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Micromagnetic simulations require the numerically challenging preservation of the Euclidean norm during the whole simulation. This can be accomplished by applying a priori length preserving methods, renormalization algorithms, or penalization strategies. The latter one includes both the penalty method and the Lagrangian multiplier. The penalty method requires the definition of a penalty parameter during the initiation of the simulation which, depending on its size, can lead to an unsatisfied constraint or stiff and difficult to solve systems of equations. The Lagrange multiplier always penalizes in problem-dependent intensity, hence, an additional degree of freedom is added to the system of equations to the drawback of higher computational costs. This paper proposes a method that utilizes a perturbed Lagrangian multiplier and an element level static condensation to condensate the additional degree of freedom. This guarantees fast simulations, and no parameter fitting in advance. Suitable numerical examples are conducted to prove the workability of the outlined scheme and to highlight the efficiency compared to the non-condensed formulation. [ABSTRACT FROM AUTHOR]

Lagrangian heuristics for discrete optimization work by modifying Lagrangian relaxed solutions into feasible solutions to an original problem. They are designed to identify feasible, and hopefully also near-optimal, solutions and have proven to be highly successful in many applications. Based on a primal-dual global optimality condition for non-convex optimization problems, we develop a meta-heuristic extension of the Lagrangian heuristic framework. The optimality condition characterizes (near-)optimal solutions in terms of near-optimality and near-complementarity measures for Lagrangian relaxed solutions. The meta-heuristic extension amounts to constructing a weighted combination of these measures, thus creating a parametric auxiliary objective function, which is a close relative to a Lagrangian function, and embedding a Lagrangian heuristic in a search procedure in the space of the weight parameters. We illustrate and make a first assessment of this meta-heuristic extension by applying it to the generalized assignment and set covering problems. Our computational experience show that the meta-heuristic extension of a standard Lagrangian heuristic can significantly improve upon solution quality.  [ABSTRACT FROM AUTHOR]

The Weierstrass' theory of one-dimensional Lagrangian systems and a quasi-continuum approach are employed to study the propagation of solitary waves in tensegrity mass-spring chains, which exhibit softening-type elastic response in the large displacement regime and are subject to external pre-compression. The presented study analytically derives the shape of the traveling rarefaction pulses, and limiting values of the speeds of such pulses. Use is made of a tensegrity-like interaction potential that captures the main features of the real force-displacement response of the examined units. The Weierstrass approach is validated through numerical applications that establish comparisons between the theory developed in the present work and previous results available in literature. [ABSTRACT FROM AUTHOR]

AbstractResearch in topology optimization with stress constraints has shown that defining the stress constraint locally yields better performance compared to global methods. However, an examination of the formulas for local stress constraints reveals a limitation - the constraint is only satisfied at points where it is explicitly defined, failing to guarantee satisfaction across the entire design domain. To address this shortcoming, this paper proposes utilizing an integral form of the stress constraint. The integral formulation theoretically ensures that the stress constraint is satisfied over whole design domain. The objective is to minimize the mass of plane stress structures using reaction-diffusion level set method while incorporating local stress constraints in this integral form. The methodology utilizes finite element approximations for geometry and displacements, defining local stress constraints through an integral formulation. A Lagrangian function combines objective and constraint functions, with sensitivity analysis performed during optimization based on level set function changes. Structural boundaries are updated using the Hamilton-Jacobi equation. The paper presents numerical examples with varying loads and support conditions to demonstrate the effectiveness of this integral stress constraint approach. Results illustrate the capability of the proposed method to generate optimal topologies while satisfying stress constraints across the entire design space. [ABSTRACT FROM AUTHOR]

It is well known that k -contact geometry is a suitable framework to deal with non-conservative field theories. In this paper, we study some relations between solutions of the k -contact Euler–Lagrange equations, symmetries, dissipation laws and Newtonoid vector fields. We review the k -contact Euler–Lagrange equations written in terms of k -vector fields and sections and provide new results relating the solutions in both approaches. We also study different kinds of symmetries depending on the structures they preserve: natural (preserving the Lagrangian function), dynamical (preserving the solutions), and k -contact (preserving the underlying geometric structures) symmetries. For some of these symmetries, we provide Noether-like theorems relating symmetries and dissipation laws. We also analyze the relation between k -contact symmetries and Newtonoid vector fields. Throughout the paper, we will use the damped vibrating string as our main illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, based on Lagrangian functions of the optimization problem we develop a sketch-and-project method for solving the linear matrix equation A X B = C by introducing three parameters. A thorough convergence analysis on the proposed method is explored in details. A lower bound on the convergence rate and some convergence conditions are derived. By varying three parameters in the new method and convergence theorems, an array of well-known algorithms and their convergence results are recovered. Finally, numerical experiments are given to illustrate the effectiveness of recovered methods. [ABSTRACT FROM AUTHOR]

In recent years, coupled shear walls have been used more often because they provide a significant percentage of the total stiffness of a tall building total horizontal bracing system. This article presents an analytical procedure for the static and dynamic analysis of coupled shear walls of a bay using a continuous model type – CTB of three fields. The coupled shear wall is modeled as a continuous structure consisting of the parallel coupling of a Timoshenko extensible beam and a shear beam. Using a variational approach, the equilibrium equations, constitutive laws, and boundary conditions are derived by applying Hamilton's principle to the relevant Lagrangian function. Closed-form solutions are provided for the static deflection of the coupled shear wall under general profile lateral loads and the periods of vibration of the coupled shear wall under vertical loads uniformly distributed along its height. A validation of the static analysis was carried out, and to find the range of validity of the continuous model and proposed solution methods, a parametric analysis was carried out. The results are encouraging and show that the proposed analytical solutions have excellent correspondence with respect to the finite element method; they are accurate, reliable and involve low computational cost at a preliminary stage, and can be used as a verification method at the final stage of structural analysis and design. [ABSTRACT FROM AUTHOR]

Small-scale motions in turbulent flows play a significant role in various small-scale processes, such as particle relative dispersion and collision, bubble or droplet deformation, and orientation dynamics of non-sphere particles. Recovering the small-scale flows that cannot be resolved in large eddy simulation (LES) is of great importance for such processes sensitive to the small-scale motions in turbulent flows. This study proposes a subgrid-scale model for recovering the small-scale turbulent velocity field based on the artificial neural network (ANN). The governing equations of small-scale turbulent velocity are linearized, and the pressure gradient and the nonlinear convection term are modeled with the aid of the ANN. Direct numerical simulation (DNS) and filtered direct numerical simulation (FDNS) provide the data required for training and validating the ANN. The large-scale velocity and velocity gradient tensor are selected as inputs for the ANN model. The linearized governing equations of small-scale turbulent velocity are numerically solved by coupling the large-scale flow field information. The results indicate that the model established by the ANN can accurately recover the small-scale velocity lost in FDNS due to filtering operation. With the ANN model, the flow fields at different Reynolds numbers agree well with the DNS results regarding velocity field statistics, flow field structures, turbulent energy spectra, and two-point, two-time Lagrangian correlation functions. This study demonstrates that the proposed ANN model can be applied to recovering the small-scale velocity field in the LES of isotropic turbulent flows at different Reynolds numbers. [ABSTRACT FROM AUTHOR]

In this paper, a partial Bregman alternating direction method of multipliers (ADMM) with a general relaxation factor α ∈ (0 , 1 + 5 2) is proposed for structured nonconvex and nonsmooth optimization, where the objective function is the sum of a nonsmooth convex function and a smooth nonconvex function without coupled variables. We add a Bregman distance to alleviate the difficulty of solving the nonsmooth subproblem. For the smooth subproblem, we directly perform a gradient descent step of the augmented Lagrangian function, which makes the computational cost of each iteration of our method very cheap. To our knowledge, the nonconvex ADMM with a relaxation factor α ≠ 1 in the literature has never been studied for the problem under consideration. Under some mild conditions, the boundedness of the generated sequence, the global convergence and the iteration complexity are established. The numerical results verify the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we study the nonconvex constrained composition optimization, in which the objective contains a composition of two expected-value functions whose accurate information is normally expensive to calculate. We propose a STochastic nEsted Primal-dual (STEP) method for such problems. In each iteration, with an auxiliary variable introduced to track the inner layer function values we compute stochastic gradients of the nested function using a subsampling strategy. To alleviate difficulties caused by possibly nonconvex constraints, we construct a stochastic approximation to the linearized augmented Lagrangian function to update the primal variable, which further motivates to update the dual variable in a weighted-average way. Moreover, to better understand the asymptotic dynamics of the update schemes we consider a deterministic continuous-time system from the perspective of ordinary differential equation (ODE). We analyze the Karush-Kuhn-Tucker measure at the output by the STEP method with constant parameters and establish its iteration and sample complexities to find an \epsilon-stationary point, ensuring that expected stationarity, feasibility as well as complementary slackness are below accuracy \epsilon. To leverage the benefit of the (near) initial feasibility in the STEP method, we propose a two-stage framework incorporating a feasibility-seeking phase, aiming to locate a nearly feasible initial point. Moreover, to enhance the adaptivity of the STEP algorithm, we propose an adaptive variant by adaptively adjusting its parameters, along with a complexity analysis. Numerical results on a risk-averse portfolio optimization problem and an orthogonal nonnegative matrix decomposition reveal the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We have presented a novel lattice Boltzmann approach for the non‐uniform rectangular mesh based on the radial basis function approximation (RBF‐LBM). The non‐uniform rectangular mesh is a good option for local grid refinement, especially for the wall boundaries and flow areas with intensive change of flow quantities. Which allows, the total number of grid cells to be reduced and so the computational cost, therefore improving the computational efficiency. But the grid structure of the non‐uniform rectangular mesh is no longer applicable to the classic lattice Boltzmann method (CLBM), which is based on the famous BGK collision‐streaming evolution. This is why the present study is inspired by the idea of the interpolation‐supplemented LBM (ISLBM) methodology. The ISLBM algorithm is improved in the present manuscript and developed into a novel LBM approach through the radial basis function approximation instead of the Lagrangian interpolation scheme. The new approach is validated for both steady states and unsteady periodic solutions. The comparison between the radial basis function approximation and the Lagrangian interpolation is discussed. It is found that the novel approach has a good performance on computational accuracy and efficiency. Proving that the non‐uniform rectangular mesh allows grid refinement while obtaining precise flow predictions. [ABSTRACT FROM AUTHOR]

A coupled Computational Fluid Dynamics-Discrete Element Method (CFD-DEM) approach is used to calculate the interaction of a flexible rag transported by a fluid current with a fixed solid cylinder. More specifically a hybrid Eulerian-Lagrangian approach is used with the rag being modeled as a set of interconnected particles. The influence of various parameters is considered, namely the inlet velocity (1.5, 2.0, and 2.5 m/s, respectively), the angle formed by the initially straight rag with the flow direction (45°, 60° and 90°, respectively), and the inlet position (90, 100, and 110 mm, respectively). The results show that the flow rate has a significant impact on the permeability of the rag. The higher the flow rate, the higher the permeability and the rag speed difference. The angle has a minor effect on rag permeability, with 45° being the most favorable angle for permeability. The inlet position has a small impact on rag permeability, while reducing the initial distance between the rag an the cylinder makes it easier for rags to pass through. [ABSTRACT FROM AUTHOR]

Users andedge servers arenot fullymutuallytrusted inmobile edge computing(MEC), andhenceblockchaincanbe introduced to provide trustable MEC. In blockchain-based MEC, each edge server functions as a node in bothMEC and blockchain, processing users' tasks and then uploading the task related information to the blockchain. That is, each edge server runs both users' offloaded tasks and blockchain tasks simultaneously.Note that there is a trade-off between the resource allocation for MEC and blockchain tasks. Therefore, the allocation of the resources of edge servers to the blockchain and the MEC is crucial for the processing delay of blockchain-based MEC. Most of the existing research tackles the problem of resource allocation in either blockchain or MEC,which leads to unfavorable performance of the blockchain-based MEC system. In this paper, we study how to allocate the computing resources of edge servers to the MEC and blockchain tasks with the aim to minimize the total system processing delay. For the problem, we propose a computing resource Allocation algorithm for Blockchain-based MEC (ABM) which utilizes the Slater's condition, Karush-Kuhn-Tucker (KKT) conditions, partial derivatives of the Lagrangian function and subgradient projectionmethod to obtain the solution. Simulation results show that ABM converges and effectively reduces the processing delay of blockchain-based MEC. [ABSTRACT FROM AUTHOR]

This article is concerned with a class of nonsmooth semi-infinite programming problems on Hadamard manifolds (abbreviated as, (NSIP)). We introduce the Guignard constraint qualification (abbreviated as, (GCQ)) for (NSIP). Subsequently, by employing (GCQ), we establish the Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary optimality conditions for (NSIP). Further, we derive that the Lagrangian function associated with a fixed Lagrange multiplier, corresponding to a known solution, remains constant on the solution set of (NSIP) under geodesic pseudoconvexity assumptions. Moreover, we derive certain characterizations of the solution set of the considered problem (NSIP) in the framework of Hadamard manifolds. We provide illustrative examples that highlight the importance of our established results. To the best of our knowledge, characterizations of the solution set of (NSIP) using Clarke subdifferentials on Hadamard manifolds have not been investigated before. [ABSTRACT FROM AUTHOR]

We address the problem of minimizing a smooth function under smooth equality constraints. Under regularity assumptions on these constraints, we propose a notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization. Using a smooth exact penalty function known as Fletcher's augmented Lagrangian, we propose an algorithm to minimize the penalized cost function which reaches ε -approximate second-order critical points of the original optimization problem in at most O (ε - 3) iterations. This improves on current best theoretical bounds. Along the way, we show new properties of Fletcher's augmented Lagrangian, which may be of independent interest. [ABSTRACT FROM AUTHOR]

K-essence theories are usually studied in the framework of a single scalar field ϕ. Namely, the Lagrangian of K-essence is the function of the single scalar field ϕ and its covariant derivative. However, in this paper, we explore a double-field pure K-essence, i.e., the corresponding Lagrangian is the function of covariant derivatives of double scalar fields without a dependency on scalar fields themselves. This is why we call it double-field pure K-essence. The novelty of this K-essence is that its Lagrangian contains the quotient term of the kinetic energies from the two scalar fields. This results in the presence of many interesting features; for example, the equation of state can be arbitrarily small and arbitrarily large. In comparison, the range of the equation of state for quintessence is − 1 to + 1 . Interestingly, this novel K-essence can play the role of an inflation field, dark matter, or dark energy by appropriately selecting the expressions of Lagrangian. [ABSTRACT FROM AUTHOR]

We aim to derive a super-radiance solution of coherent light and sound waves involving water degrees of freedom in the environment of a microtubule. We introduce a Lagrangian density functional of quantum electrodynamics with non-relativistic charged bosons as a model of quantum brain dynamics (QBD) involving water molecular conformational states and photon fields. We also introduce the model of charged boson fields (water degrees of freedom) coupled with phonons. Both optical and acoustic super-radiance solutions are derived in our approach. An acoustic super-radiance mechanism involving information transfer is proposed as an additional candidate to solve the binding problem and to achieve acoustic holography. Our results can be applied to achieve holographic memory storage and information processing in QBD. [ABSTRACT FROM AUTHOR]

In this work, we derived a new type model for spatial Hill's system considering the created perturbation by the parameter effect of the continuation fractional potential. The new model is considered a reduced system from the restricted three-body problem under the same effect for describing Hill's problem. We identified the associated Lagrangian and Hamiltonian functions of the new system, and used them to verify the existence of the new equations of motion. We also proved that the new model has different six valid solutions under different six symmetries transformations as well as the original solution, where the new model is an invariant under these transformations. The several symmetries of Hill's model can extremely simplify the calculation and analysis of preparatory studies for the dynamical behavior of the system. Finally, we confirm that these symmetries also authorize us to explore the similarities and differences among many classes of paths that otherwise differ from the obtained trajectories by restricted three-body problem. [ABSTRACT FROM AUTHOR]

The energy transport in aero-acoustics is investigated in the Lagrangian frame. First, based on finite-time Lyapunov exponent (FTLE) and momentum potential theory, a Lagrangian approach is proposed to identify transport barriers of acoustic energy. Specifically, the method, named relative flux gradient (RFG), is presented in detail. Then, to verify the method, it is applied to analytical fields, showing that it could reveal the wavefronts and energy transport barriers depending on the time interval of computation. Moreover, RFG is applied to analyze a simulated flow field of an open cavity flow, and the results are compared with the Lagrangian coherent structures identified by FTLE, demonstrating great similarity. Importantly, the differences between the structures are further analyzed, illustrating several transport channels that correspond to the Rossiter mode and showing a complex interaction between acoustic and vorticity modes. Finally, the relationship between the identified transport barriers and the acoustic behaviors in Eulerian frame is studied in detail. The results show that the transport barriers identified by RFG significantly impact the orbits in phase space, and in particular, RFG has the potential to illustrate and analyze the transport of acoustic energy in complex flow fields in a quantitative way: one method for direct analysis of acoustic phenomena in complex flow regions. [ABSTRACT FROM AUTHOR]

We propose an efficient primal-dual interior-point relaxation algorithm based on a smoothing barrier augmented Lagrangian, called IPRSDP, for solving semidefinite programming problems in this paper. The IPRSDP algorithm has three advantages over classical interior-point methods. Firstly, IPRSDP does not require the iterative points to be positive definite. Consequently, it can easily be combined with the warm-start technique used for solving many combinatorial optimization problems, which require the solutions of a series of semidefinite programming problems. Secondly, the search direction of IPRSDP is symmetric in itself, and hence the symmetrization procedure is not required any more. Thirdly, with the introduction of the smoothing barrier augmented Lagrangian function, IPRSDP can provide the explicit form of the Schur complement matrix. This enables the complexity of forming this matrix in IPRSDP to be comparable to or lower than that of many existing search directions. The global convergence of IPRSDP is established under suitable assumptions. Numerical experiments are made on the SDPLIB set, which demonstrate the efficiency of IPRSDP. [ABSTRACT FROM AUTHOR]

Generalized nonlinear programming is considered without any convexity assumption, capturing a variety of problems that include nonsmooth objectives, combinatorial structures, and set-membership nonlinear constraints. We extend the augmented Lagrangian framework to this broad problem class, pre-serving an implicit formulation and introducing auxiliary variables merely as a formal device. This, however, gives rise to a generalized augmented Lagrangian function that lacks regularity. Based on parametric optimization, we develop a tailored stationarity concept to better qualify the iterates, generated as approximate solutions to a sequence of subproblems. Using this variational characterization and the lifted representation, asymptotic properties and convergence guarantees are established for a safeguarded augmented Lagrangian scheme. Numerical examples showcase the modelling versatility gained by dropping convexity assumptions and the practical benefits of the advocated implicit approach. [ABSTRACT FROM AUTHOR]

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method. Funding: This work was supported by the National Science Foundation [Grant CNS-1545046] and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research [Grant DE-AC02-06CH11347]. [ABSTRACT FROM AUTHOR]

This article focuses on three main contributions. Firstly, we provide an in-depth overview of the nonlocal Lagrangian formalism. Secondly, we introduce an extended version of the second Noether's theorem tailored for nonlocal Lagrangians. Finally, we apply both the formalism and the extended theorem to the context of non-commutative U(1) gauge theory, including its Hamiltonian and quantization, showcasing their practical utility. [ABSTRACT FROM AUTHOR]

We evaluate here the quantum gravity partition function that counts the dimension of the Hilbert space of a simply connected spatial region of a fixed proper volume in the context of Lovelock gravity, generalizing the results for Einstein gravity. It is found that there are sphere saddle metrics for a partition function at a fixed spatial volume in Lovelock theory. Those stationary points take exactly the same forms as in Einstein gravity. The logarithm of Z corresponding to a zero effective cosmological constant indicates that the Bekenstein–Hawking entropy of the boundary area and that corresponding to a positive effective cosmological constant points to the Wald entropy of the boundary area. We also show the existence of zeroth-order phase transitions between different vacua, a phenomenon distinct from Einstein gravity. [ABSTRACT FROM AUTHOR]

Additive manufacturing (AM) has emerged as a highly promising manufacturing technique, offering unprecedented possibilities for creating complex geometries and functional structures. However, harnessing the full potential of AM requires the development of a robust computational framework capable of capturing the intricate multi-scale and multi-physics nature of the process. The constitutive and structural responses encountered in AM are particularly challenging to reproduce due to the complex behavior of the material involved. This research aims to address these challenges by presenting a comprehensive computational approach that incorporates a material model capable of accurately representing the behavior of different phases occurring during AM. To achieve this, the finite element method, using the Lagrangian framework in the implicit time scheme, is employed through the widely adopted ABAQUS software. Computational implementation is facilitated using the FORTRAN programming language. By employing weakly coupled thermal and mechanical constitutive equations, the framework enables the analysis of thermal stresses, strains, and displacements during realistic solidification processes, which inherently involve highly nonlinear constitutive relations. Through a series of numerical examples, the capabilities of the proposed model are demonstrated across various computational scales, particularly during the rapid melting and solidification phases. These simulations reveal the formation of residual stresses, which can lead to part distortion and have detrimental effects on the mechanical properties of the manufactured components. This research contributes to the advancement of additive manufacturing by providing a reliable computational tool that integrates the complex interplay between thermal and mechanical phenomena. The developed framework enhances our understanding of the AM process, offering valuable insights into the factors influencing the structural integrity and performance of additively manufactured parts. [ABSTRACT FROM AUTHOR]

The existence of L2–normalized solutions is studied for the equation − Δ u + μ u = f (x , u) in R N , ∫ R N u 2 d x = m. Here m > 0 and f(x, s) are given, f(x, s) has the L2-subcritical growth and (μ, u) ∈ R × H1(RN) are unknown. In this paper, we employ the argument in Hirata and Tanaka ("Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches," Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka ("Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches," Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry. [ABSTRACT FROM AUTHOR]

This study delves into addressing the challenge of resolving the Unit Commitment (UC) problem, which focuses on enhancing the efficiency of production units and devising their operational schedules to accommodate fluctuations in consumption spanning from a day to a month. Given the intricate, combinatorial, and nonlinear constraints associated with each production unit, this study advocates an optimization approach rooted in fuzzy logic. A Langrangian function was established to simplify the UCP and to transform the different inequality into a linear unconstrained problem. The choice of fuzzy inputs was established using the partial derivatives of a Lagrangian function as a function of the powers injected into each node of the electrical network. This combination of the Lagrangian function and the input of the fuzzy regulator made it possible to control the different constraints in the total production cost function and to improve the operating efficiency of the different production units. This method was effectively applied to a 14-bus IEEE power network encompassing 5 generating units, to address the UC problem by optimizing generator load capacity (LCG) and minimizing Incremental Losses (IL). The numerical processing of the fuzzy linguistic variables was implemented using Mamdani-type fuzzy rules. This strategy stands out for its robust exploratory capability, facilitating the identification of optimal solutions to reduce production costs while ensuring optimal planning of production units. [ABSTRACT FROM AUTHOR]

Reconstruction of Lagrangian functions present in the action of modified gravity theories is one of the most captivating subjects in recent cosmology. This paper reconstructs the form of Lagrangian function of some teleparallel theories based on a recently proposed holographic dark energy model namely new Tsallis HDE. For this purpose, we assume ordinary matter contents as perfect fluid with background geometry as flat FRW spacetime. On the first place, we reconstruct f (T) function by taking some well-known IR cutoff choices into account like Hubble radius, event horizon, Granda–Oliveros cutoff and conformal time as different cases. Moreover, we consider two entropy corrected forms of new Tsallis HDE with the simplest choice for IR cutoff. We explore the graphical behavior of different cosmic quantities, e.g. dark energy EoS parameter, the generalized second law of thermodynamics, the null energy condition, speed of sound and trajectories in ω T − ω T ′ plane for these reconstructed solutions and discuss their viability. Then we extend our work to f (T , B) theory and reconstruct the Lagrangian function for new Tsallis HDE with three simple choices of IR cutoff (Hubble radius and event horizon), and Granda–Oliveros cutoff and discuss the significance of obtained solutions graphically through same cosmic measures. It is found that presented models are consistent with the recent cosmic data values, however, stability can be achieved in some certain cases only. [ABSTRACT FROM AUTHOR]

(1) Background--New quality productive forces is an important driving force for economic and social development. New quality productive forces can help to promote the supply of diversified products and services in the development of forest health care industry and the deep integration of forest health care industry and other related industries to form an industrial chain. In the face of challenges such as the balance between environmental protection and sustainable utilization of resources, the improvement of health care service quality, and the improvement of relevant policies and industry standards, the introduction of new quality productive forces provides new opportunities for the development of forest health care industry. (2) Methods--Four factors of scientific and technological innovation, talents cultivation, management innovation and green productivity were selected as the independent variables to build a model of forest health care industry growth optimization. By constructing Lagrangian function to introduce constraint conditions, this paper used Lagrange multiplier method to balance the objective functions and constraint conditions, obtain the optimal solution of each variable, and maximize the growth rate G of forest health care industry. In the process of solving the model, the influence of four scenarios on industrial growth rate and key driving factors was investigated by numerical method to reveal the decisive influence of policy choice on industrial potential. Time was dynamically introduced into the mathematical model of forest health care industry growth, and a dynamic system model was constructed. Python numerical method was used to simulate the system and analyse its sensitivity, so as to explore the impact of changes of the key parameters in the model on industrial growth rate and green productivity. (3) Results--First, scientific and technological innovation, management innovation and green productivity are the core elements of new quality productive forces, which play a decisive role in the high-quality development of forest health care industry. Second, scientific and technological innovation plays a key role in improving service efficiency, optimizing customer experience and developing new health care products. Third, some measures of management innovation can effectively improve the operational efficiency and market adaptability of the organization, thus enhancing the overall competitiveness of the industry. Fourth, green productivity plays a core role in the development of forest health care industry. The use of environmentally friendly materials, energy-saving technologies and the design of eco-friendly recreational activities contribute to the sustainable development of the forest health care industry. (4) Conclusions and Discussions--Based on the above research conclusions, four paths are proposed to enable the high-quality development of forest health care industry with new quality productive forces; First, improve the service quality and operational efficiency of forest health care and promote the transformation and upgrading of the industry through the application of cutting-edge technologies, intelligent management, research and development of new products and other scientific and technological innovation means. Second, promote the deep integration of forest health care industry with tourism, pharmaceutical industry, sports industry, elderly care industry and other industries. Third, introduce modern management concepts, innovate talents cultivation modes, optimize resources allocation and operation modes, and standardize management system. Fourth, give full play to the leading role of green productivity by strengthening environmental protection concepts and technologies and supporting industrial green development. [ABSTRACT FROM AUTHOR]

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., "An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model," J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works. [ABSTRACT FROM AUTHOR]

The development and deployment of the next generation of wind energy systems calls for simulation tools that model the entire wind farm while balancing accuracy and computational cost. A full-system wind farm simulation must consider the atmospheric inflow, the wakes and consequent response of the multiple turbines, and the implementation of the appropriate farm-collective control strategies that optimize the entire wind farm's output. In this article, we present a novel vortex lattice model that enables the effective representation of the complex vortex wake dynamics of the turbines in a farm subject to transient inflow conditions. This work extends the capabilities of our multi-physics suite, CODEF, to include the capability to simulate the wakes and the high-fidelity aeroelastic response of multiple turbines in a wind farm. Herein, we compare the results of our GVLM technique with the LiDAR measurements obtained at Sandia National Laboratories' SWiFT facility. The comparison shows remarkable similarities between the simulation and field measurements of the wake velocity. These similarities demonstrate our model's capabilities in capturing the entire wake of a wind turbine at a significantly reduced computational cost as compared to other techniques. [ABSTRACT FROM AUTHOR]

First-order sufficient optimality conditions in terms of Lagrangian functions and Lagrange multipliers for nondifferentiable minimax programs and vector optimization problems in an Asplund space setting are obtained in this paper. Two illustrative examples are given. Our results implement the first-order necessary optimality conditions of Chuong and Kim (Ann Oper Res 251:73–87, 2017). [ABSTRACT FROM AUTHOR]

We revisit the notion of parametrization invariance while introducing certain weakened notions of invariance in the calculus of variations. In this work, we employ a straightforward approach in the classical setting and mostly restrict attention to functionals on one-dimensional domains. We establish a connection between parametrization invariant functionals and functionals embodying a weaker notion of invariance of their Lagrangian; we term this notion as T - Lagrangian analogous to the well-known idea of null Lagrangian. However, the Euler–Lagrange operator of a T -Lagrangian vanishes only along the tangential direction in the configuration space. On one-dimensional domain and for first- and second-order theories, we show that functional described by such a Lagrangian is necessarily a parametrization invariant functional modulo null Lagrangian. Keeping the motivation for partial differential equations, we also introduce and explore the notion of N - Lagrangian, with an invariance complementary to the case of T - Lagrangian, whose Euler–Lagrange operator vanishes along normal directions. We find that in a one-dimensional setting, every N -Lagrangian is simply a null Lagrangian. [ABSTRACT FROM AUTHOR]

Operations in areas of importance to society are frequently modeled as mixed-integer linear programming (MILP) problems. While MILP problems suffer from combinatorial complexity, Lagrangian Relaxation has been a beacon of hope to resolve the associated difficulties through decomposition. Due to the non-smooth nature of Lagrangian dual functions, the coordination aspect of the method has posed serious challenges. This paper presents several significant historical milestones (beginning with Polyak's pioneering work in 1967) toward improving Lagrangian Relaxation coordination through improved optimization of non-smooth functionals. Finally, this paper presents the most recent developments in Lagrangian Relaxation for fast resolution of MILP problems. The paper also briefly discusses the opportunities that Lagrangian Relaxation can provide at this point in time. [ABSTRACT FROM AUTHOR]

The past few decades have witnessed a growing popularity in Eulerian–Lagrangian solvers due to their significant potential for simulating aerodynamic flows, particularly in cases involving strong body–vortex interactions. In this hybrid approach, the two component solvers are mutually coupled in a two-way fashion. Initially, the Lagrangian solver can supply boundary conditions to the Eulerian solver, while the Eulerian solver functions as a corrector for the Lagrangian solution in regions where the latter cannot achieve high accuracy. To utilize such tools effectively, it is vital for them to be capable of handling dynamic mesh movements. This study builds upon the previous research conducted by our team and extends the capabilities of the hybrid solver to handle dynamic meshes. While OpenFOAM, the Eulerian component of this hybrid code, incorporates built-in dynamic mesh properties, certain modifications are necessary to ensure its compatibility with the Lagrangian solver. More specifically, the evolution algorithm of the pimpleFOAM solver needs to be divided into two discrete steps: first, updating the mesh, and later, evolving the solution. This division enables a proper coupling between pimpleFOAM and the Lagrangian solver as an intermediate step. Therefore, the primary objective of this specific paper is to adapt the OpenFOAM solver to meet the demands of the hybrid solver and subsequently validate that the hybrid solver can effectively address dynamic mesh challenges using this approach. This approach introduces a pioneering method for conducting dynamic mesh simulations within the OpenFOAM framework, showcasing its potential for broader applications. To validate the approach, various test cases involving dynamic mesh movements are employed. Specifically, all these cases employ the Lamb–Oseen diffusing vortex, but each case incorporates different types of mesh movements, including translational, rotational, oscillational, and combinations thereof. The results from these cases demonstrate the effectiveness of the proposed OpenFOAM algorithm, with the maximum relative errors —when compared to the analytical solution across all presented cases—capped at 2.0 % for the worst-case scenario. This affirms the algorithm's capability to successfully handle dynamic mesh simulations with the proposed solver. [ABSTRACT FROM AUTHOR]

2,4-dinitroanisole (DNAN) is a good replacement for 2,4,6-trinitrotoluene (TNT) in melt-cast explosives due to its superior insensitivity. With the increasing use of DNAN-based melt-cast explosives, the prediction of reaction violence and hazard assessment of the explosives subjected to shock is of great significance. This study investigated the shock initiation characteristics for a DNAN-based melt-cast explosive, DHFA, using the one-dimensional Lagrangian apparatus. The embedded manganin gauges in the apparatus record the pressure histories at four Lagrangian positions and show that shock-todetonation transition in DHFA needs a high input shock pressure. The experimental data are analyzed to calibrate the Ignition and Growth model. The calibration is performed using an objective function based on both pressure history and the arrival time of shock. Good agreement between experimental and calculated pressure histories indicates the high accuracy of the calibrated parameters with the optimization method. [ABSTRACT FROM AUTHOR]

Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics, and data analytics. In this paper, based on the augmented Lagrangian function, we introduce a flexible first-order primal-dual method, to be called nonconvex auxiliary problem principle of augmented Lagrangian (NAPP-AL), for solving a class of nonlinearly constrained nonconvex and nonsmooth optimization problems. We demonstrate that NAPP-AL converges to a stationary solution at the rate of o(1/k) , where k is the number of iterations. Moreover, under an additional error bound condition (to be called HVP-EB in the paper) with exponent θ∈(0,1) , we further show the global convergence of NAPP-AL. Additionally, if θ∈(0,12] , then we furthermore show that the convergence rate is in fact linear. Finally, we show that the well-known Kurdyka-Łojasiewicz property and the Hölderian metric subregularity imply the aforementioned HVP-EB condition. We demonstrate that under mild conditions, NAPP-AL can also be interpreted as a variant of the forward-backward operator splitting method in this context. Funding: This work was supported by the National Natural Science Foundation of China [Grant 71871140]. [ABSTRACT FROM AUTHOR]

This paper studies a steelmaking-continuous casting scheduling problem with different processing routes. We model this problem as a mixed-integer nonlinear programming problem. Next, Lagrangian relaxation approach is introduced to solve this problem by relaxing the coupling constraints. Due to the nonseparability in Lagrangian functions, we design an improved concave–convex procedure to decompose the Lagrangian relaxation problem into three tractable subproblems and analyse the convergence of the improved concave–convex procedure under some assumptions. Furthermore, we present an effective surrogate subgradient algorithm with global convergence to solve the Lagrangian dual problem. Lastly, computational experiments on the practical production data show the effectiveness of the proposed surrogate subgradient method for solving this steelmaking-continuous casting scheduling problem. [ABSTRACT FROM AUTHOR]

This paper introduces an innovative approach that seamlessly integrates a Genetic Algorithm (GA)-based method for generating and optimizing gait patterns with inverse dynamics analysis within the realm of personalized robot-assisted rehabilitation/training. Unlike conventional methods reliant on experimental data, our proposed approach enables the estimation and prediction of essential biomechanical information, including joint loads (joint force and torque). It achieves this by harnessing data derived from a customized musculoskeletal model for each user. By incorporating Lagrangian dynamics into the same platform used for gait pattern generation, the obtained gait pattern serves as a direct reference value for joint torque in applications such as robotic rehabilitation/training using wearable exoskeleton robots. Even in medical scenarios where only gait patterns are available using biomechanical information, our method can effectively estimate these parameters. We validated our method using a motion capture (MoCap) dataset as ground truth. This comparison assessed how well our estimations of joint angles, torques, and ground reaction forces (GRFs) matched actual human movement data. The results demonstrated high similarity, with an average Pearson Correlation Coefficient (PCC) value of 0.986 for joint angles, 0.748 for joint torques, and 0.975 for GRFs.

A wide family of position‐dependent mass damped oscillators affected by an external potential is investigated. First, a Lagrangian formulation is introduced for the corresponding problem. The Lagrangian function is time‐dependent, and the problem cannot be approached with classical procedures because it lacks variational symmetries. Therefore, the variational λ$$ \lambda $$‐symmetry method is applied to find exact solutions. Variational λ$$ \lambda $$‐symmetries are determined for a family of potential functions, which lead to a one‐parameter family of exact solutions. The results are applied to particular examples corresponding to some interesting mass functions reported in the previous literature. [ABSTRACT FROM AUTHOR]

• A differential exact penalty function exploiting the topology optimization penalty principle is proposed. • New active and loss functions are devised to treat the violation of equality and inequality constraints. • The derived optimization conditions without dual and slack variables coincide with the canonical KKT conditions. • The derivatives of the proposed loss functions can be interpreted as the traditional Lagrange multipliers. • The appearance of the first order partial derivatives in the Hessian matrix is unveiled. A smooth penalty function method which does not involve dual and slack implicit variables to formulate constrained optimization conditions is proposed. New active and loss functions are devised to independently and adaptively handle the violation of each constraint function. A single unconstrained minimization of the proposed penalty function produces a solution to the original optimization problem. The constrained optimization conditions depending only on the primal variable are derived and reduce to the canonical Karush-Kuhn-Tucker(KKT) conditions in a much more general framework. The derivative of the proposed loss functions with respect to constraint function can be interpreted as the Lagrange multiplier of the traditional Lagrangian function. The appearance of the first-order Hessian information is unveiled. This is in contrast to the existing works where the classical Lagrange Hessian contains only second order derivatives. Finally, some numerical examples including medium-scale stress constrained topology optimization and scenario based reliability design problems are presented to demonstrate the effectiveness of the proposed methodology. [ABSTRACT FROM AUTHOR]

This paper uses an augmented Lagrangian method based on an inexact exponential penalty function to solve constrained multiobjective optimization problems. Two algorithms have been proposed in this study. The first algorithm uses a projected gradient, while the second uses the steepest descent method. By these algorithms, we have been able to generate a set of nondominated points that approximate the Pareto optimal solutions of the initial problem. Some proofs of theoretical convergence are also proposed for two different criteria for the set of generated stationary Pareto points. In addition, we compared our method with the NSGA-II and augmented the Lagrangian cone method on some test problems from the literature. A numerical analysis of the obtained solutions indicates that our method is competitive with regard to the test problems used for the comparison. [ABSTRACT FROM AUTHOR]

The purpose of this research is to investigate a strong-form meshless method for experimentally and numerically analyzing hyperelastic plates subjected to concentrated loading. The silicone hyperelastic plate was formulated utilizing Mooney-Rivlin strain energy function and Lagrangian strains. The displacements and Lagrangian strains of the plate are also calculated using the first-order shear deformation plate theory (FSDPT). The hyperelastic plate's governing equations are deduced in strong form and analyzed using the logarithmic thin-plate spline radial basis function (LTPS-RBF) and the meshless method for the first-time. The nonlinear system of governing equations is solved using the arc-length continuation algorithm. In the meshless method, the concentrated loading is modeled using the Kronecker delta function in the expression of external work. A silicone plate is employed experimentally to evaluate the accuracy of the meshless method's results. Furthermore, the finite element method is utilized to compare the contours obtained from the meshless method. Concentrated loading is considered at various points to validate the meshless methods and results from the experimental analysis. The most significant difference between the meshless method and the experimental results is less than 15%. The results indicate that the meshless method is highly accurate when analyzing hyperelastic plates subjected to concentrated loading. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a dampened proximal alternating direction method of multipliers (DP.ADMM) for solving linearly constrained nonconvex optimization problems where the smooth part of the objective function is nonseparable. Each iteration of DP.ADMM consists of (i) a sequence of partial proximal augmented Lagrangian (AL) updates, (ii) an under-relaxed Lagrange multiplier update, and (iii) a novel test to check whether the penalty parameter of the AL function should be updated. Under a basic Slater point condition and some requirements on the dampening factor and under-relaxation parameter, it is shown that DP.ADMM obtains an approximate first-order stationary point of the constrained problem in O(ε-3) iterations for a given numerical tolerance ε > O. One of the main novelties of the paper is that convergence of the method is obtained without requiring any rank assumptions on the constraint matrices. [ABSTRACT FROM AUTHOR]

Support-vector machines (SVMs) have been successfully employed to diagnose neurological disorders like epilepsy and sleep disorders via electroencephalogram (EEG) signal classification. However, EEG signals are susceptible to noise and outliers while recording. Thus, classification of EEG signals require efficient methods that is robust to noise and outliers. SVMs suffer in the presence of noise and outliers as it treats each sample to be of equal importance. Fuzzy membership function has been successful in dealing with noise and outliers. Fuzzy support vector machine uses fuzzy membership to give the appropriate weight to each sample for reducing the effect of noise and outliers. To reduce the effect of noise and outliers, intuitionistic fuzzy twin support vector machines (IFTWSVM) uses both membership and non-membership weights to subsidise the effect of outliers. Moreover, IFTWSVM solves smaller size quadratic programming problems which makes it comparatively efficient than fuzzy support vector machines. However, IFTWSVM suffers as (i) it involves the computation of matrix inverses which makes it intractable as the size of the data increases. (ii) it solves different problems for linear and nonlinear cases, and nonlinear case with linear kernel does not reduce to the linear case. Thus, it can only solve the approximate formulation not the exact formulation. To overcome these issues, we propose improved intuitionistic fuzzy twin support vector machine (IIFTWSVM). The proposed IIFTWSVM introduces different Lagrangian functions to avoid the computation of matrix inverses. Also, in nonlinear case, kernel trick is applied directly and hence, the exact formulation is solved. We employed the proposed IIFTWSVM for the classification of EEG signals. The experimental results and statistical analysis show that the proposed IIFTWSVM model is better compared to the given baseline models. Also, we evaluate the performance of the models on KEEL datasets to check the robustness of the proposed IIFTWSVM model. [ABSTRACT FROM AUTHOR]

Twin support vector regression (TSVR) has been widely applied in regression problems. TSVR seeks a pair of ε -insensitive proximal planes by solving two support vector machine type problems. TSVR assumes that the matrices appearing in the dual formulation are positive definite. However, in real-world scenarios, such an assumption may not be fulfilled and, hence, leads to suboptimal performance. ε -Twin support vector regression (ε -TSVR) improved the TSVR by introducing the regularisation term to avoid the singularity issues. Most of the twin support vector machine models involve the computation of matrix inverses. Also, TSVR implements the empirical risk minimization principle. In this paper, we propose an improved twin support vector regression (ITSVR) for brain age estimation by introducing different Lagrangian functions for the primal problems of the TSVR. The proposed ITSVR implements the structural risk minimization principle and avoids the computation of the matrix inverses. To solve the optimization problem more efficiently, we used successive overrelaxation (SOR) technique. We evaluated the proposed ITSVR on cognitively healthy subjects, mild cognitive impairment subjects and Alzheimer's disease subjects. The experimental results demonstrate that the proposed ITSVR has superior performance compared to the baseline models for brain age estimation. [ABSTRACT FROM AUTHOR]

In this work, a virtual body-fitted grid is introduced into the velocity correction-based immersed boundary method (IBM) to simulate incompressible flows. The impact of the immersed boundary is indirectly transmitted to the flow field via a virtual body-fitted grid. In this method, the fractional step technique consisting of the predictor and the corrector is adopted. The prediction step is executed on the Eulerian mesh, and the correction step is done on the virtual grid to fulfill the no-slip boundary condition. After the correction step, the corrected velocity field on the virtual grid is then assigned to that on the Eulerian mesh to update the flow field. Being able to adjust the grid spacing flexibly, the virtual body-fitted grid alleviates the shortcomings of the conventional IBM that uses the smooth Dirac delta function to associate Lagrangian points with their surrounding Eulerian points. As a result, the present method is easy to apply to non-uniform Cartesian grids, which is inapplicable to the conventional IBM with the smooth Dirac delta function. Numerical experiments concerning flow past a circular cylinder and a NACA0012 airfoil demonstrate the advantages of the present method, i.e., fewer Lagrangian points are required to avoid the streamline penetration of boundary and the range of "diffuse interface" can be narrowed by reducing the normal grid spacing of the virtual body-fitted grid to improve numerical results on a coarse mesh. In addition, an accuracy assessment on the decaying vortex problem reveals that the present IBM has a second-order accuracy. [ABSTRACT FROM AUTHOR]

In this work we propose new proofs of some classical results of nonlinear programming milestones, in particular for the Kuhn-Tucker conditions and Lagrangian methods and functions. This study is concerned with some interesting features found in the well-known tools and methods, connecting them with a technical analysis of the "Maximal Margin Classifier" designed specifically for linearly separable data, while referring to the condition in which data can be separated linearly by using a hyperplane. In this context of analysis, we technically point out the centrality played by these mathematical tools when obtaining robustness in Machine Learning procedures analyzing some support vector machine (SVM) models, as they are used in various contexts and applications (e.g., Soft Margin SVM and Maximum Margin SVM). This paper represents the first study reinforcing the ongoing Machine Learning Modeling and the research project we will launch in the near future on this fascinating frame of analysis. In this work we examine the problem of estimating the bias into a decision-making process. A new decision function algorithm is introduced as well. [ABSTRACT FROM AUTHOR]