Search Results (76)

In this paper, we study a class of optimization problems with separable constraint structures, characterized by a combination of convex and nonconvex constraints. To handle these two distinct types of constraints, we introduce a partial augmented Lagrangian function by retaining nonconvex constraints while relaxing convex constraints into the objective function. Specifically, we employ the Moreau envelope for the convex term and apply second-order variational geometry to analyze the nonconvex term. For this partial augmented Lagrangian function, we study its saddle points and establish their relationship with KKT conditions. Furthermore, second-order optimality conditions are developed by employing tools such as second-order subdifferentials, asymptotic second-order tangent cones, and second-order tangent sets. Full article

The quest to multiply two large matrices as fast as possible is one that has already intrigued researchers for several decades. However, the ‘optimal’ algorithm for a certain problem size is still not known. The fast matrix multiplication (FMM) problem can be formulated as a non-convex optimization problem—more specifically, as a challenging tensor decomposition problem. In this work, we build upon a state-of-the-art augmented Lagrangian algorithm, which formulates the FMM problem as a constrained least squares problem, by incorporating a new, generalized cyclic symmetric (CS) structure in the decomposition. This structure decreases the number of variables, thereby reducing the large search space and the computational cost per iteration. The constraints are used to find practical solutions, i.e., decompositions with simple coefficients, which yield fast algorithms when implemented in hardware. For the FMM problem, usually a very large number of starting points are necessary to converge to a solution. Extensive numerical experiments for different problem sizes demonstrate that including this structure yields more ‘unique’ practical decompositions for a fixed number of starting points. Uniqueness is defined relative to the known scale and trace invariance transformations that hold for all FMM decompositions. Making it easier to find practical decompositions may lead to the discovery of faster FMM algorithms when used in combination with sufficient computational power. Lastly, we show that the CS structure reduces the cost of multiplying a matrix by itself. Full article

System representation via computational graphs has become a cornerstone of modern machine learning, underpinning the gradient-based training of complex models. We have previously introduced the Partial Lagrangian Method—a divide-and-conquer approach that decomposes the Lagrangian into link-wise components—to derive and evaluate the equations of motion for robot systems with dynamically changing structures. That method leverages the symbolic expressiveness of computational graphs with automatic differentiation to streamline dynamic analysis. In this paper, we advance this framework by establishing a principled way to encode time-dependent differential equations as computational graphs. Our approach, which augments the state vector and applies the chain rule, constructs fully time-independent graphs directly from the Lagrangian, eliminating the erroneous time-derivative embeddings that previously required manual correction. Because our transformation is derived from first principles, it guarantees graph correctness and generalizes to any system governed by variational dynamics. We validate the method on a simple serial-link robotic arm, showing that it faithfully reproduces the standard equations of motion without graph failure. Furthermore, by compactly representing state variables, the resulting computational graph achieves a seven-fold reduction in evaluation time compared to our prior implementation. The proposed framework thus offers a more intuitive, scalable, and efficient design and analysis of complex dynamic systems. Full article

Energy management for hybrid electric commercial vehicles, involving continuous power output and discrete gear shifting, constitutes a typical mixed-integer programming (MIP) problem, presenting significant challenges for real-time performance and computational efficiency. To address this, this paper proposes a physics-informed neural network-optimized model predictive control (PINN-MPC) strategy. On one hand, this strategy simultaneously optimizes continuous and discrete states within the MPC framework to achieve the integrated objectives of minimizing fuel consumption, tracking speed, and managing battery state-of-charge (SOC). On the other hand, to overcome the prohibitively long solving time of the MIP-MPC, a physics-informed neural network (PINN) optimizer is designed. This optimizer employs the soft-argmax function to handle discrete gear variables and embeds system dynamics constraints using an augmented Lagrangian approach. Validated via hardware-in-the-loop (HIL) testing under two distinct real-world driving cycles, the results demonstrate that, compared to the open-source solver BONMIN, PINN-MPC significantly reduces computation time—dramatically decreasing the average solving time from approximately 10 s to about 5 ms—without sacrificing the combined vehicle dynamic and economic performance. Full article

The convergence rate of the augmented Lagrangian method (ALM) for solving the nonlinear semidefinite optimization problem is studied. Under the Jacobian uniqueness conditions, when a multiplier vector $(\pi ,Y)$ and the penalty parameter $\sigma$ are chosen such that $\sigma$ is larger than a threshold ${\sigma }^{*}>0$ and the ratio $\parallel (\pi ,Y)-({\pi }^{*},Y^{*})\parallel /\sigma$ is small enough, it is demonstrated that the convergence rate of the augmented Lagrange method is linear with respect to $\parallel (\pi ,Y)-({\pi }^{*},Y^{*})\parallel$ and the ratio constant is proportional to $1/\sigma$, where $({\pi }^{*},Y^{*})$ is the multiplier corresponding to a local minimizer. Furthermore, by analyzing the second-order derivative of the perturbation function of the nonlinear semidefinite optimization problem, we characterize the rate constant of local linear convergence of the sequence of Lagrange multiplier vectors produced by the augmented Lagrange method. This characterization shows that the sequence of Lagrange multiplier vectors has a Q-linear convergence rate when the sequence of penalty parameters $\left\{{\sigma }_k\right\}$ has an upper bound and the convergence rate is superlinear when $\left\{{\sigma }_k\right\}$ is increasing to infinity. Full article

In this paper, we propose a novel alternating direction method of multipliers based on acceleration technique involving two symmetrical inertial terms for a class of nonconvex optimization problems with a two-block structure. To address the nonconvex subproblem, we introduce a proximal term to reduce the difficulty of solving this subproblem. For the smooth subproblem, we employ a gradient descent method on the augmented Lagrangian function, which significantly reduces the computational complexity. Under appropriate assumptions, we prove subsequential convergence of the algorithm. Moreover, when the generated sequence is bounded and the auxiliary function satisfies Kurdyka–Łojasiewicz property, we establish global convergence of the algorithm. Finally, effectiveness and superior performance of the proposed algorithm are validated through numerical experiments in signal processing and smoothly clipped absolute deviation penalty problems. Full article

This study proposes an Index-Based Neural Network (IBNN) framework for the static analysis of truss structures, employing a Lagrangian dual optimization technique grounded in the force method. A truss is a discrete structural system composed of linear members connected to nodes. Despite their geometric simplicity, analysis of large-scale truss systems requires significant computational resources. The proposed model simplifies the input structure and enhances the scalability of the model using member and node indices as inputs instead of spatial coordinates. The IBNN framework approximates member forces and nodal displacements using separate neural networks and incorporates structural equations derived from the force method as mechanics-informed constraints within the loss function. Training was conducted using the Augmented Lagrangian Method (ALM), which improves the convergence stability and learning efficiency through a combination of penalty terms and Lagrange multipliers. The efficiency and accuracy of the framework were numerically validated using various examples, including spatial trusses, square grid-type space frames, lattice domes, and domes exhibiting radial flow characteristics. Multi-index mapping and domain decomposition techniques contribute to enhanced analysis performance, yielding superior prediction accuracy and numerical stability compared to conventional methods. Furthermore, by reflecting the structured and discrete nature of structural problems, the proposed framework demonstrates high potential for integration with next-generation neural network models such as Quantum Neural Networks (QNNs). Full article

This paper studies the safe trajectory tracking control of intelligent vehicles, which is still an open and challenging problem. A deep reinforcement learning algorithm based on augmented Lagrangian safety constraints is proposed to the lateral control of vehicle trajectory tracking. First, the tracking control of intelligent vehicles is described as a reinforcement learning process based on the Constrained Markov Decision Process (CMDP). The actor-critic neural network based reinforcement learning framework is established and the environment of reinforcement learning is designed to include the vehicle model, tracking model, road model and reward function. Secondly, the augmented Lagrangian Deep Deterministic Policy Gradient (DDPG) method is proposed for updating, in which a replay separation buffer method is used to solve the problem of sample correlation, and a neural network with the same structure is copied to solve the update divergence problem. Finally, a vehicle lateral control approach is obtained, whose effectiveness and advantages over existing results are verified through simulation results. Full article

Emerging evidence indicates that large-scale forest restoration exhibits dual hydrological effects: direct reduction of local water availability through elevated evapotranspiration (ET) and indirect augmentation of water resources via enhanced atmospheric moisture recycling. However, the quantitative assessment of these counteracting effects remains challenging due to the limited observational constraints on moisture transport. Here, we integrate the Budyko model with the Lagrangian-based UTrack moisture-tracking dataset to disentangle the direct (via ET) and indirect (via precipitation) large-scale hydrological impacts of China’s four-decade forest restoration campaign across eight major river basins. Multisource validation datasets, including gauged runoff records, hydrological reanalysis products, and satellite-derived forest cover maps, were systematically incorporated to verify the Budyko model at the nested spatial scales. Our scenario analyses reveal that during 1980–2015, extensive afforestation individually reduced China’s terrestrial water yield by −28 ± 25 mm yr⁻¹ through dominant ET increases. Crucially, atmospheric moisture recycling mechanisms attenuated this water loss by 12 ± 5 mm yr⁻¹ nationally, with marked spatial heterogeneity across the basins. In some moisture-limited watersheds in the Yellow River Basin, the negative ET effect was compensated for to a certain extent by precipitation recycling, demonstrating net positive hydrological outcomes. We conclude that China’s forest expansion imposes local water stress (direct effect) by elevating ET, while the concomitant strengthening of continental-scale moisture recycling generates compensatory water gains (indirect effect). These findings advance the mechanistic understanding of the vegetation-climate-water nexus, providing quantitative references for optimizing forestation strategies under atmospheric water connectivity constraints. Full article

Noises and outliers often degrade the final prediction performance in practical data processing. Multi-view learning by integrating complementary information across heterogeneous modalities has become one of the core techniques in the field of machine learning. However, existing methods rely on explicit-view clustering and stringent alignment assumptions, which affect the effectiveness in addressing the challenges such as inconsistencies between views, noise interference, and misalignment across different views. To alleviate these issues, we present a latent multi-view representation learning model based on low-rank embedding by implicitly uncovering the latent consistency structure of data, which allows us to achieve robust and efficient multi-view feature fusion. In particular, we utilize low-rank constraints to construct a unified latent subspace representation and introduce an adaptive noise suppression mechanism that significantly enhances robustness against outliers and noise interference. Moreover, the Augmented Lagrangian Multiplier Alternating Direction Minimization (ALM-ADM) framework enables efficient optimization of the proposed method. Experimental results on multiple benchmark datasets demonstrate that the proposed approach outperforms existing state-of-the-art methods in both clustering performance and robustness. Full article

This paper applies the Method of Successive Approximations (MSA) based on Pontryagin’s principle to solve optimal control problems with state constraints for semilinear parabolic equations. Error estimates for the first and second derivatives of the function are derived under $L^{\infty }$-bounded conditions. An augmented MSA is developed using the augmented Lagrangian method, and its convergence is proven. The effectiveness of the proposed method is demonstrated through numerical experiments. Full article

The generalized convex nearly isotonic regression problem addresses a least squares regression model that incorporates both sparsity and monotonicity constraints on the regression coefficients. In this paper, we introduce an efficient semismooth Newton-based augmented Lagrangian (Ssnal) algorithm to solve this problem. We demonstrate that, under reasonable assumptions, the Ssnal algorithm achieves global convergence and exhibits a linear convergence rate. Computationally, we derive the generalized Jacobian matrix associated with the proximal mapping of the generalized convex nearly isotonic regression regularizer and leverage the second-order sparsity when applying the semismooth Newton method to the subproblems in the Ssnal algorithm. Numerical experiments conducted on both synthetic and real datasets clearly demonstrate that our algorithm significantly outperforms first-order methods in terms of efficiency and robustness. Full article

Two-phase evaporative spray cooling technology can significantly reduce power consumption in data centre cooling applications. However, the literature lacks an established methodology for assessing the overall performance of such evaporation systems in terms of the water-energy nexus. The current study develops a Lagrangian–Eulerian computational fluid dynamics (CFD) modelling approach to examine the functionality of these two-phase evaporative spray cooling systems. To replicate a modular system, a hollow spray cone nozzle with Rosin–Rammler droplet size distribution is simulated in a turbulent convective natural-air environment. The model was validated against the available experimental data from the literature. Parametric studies on geometric, flow, and climatic conditions, namely, domain length, droplet size, water mass flow rate, temperature, and humidity, were performed. The findings indicate that at elevated temperatures and low humidity, evaporation results in a bulk temperature reduction of up to 12 °C. A specific focus on the climatic conditions of Dublin, Ireland, was used as an example to optimize the evaporative system. A new formulation for the coefficient of performance (COP) is established to assess the performance of the system. Results showed that doubling the injector water mass flow rate improved the evaporated mass flow rate by 188% but reduced the evaporation percentage by 28%, thus reducing the COP. Doubling the domain length improved the temperature drop by 175% and increased the relative humidity by 160%, thus improving the COP. The COP of the evaporation system showed a systematic improvement with a reduction in the droplet size and the mass flow rate for a fixed domain length. The evaporated system COP improves by two orders of magnitude (~90 to 9500) with the reduction in spray Sauter mean diameter (SMD) from 292 μm to 8–15 μm. Under this reduction, close to 100% evaporation rate was achieved in comparison to only a 1% evaporation rate for the largest SMD. It was concluded that the utilization of a fine droplet spray nozzle provides an effective solution for the reduction in water consumption (97% in our case) for data centres, whilst concomitantly augmenting the proportion of evaporation. Full article

Optimization problems with PDE constraints are widely used in engineering and technical fields. In some practical applications, it is necessary to smooth the control variables and suppress their large fluctuations, especially at the boundary. Therefore, we propose an elliptic PDE-constrained optimization model with a control gradient penalty term. However, introducing this penalty term increases the complexity and difficulty of the problems. To solve the problems numerically, we adopt the strategy of “First discretize, then optimize”. First, the finite element method is employed to discretize the optimization problems. Then, a heterogeneous strategy is introduced to formulate the augmented Lagrangian function for the subproblems. Subsequently, we propose a three-block inexact heterogeneous alternating direction method of multipliers (three-block ihADMM). Theoretically, we provide a global convergence analysis of the three-block ihADMM algorithm and discuss the iteration complexity results. Numerical results are provided to demonstrate the efficiency of the proposed algorithm. Full article

The research of multimodal emotion recognition based on facial expressions, speech, and body gestures is crucial for oncoming intelligent human–computer interfaces. However, it is a very difficult task and has seldom been researched in this combination in the past years. Based on the GEMEP and Polish databases, this contribution focuses on trimodal emotion recognition from facial expressions, speech, and body gestures, including feature extraction, feature fusion, and multimodal classification of the three modalities. In particular, for feature fusion, two novel algorithms including supervised least squares multiset kernel canonical correlation analysis (SLSMKCCA) and sparse supervised least squares multiset kernel canonical correlation analysis (SSLSMKCCA) are presented, respectively, to carry out efficient facial expression, speech, and body gesture feature fusion. Different from the traditional multiset kernel canonical correlation analysis (MKCCA) algorithms, our SLSKMCCA algorithm is a supervised version and is based on the least squares form. The SSLSKMCCA algorithm is implemented by the combination of SLSMKCCA and a sparse item (L1 Norm). Moreover, two effective solving algorithms for SLSMKCCA and SSLSMKCCA are presented in addition, which use the alternated least squares and augmented Lagrangian multiplier methods, respectively. The extensive experimental results on the popular public GEMEP and Polish databases show that the recognition rate of multimodal emotion recognition is superior to bimodal and monomodal emotion recognition on average, and our presented SLSMKCCA and SSLSMKCCA fusion methods both obtain very high recognition rates, especially for the SSLSMKCCA fusion method. Full article