Lagrange Multiplier Technique Research Articles

Constrained optimization approach for magnetohydrostatic equilibria on the solar atmosphere

The magnetic field plays an essential role in the evolution of structures and the description of events in the solar atmosphere. Several models have been developed to reconstruct the magnetic field, due to the impossibility of its direct measurement in the solar corona. The model proposed here extrapolates the photospheric magnetogram data up to the corona using a constrained optimization method. In the upper photosphere and chromosphere, both the magnetic and nonmagnetic forces must be taken into account, and the magnetic field reconstruction must be done considering the plasma pressure and density. This is done by applying the Lagrange multiplier technique, as the constrained optimization method, to compute the magnetic field, plasma pressure, and density in magnetohydrostatic equilibria. This approach has previously been introduced to reconstruct a nonlinear force-free magnetic field. For this work we extended it to a more realistic issue to reconstruct the magnetic field and calculate the plasma pressure and density in a magnetohydrostatic environment. Our approach was to use the constrained optimization method, which is computationally more efficient and easy to implement. The Lagrange multiplier technique is a powerful mathematical tool that has been successfully applied to many areas of physics. We sought to minimize a Lagrangian, which minimizes the divergence term subject to the constraint magnetohydrostatic equilibrium equation. The plasma parameters and magnetic field were eventually computed following the iteration scheme along with appropriate boundary data. In our previous work, we applied Lagrange multiplier techniques to reconstruct a force-free magnetic field for the solar atmosphere. For this wok, we extended the same optimization technique to extrapolate magnetic field and plasma parameters in magnetohydrostatic equilibria. The results for the magnetic field and plasma parameters were calculated and compared with those obtained by other models in the magnetohydrostatic equilibrium environment as well as the semi-analytical solution as a reference model. A force-free magnetic field and a suitable distribution for pressure and density were used as the initial input for their corresponding evolution equations. After $20,000$ iterations, the convergence of the Lagrangian in our model was slightly better than that of the comparison model. The indicators such as the relative magnetic energy and magnetic field lines were investigated, which are in agreement with the reference model compared to the comparison model.

Greening the Gulf: A Deep-Dive into the Synergy Between Natural Resources, Institutional Quality, Foreign Direct Investment, and Pathways to Environmental Sustainability

Environmental quality is a global concern, especially in Gulf Cooperation Council (GCC) countries where abundant mineral resources, economic growth, and globalization have strained the environment through urbanization and resource exploitation. This study examines the impact of globalization (GLOL), urbanization (URBN), natural resource extraction (NRER), institutional quality (INSQ), and foreign direct investment (FDI) on environmental quality in GCC countries from 1999 to 2021. Cross-sectional dependence (CSD) was assessed using the Lagrange Multiplier (LM) and cross-dependence (CD) techniques, and stationarity was confirmed with the Levin–Lin–Chu test. The Augmented Dickey–Fuller (ADF) co-integration test verified long-term relationships, and Pooled Mean Group Autoregressive Distributed Lag (PMG-ARDL) methodology assessed short- and long-term effects. Our findings show that FDI, GLOL, and INSQ have negative long-term impacts on environmental quality, while NRER and URBN are beneficial. In the short term, FDI and INSQ improve green quality, while GLOL, URBN, and NRER have detrimental effects. Policy recommendations include discouraging FDI in non-renewable projects, promoting sustainable FDI, addressing income inequality to improve environmental quality, and investing in urban development to reduce ecological footprints (ECFTs) and enhance environmental quality in GCC countries.

Optimizing Stand-Alone Microgrids with Lagrange Multiplier Technique: A Cost-Effective and Sustainable Solution for Rural Electrification

Optimizing Stand-Alone Microgrids with Lagrange Multiplier Technique: A Cost-Effective and Sustainable Solution for Rural Electrification

Why the action?

In this note, I present two variations of the principle of stationary action in order to provide alternative ways for students to think about the action that they may find more intuitive than the traditional approach. These methods involve Lagrange multiplier techniques and are, therefore, best suited for students who are encountering the stationary action principle for a second or third time, perhaps in an advanced undergraduate mathematical methods course or a graduate classical mechanics course.

Partially and fully implicit multi-step SAV approaches with original dissipation law for gradient flows

The scalar auxiliary variable (SAV) approach was considered by Shen et al. in Shen et al. (2019) and has been widely used to simulate a series of gradient flows. However, the SAV-based schemes are known for the stability of a ‘modified’ energy. In this paper, we construct a series of modified SAV approaches with unconditional energy dissipation law based on several improvements to the classic SAV approach. Firstly, by introducing the three-step technique, we can reduce the number of constant coefficient linear equations that need to be solved at each time step, while retaining all of its other advantages. Secondly, the addition of energy-optimal technique and Lagrange multiplier technique can make the numerical schemes have the advantage of preserving the original energy dissipation. Thirdly, we use the first-order approximation of the energy balance equation in the GSAV approach, instead of discretizing the dynamic equation of the auxiliary variable, so that we can construct the high-order unconditional original energy stable numerical schemes. Finally, representative numerical examples show that the efficiency and accuracy of the proposed schemes are improved.

Core allocation to minimize total flow time in a multicore system in the presence of a processing time constraint

Data centers are a vital and fundamental infrastructure component of the cloud. The requirement to execute a large number of demanding jobs places a premium on processing capacity. Parallelizing jobs to run on multiple cores reduces execution time. However, there is a decreasing marginal benefit to using more cores, with the speedup function quantifying the achievable gains. A critical performance metric is flow time. Previous results in the literature derived closed-form expressions for the optimal allocation of cores to minimize total flow time for a power-law speedup function if all jobs are present at time 0. However, this work did not place a constraint on the makespan. For many diverse applications, fast response times are essential, and latency targets are specified to avoid adverse impacts on user experience. This paper is the first to determine the optimal core allocations for a multicore system to minimize total flow time in the presence of a completion deadline (where all jobs have the same deadline). The allocation problem is formulated as a nonlinear optimization program that is solved using the Lagrange multiplier technique. Closed-form expressions are derived for the optimal core allocations, total flow time, and makespan, which can be fitted to a specified deadline by adjusting the value of a single Lagrange multiplier. Compared to the unconstrained problem, the shortest job first property for optimal allocation is maintained; however, a number of other properties require revising and other properties are only retained in a modified form (such as the scale-free and size-dependence properties). It is found that with a completion deadline the optimal solution may contain groups of simultaneous completions. In general, all possible patterns of single- and group-completion need to be considered, producing an exponential search space. However, the paper determines analytically that the optimal completion pattern consists of a sequence of single completions followed by a single group of simultaneous completions at the end, which reduces the search space dimension to being linear. The paper validates the Lagrange multiplier approach by verifying constraint qualifications.

Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method

This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hybridization. The method yields approximations of all variables, including the solution and gradient, with optimal order. Furthermore, the matrix representing the final linear algebra systems is not only symmetric but also positive definite. Numerical examples convincingly demonstrate the effectiveness of the hybrid mixed finite element method in addressing elliptic interface problems.

Optimal solution of nonlinear 2D variable-order fractional optimal control problems using generalized Bessel polynomials

This study aims to propose a new optimization method based on the generalized Bessel polynomials (GBPs) as a class of basis functions for a category of nonlinear two-dimensional variable-order fractional optimal control problems (N-2D-VOFOCPs) involved in fractional-order dynamical systems and Caputo derivatives. For the optimal solution of such problems, the optimization method is developed on the basis of operational matrices (OMs) scheme of derivatives, 2D Gauss–Legendre quadrature rule, and Lagrange multiplier technique. The state and control functions are expanded in terms of the GBPs to reduce the complexity of these problems. The proposed method focuses on a system of nonlinear algebraic equations in the process of finding solution to the problems. The convergence of the method based on GBPs is proved, and the accuracy of the method is analyzed by solving several examples.

KinFit: A Kinematic Fitting Package for Hadron Physics Experiments

A kinematic fitting package, KinFit, based on the Lagrange multiplier technique has been implemented for generic hadron physics experiments. It is particularly suitable for experiments where the interaction point is unknown, such as experiments with extended target volumes. The KinFit package includes vertex finding tools and fitting with kinematic constraints, such as mass hypothesis and four-momentum conservation, as well as combinations of these constraints. The new package is distributed as an open source software via GitHub. This paper presents a comprehensive description of the KinFit package and its features, as well as a benchmark study using Monte Carlo simulations of the pprightarrow pK^+Lambda rightarrow pK^+ppi ^- reaction. The results show that KinFit improves the parameter resolution and provides an excellent basis for event selection.

Analytical assessment of heat transfer due to Williamson hybrid nanofluid (MoS2 + ZnO) with engine oil base material due to stretched sheet

The efficiency of fuel consumptions and improved performances is real challenge in the vehicle engines. Recent advances in thermal engineering suggested various tools to improve the capacitance of engine oil for which the interaction of nanoparticles is more dynamical. The aim of current research is to enhance the thermal aspects of engine oil with suspension of Williamson hybrid nanofluid. The hybrid nanofluid is the suspension of molybdenum disulfide MoS2 and zinc oxide ZnO. In order to increase the heating capability of system, external heat source and viscous dissipation features are contributed. The flow analysis is based on moving surface contains porous medium. After modelling the problem, the analytical outcomes of attributed with variational iteration method (VIM). The computational simulations are performed with help of Lagrange Multiplier technique. The impact of numerous physical parameters on skin fraction, Nusselt numbers, temperature distribution and velocity profile is exhibited.

Applying the Lagrange multiplier technique to reconstruct a force-free magnetic field

Context. The magnetic field in the solar corona can be reconstructed by extrapolating the data obtained by measuring the magnetic field in the photosphere. It is widely thought that the dynamics of the solar corona is governed by the magnetic field. The magnetic field in the corona can therefore be reconstructed using the physics of force-free fields. Several models have been developed so far to reconstruct the magnetic field, but shortcomings in this regard remain. Therefore, alternative models in this respect can still be proposed. Here we apply the Lagrange multiplier technique to render an optimization model. Aims. The main aim of the present paper is to propose a method of constrained optimization using the vector Lagrange multiplier and compare the results with those of preexisting models. Methods. Our main focus is on the conceptual modification of the optimization models. Since these models are computationally efficient and easy to implement, any possible progressive step would be welcome. The Lagrange multiplier technique is a powerful mathematical tool that has been successfully applied to many areas in physics. It may serve this purpose. In the absence of nonmagnetic forces such as pressure, gravity, and dissipative forces, the coronal medium is dominated by magnetic force. Thus, the function that is considered to be minimized may include a divergence term subject to the constraint force-free term, which yields a solenoidal and force-free magnetic field by an iterative process. Results. The numerical analysis of the proposed model was conducted through an artificial magnetogram and an observational vector magnetogram obtained by SDO/HMI images. The results obtained confirm that (i) the Lagrangian to be minimized in the present model converges slightly faster toward zero, at least for initial iteration steps, (ii) the energy variation during the optimization is compatible with the variations in previous studies, (iii) the numerical results seem to be compatible with a semi-analytical solution as the test case, and (iv) the model is applied to a real magnetogram, and relevant quantities such as magnetic energy content, the current-weighted angle between the current density vector and magnetic field, and the fractional flux errors are computed. Conclusions. The methods and techniques that convert a constrained problem into an unconstrained one are the penalty method, the barrier method, the augmented Lagrangian method, and the Lagrange multiplier technique, for instance. We have employed the Lagrange multiplier technique, by which any constrained condition could be added to the Lagrangian by an appropriate Lagrange multiplier. In our case, the constraint is the force-freeness of the magnetic field and is therefore a special case. The method has the following advantages: (i) The convergence rate is slightly higher for the initial iteration steps, which may help us for time-series events, while several magnetograms must be considered and limited steps of iteration may be needed. (ii) The Lagrangian is introduced based on the Lagrange multiplier technique, which facilitates first fixing a physical compromise such as a divergence-free condition, and subsequently adding any given constraint term. (iii) The quantities obtained by the constrained optimization with the vector Lagrange multiplier model, that is, the relative magnetic energy, the ratio of the magnetic energy to its initial value, the angle between the electric current and the magnetic field, fractional flux errors, the normalized vector error, the vector correlation, and the Cauchy-Schwarz indicator, are comparable with those of the comparison models considered in this article.

Nonlinear static analysis of multi-segmented toroidal shells under external pressure

This paper investigates the nonlinear static responses of multi-segmented toroidal shells with egg-shaped cross sections. The discontinuity effect in terms of displacement and the slope at the toroidal shell edge must be constrained by the Lagrange multiplier technique. The structural shape of the multi-segmented toroidal shells can be described by the differential geometry of the shell with the six-center-combined toroidal shells. The energy functional of the multi-segmented toroidal shells can be written in the appropriate form based on the principle of virtual work. The numerical results of the displacement responses and the volume change can be obtained by the nonlinear finite element method via the fifth-order polynomial shape function and a direct iterative method. In this study, the Lagrange multiplier is the internal force at the toroidal shell junction sustaining the smooth curve of the multi-segmented toroidal shells under several loads. Finally, the results of the volume change and the displacement responses for various geometric parameters are presented and discussed.

Large displacement analysis of egg-shaped containment shells using Lagrange multipliers

In this paper, a large displacement analysis of three-center-combined spherical shells, used as egg-shaped containment structures, is performed using the first and second fundamental forms of the shell surface with two different radii of curvature. The energy functional of the three-center-combined spherical shells system is derived using a variational formulation, and is written in the appropriate form to reduce the computation time in an iterative procedure. The numerical results in terms of displacement, membrane stress, and bending moment resultants are obtained by nonlinear finite element analysis with a fifth-order polynomial shape function. At the spherical shell edge, the Lagrange multipliers technique is used to enforce the discontinuity effect. The main conclusions are that the displacement responses, membrane stress, and bending moment resultants depend on the geometric parameters of the three-center-combined spherical shells. The Lagrange multipliers correspond to the internal forces due to the displacement and the slope at the shell edge in order to sustain the smooth curve for deformed configuration of the three-center-combined spherical shells under uniform pressure. The volume capacity of the three-center-combined spherical shells are more than the spherical shells with a constant radius. Finally, the results indicate that the linear assumption is good enough for all practical purposes. However, the large displacement analysis becomes significant on the tangential displacement, meridional, and circumferential moments under the large value of uniform pressure.

Representation learning of knowledge graphs with correlation-based methods

Knowledge graphs (KGs) are intrinsically incomplete. Knowledge representation learning methods deal with embedding components of a knowledge graph (i.e., entities and relations) into a low-dimensional vector space with the aim of predicting the missing links. The KG Embedding models use gradient descent-based optimization techniques to optimize their parameters, but they suffer from late convergence, making them inefficient for dynamic knowledge graphs. Moreover, generating and modeling negative samples has a great impact on the accuracy of the prediction model.To address these challenges, we propose a correlation-based knowledge representation learning (CKRL) method. We utilize three semantic spaces, namely entity, relation, and canonical spaces. The parameter values of these spaces, including transformation matrices and embedding of KG components, are determined by defining four optimization problems. Adapting least-squares and Lagrange multiplier techniques, these optimization problems are solved by time-efficient direct methods. To model positive and negative samples, we use two distinct relation spaces. Negative samples are generated by defining the concept of relational patterns used to measure the similarity between positive and negative samples. Evaluation results on link prediction task and training time analysis demonstrate the superiority of the proposed method in comparison with the state-of-the-art methods.

Utility Maximization Analysis of an Organization: A Mathematical Economic Procedure

In the society utility is the vital concept, especially in mathematical economics. It is considered as the tendency of an object or action that increases or decreases overall happiness. In social sciences, the property of a commodity that enables to satisfy human wants is called utility. This paper has tried to operate utility maximization policy of an organization by considering two constraints: budget constraint and coupon constraint. To develop the maximization policy of utility function, the techniques of multivariate calculus are used. In this study four commodity variables are used to operate the mathematical analysis efficiently. In this article Lagrange multiplier technique is applied to achieve optimal result throughout the study.

Three interior penalty DG methods for stationary incompressible magnetohydrodynamics

In this paper, we propose and analyze three interior penalty DG methods for the stationary incompressible magnetohydrodynamics (MHD) equations. We use div-conforming Brezzi–Douglas–Marini (BDM) elements to discretize the velocity field and therefore the approximation of the velocity field is exactly divergence-free. The magnetic field is approximated by curl-conforming Nédélec elements. In the mixed formulation, the zero mean-valued constraint of the pressure is enforced via a Lagrange multiplier technique. The stability and convergence of the proposed methods are proved in a general Lipschitz domain. Finally, a series of numerical experiments are presented to verify the theoretical results.

Importance of Total Coupon in Utility Maximization: A Sensitivity Analysis

This study takes an attempt to discuss utility maximization policies. The property of a commodity that enables it to satisfy human wants is called utility. The sensitivity analysis is included in the operation to show optimal policy of an organization. This study deals with four commodities and two constraints, such as budget constraint, and coupon constraint. The economic predictions of future production are necessary for the sustainable production. In the study Lagrange multipliers technique is applied to operate 6×6 bordered Hessian and 6×10 Jacobian appropriately. The sensitivity analysis between commodity and total budget are discussed with detail mathematical analysis.

A Three-Field Variational Formulation for a Frictional Contact Problem with Prescribed Normal Stress

In the present work, we address a nonlinear boundary value problem that models frictional contact with prescribed normal stress between a deformable body and a foundation. The body is nonlinearly elastic, the constitutive law being a subdifferential inclusion. We deliver a three-field variational formulation by means of a new variational approach governed by the theory of bipotentials combined with a Lagrange-multipliers technique. In this new approach, the unknown of the mechanical model is a triple consisting of the displacement field, a Lagrange multiplier related to the friction force and the Cauchy stress tensor. We obtain existence, uniqueness, boundedness and convergence results.

Collocation Method for Optimal Control of a Fractional Distributed System

In this paper, a collocation method based on the Jacobi polynomial is proposed for a class of optimal-control problems of a fractional distributed system. By using the Lagrange multiplier technique and fractional variational principle, the stated problem is reduced to a system of fractional partial differential equations about control and state functions. The uniqueness of this fractional coupled system is discussed. For spatial second-order derivatives, the proposed method takes advantage of Jacobi polynomials with different parameters to approximate solutions. For a temporal fractional derivative in the Caputo sense, choosing appropriate basis functions allows the collocation method to be implemented easily and efficiently. Exponential convergence is verified numerically under continuous initial conditions. As a particular example, the relation between the state function and the order of the fractional derivative is analyzed with a discontinuous initial condition. Moreover, the numerical results show that the integration of the state function will decay as the order of the fractional derivative decreases.

On the Frictional Contact Problem of p(x)-Kirchhoff Type

In this article we consider a class of frictional contact problem of p(x)-Kirchhoff type, on a bounded domain Ω⊆R2. Using an abstract Lagrange multiplier technique and the Schauder’s fixed point theorem we establish the existence of weak solutions. Furthermore, we also obtain the uniqueness of the solution assuming that the datum f1 satisfies a suitable monotonicity condition.