Lagrange Multiplier Technique Research Articles

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.

Two-Level Finite Element Iterative Algorithm Based on Stabilized Method for the Stationary Incompressible Magnetohydrodynamics

In this paper, based on the stabilization technique, the Oseen iterative method and the two-level finite element algorithm are combined to numerically solve the stationary incompressible magnetohydrodynamic (MHD) equations. For the low regularity of the magnetic field, when dealing with the magnetic field sub-problem, the Lagrange multiplier technique is used. The stabilized method is applied to approximate the flow field sub-problem to circumvent the inf-sup condition restrictions. One- and two-level stabilized finite element algorithms are presented, and their stability and convergence analysis is given. The two-level method uses the Oseen iteration to solve the nonlinear MHD equations on a coarse grid of size H, and then employs the linearized correction on a fine grid with grid size h. The error analysis shows that when the grid sizes satisfy , the two-level stabilization method has the same convergence order as the one-level one. However, the former saves more computational cost than the latter one. Finally, through some numerical experiments, it has been verified that our proposed method is effective. The two-level stabilized method takes less than half the time of the one-level one when using the second class Nédélec element to approximate magnetic field, and even takes almost a third of the computing time of the one-level one when adopting the first class Nédélec element.

Profit Maximization Strategy in An Industry: A Sustainable Procedure

This article tries to discuss profit maximization policies by using four variable inputs, such as capital, labor, principal raw materials, and other inputs in an industry, where mathematical economic models are applied by considering budget constraint. For the sustainable production, every industry should apply scientific method, such as mathematical techniques to obtain more accurate results. In the study Cobb-Douglas production function is analyzed with detail mathematical analysis. The sensitivity analysis is included in the operation to show profit maximization that is a pivotal goal of the industry. The economic predictions of future production are provided through the comparative statics to become sure of profit maximization before starting of production in the industry. In the study Lagrange multiplier technique is applied to achieve optimal result in every step of industrial operation.

Determination of an Extremal in Two-Dimensional Variational Problems Based on the RBF Collocation Method

This paper introduces a direct method derived from the global radial basis function (RBF) interpolation over arbitrary collocation nodes occurring in variational problems involving functionals that depend on functions of a number of independent variables. This technique parameterizes solutions with an arbitrary RBF and transforms the two-dimensional variational problem (2DVP) into a constrained optimization problem via arbitrary collocation nodes. The advantage of this method lies in its flexibility in selecting between different RBFs for the interpolation and parameterizing a wide range of arbitrary nodal points. Arbitrary collocation points for the center of the RBFs are applied in order to reduce the constrained variation problem into one of a constrained optimization. The Lagrange multiplier technique is used to transform the optimization problem into an algebraic equation system. Three numerical examples indicate the high efficiency and accuracy of the proposed technique.

Necessary optimality conditions of fractional-order discrete uncertain optimal control problems

This paper is primarily dedicated to the necessary conditions of optimal control for uncertain discrete-time systems of fractional-order governed by left Riemann-Liouville-like fractional-order nabla difference equations. First, an existence result of solutions for backward right Riemann-Liouville-like fractional-order nabla difference equations is derived from the Banach contraction mapping theorem. Besides, necessary optimality conditions of these fractional-order uncertain optimal control problems are proposed by the variational method and Lagrange multiplier technique. Finally, a special LQ optimal control problem for uncertain discrete-time systems of fractional-order is solved by necessary conditions of optimality derived.

Linear and weakly nonlinear dominant dynamics in a boundary layer flow

The aim of this paper is to investigate the linear and weakly nonlinear dynamics in flow over a flat-plate with leading edge. Linear optimal and suboptimal inflow perturbations are obtained using a Lagrangian multiplier technique. In particular, the suboptimal inflow conditions and the corresponding downstream responses are investigated in detail for the first time. Unlike the suboptimal dynamics reported in other canonical cases such as the backward-facing step flow, the growth rate of the suboptimal perturbation is in the same order as the optimal one, and both of them depend on the lift-up mechanism even though they are orthogonal. The suboptimal mode has an additional layer of vorticity that penetrates into the boundary layer farther downstream, generating a second patch of high- and low-speed streaks. The farther suboptimal ones spread to the free-stream without entering the boundary layer. The weakly nonlinear dynamics are examined by decomposing the flow field into multiple orders of perturbations using the Volterra series. Small structures in the higher order perturbations mainly concentrate in the region farther away from wall, suggesting a mechanism of outward perturbation developments, which is opposite with the well reported inward development of perturbations, i.e., from free-stream to boundary layer. The significance of these modes is then demonstrated through a prediction of flow field from the inflow condition by exploiting the orthogonality of the modes.

The nexus of financial development, technological innovation, institutional quality, and environmental quality: evidence from OECD economies.

The present study investigates the effect of institution quality, technological innovation, and financial development on environment quality using 37 OECD nations from 1998 to 2018. The cross-sectional dependence (CD) and Lagrange multiplier (LM) techniques are used to measure the cross-sectional dependence. The second-generation panel unit root tests and panel cointegration tests are applied to examine the unit-root properties and long-run association existence between variables. Finally, we employed the two-step (SYS-GMM) methodology to estimate the coefficient values. The findings showed that financial development has a positive effect on selected carbon (CO2) emission dimensions. When the moderating term is introduced, it was identified that institutional quality and technology innovation conditioning effects are crucial between financial development and CO2 emission. Our evidence-based study provides significant results for technology innovation and institutional quality moderating role in reducing CO2 emissions in OECD economies. Our findings are also robust to alternative measures, which could be useful for policymakers to formulate long-term and short-term strategies and policies for a better sustainable environment.

Multi‐patch variational differential quadrature method for shear‐deformable strain gradient plates

AbstractThe integration of generalized differential quadrature techniques and finite element (FE) methods has been developed during the past decade for engineering problems within classical continuum theories. Hence, the main objective of the present study is to propose a novel numerical strategy called the multi‐patch variational differential quadrature (VDQ) method to model the structural behavior of plate structures obeying the shear deformation plate theory within the strain gradient elasticity theory. The idea is to divide the two‐dimensional solution domain of the plate model into sub‐domains, called patches, and then to apply the VDQ method along with the FE mapping technique for each patch. The formulation is presented in a weak form and due to the ‐continuity requirements the corresponding compatibility conditions are applied through the patch interfaces. The Lagrange multiplier technique and the penalty method are implemented to apply the higher‐order compatibility conditions and boundary conditions, respectively. To show the efficiency of the proposed method, numerical results are provided for plate structures with both regular and irregular solution domains. The provided numerical examples demonstrate the applicability and accuracy of the method in predicting the bending and vibration behavior of plate structures following the higher‐order plate model.