 2018), recent years have seen a surge in learning-based Neural Combinatorial Optimization (NCO) approaches for solving CO problems, including the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP). 1

The recent success of Shampoo in the AlgoPerf contest has sparked renewed interest in Kronecker-factorization-based optimization algorithms for training neural networks. Despite its success, Shampoo relies heavily on several heuristics such as learning rate grafting and stale preconditioning to achieve performance at-scale. These heuristics increase algorithmic complexity, necessitate further hyperparameter tuning, and lack theoretical justification. This paper investigates these heuristics from the angle of Frobenius norm approximation to full-matrix Adam and decouples the preconditioner's eigenvalues and eigenbasis updates. We show that grafting from Adam mitigates the staleness and mis-scaling of the preconditioner's eigenvalues and how correcting the eigenvalues directly can eliminate the need for learning rate grafting. To manage the error induced by infrequent eigenbasis computations, we propose an adaptive criterion for determining the eigenbasis computation frequency motivated by terminating a warm-started QR algorithm. This criterion decouples the update frequency of different preconditioner matrices and enables us to investigate the impact of approximation error on convergence. These practical techniques offer a principled angle towards removing Shampoo's heuristics and developing improved Kronecker-factorization-based training algorithms.

We present a generic Branch-and-Bound procedure designed to solve L0-penalized optimization problems. Existing approaches primarily focus on quadratic losses and construct relaxations using "Big-M" constraints and/or L2-norm penalties. In contrast, our method accommodates a broader class of loss functions and allows greater flexibility in relaxation design through a general penalty term, encompassing existing techniques as special cases. We establish theoretical results ensuring that all key quantities required for the Branch-and-Bound implementation admit closed-form expressions under the general blanket assumptions considered in our work. Leveraging this framework, we introduce El0ps, an open-source Python solver with a plug-and-play workflow that enables user-defined losses and penalties in L0-penalized problems. Through extensive numerical experiments, we demonstrate that El0ps achieves state-of-the-art performance on classical instances and extends computational feasibility to previously intractable ones.

Stochastic Frank-Wolfe is a classical optimization method for solving constrained optimization problems. On the other hand, recent optimizers such as Lion and Muon have gained quite significant popularity in deep learning. In this work, we provide a unifying perspective by interpreting these seemingly disparate methods through the lens of Stochastic Frank-Wolfe. Specifically, we show that Lion and Muon with weight decay can be viewed as special instances of a Stochastic Frank-Wolfe, and we establish their convergence guarantees in terms of the Frank-Wolfe gap, a standard stationarity measure in non-convex optimization for Frank-Wolfe methods. We further find that convergence to this gap implies convergence to a KKT point of the original problem under a norm constraint for Lion and Muon. Moreover, motivated by recent empirical findings that stochastic gradients in modern machine learning tasks often exhibit heavy-tailed distributions, we extend Stochastic Frank-Wolfe to settings with heavy-tailed noise by developing two robust variants with strong theoretical guarantees, which in turn yields new variants of Lion and Muon.

Abstract-- In this work, we present a method for minimizing the time required for a redundant dual-arm robot to follow a desired relative Cartesian path at constant path speed by optimizing its joint trajectories, subject to position, ve locity, and acceleration limits. The problem is reformulated as a bi - level optimization whose lower level is a convex, closed-fo rm subproblem that maximizes path speed for a fixed trajectory, while the upper level updates the trajectory using a single-chain kinematic formulation and the subgradient of the lower-lev el value. Numerical results demonstrate the effectiveness of the proposed approach. In several industrial tasks performed by robots, the Cartesian path to be followed by the end-effector requires fewer degrees of freedom than what the manipulator possesses. In many applications [1]-[5], rotation around the attached tool center point (TCP) is considered a free axis, resulting in a kinematically redundant system [6], [7].

Fuzzy clustering, which allows an article to belong to multiple clusters with soft membership degrees, plays a vital role in analyzing publication data. This problem can be formulated as a constrained optimization model, where the goal is to minimize the discrepancy between the similarity observed from data and the similarity derived from a predicted distribution. While this approach benefits from leveraging state-of-the-art optimization algorithms, tailoring them to work with real, massive databases like OpenAlex or Web of Science - containing about 70 million articles and a billion citations - poses significant challenges. We analyze potentials and challenges of the approach from both mathematical and computational perspectives. Among other things, second-order optimality conditions are established, providing new theoretical insights, and practical solution methods are proposed by exploiting the structure of the problem. Specifically, we accelerate the gradient projection method using GPU-based parallel computing to efficiently handle large-scale data.

Reconstructing 3D scenes from sparse viewpoints is a long-standing challenge with wide applications. Recent advances in feed-forward 3D Gaussian sparse-view reconstruction methods provide an efficient solution for real-time novel view synthesis by leveraging geometric priors learned from large-scale multi-view datasets and computing 3D Gaussian centers via back-projection. Despite offering strong geometric cues, both feed-forward multi-view depth estimation and flow-depth joint estimation face key limitations: the former suffers from mislocation and artifact issues in low-texture or repetitive regions, while the latter is prone to local noise and global inconsistency due to unreliable matches when ground-truth flow supervision is unavailable. To overcome this, we propose JointSplat, a unified framework that leverages the complementarity between optical flow and depth via a novel probabilistic optimization mechanism. Specifically, this pixel-level mechanism scales the information fusion between depth and flow based on the matching probability of optical flow during training. Building upon the above mechanism, we further propose a novel multi-view depth-consistency loss to leverage the reliability of supervision while suppressing misleading gradients in uncertain areas. Evaluated on RealEstate10K and ACID, JointSplat consistently outperforms state-of-the-art (SOTA) methods, demonstrating the effectiveness and robustness of our proposed probabilistic joint flow-depth optimization approach for high-fidelity sparse-view 3D reconstruction.