• Journal of the Computational Structural Engineering Institute of Korea
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• v.25 no.6
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• pp.469-476
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• 2012

In this paper, a finite element domain decomposition method using local and mixed Lagrange multipliers for a large scal structural analysis is presented. The proposed algorithms use local and mixed Lagrange multipliers to improve computational efficiency. In the original FETI method, classical Lagrange multiplier technique was used. In the dual-primal FETI method, the interface nodes are used at the corner nodes of each sub-domain. On the other hand, the proposed FETI-local analysis adopts localized Lagrange multipliers and the proposed FETI-mixed analysis uses both global and local Lagrange multipliers. The numerical analysis results by the proposed algorithms are compared with those obtained by dual-primal FETI method.

• Communications of Mathematical Education
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• v.37 no.1
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• pp.65-84
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• 2023

The method of Lagrange multipliers, one of the most fundamental algorithms for solving equality constrained optimization problems, has been widely used in basic mathematics for artificial intelligence (AI), linear algebra, optimization theory, and control theory. This method is an important tool that connects calculus and linear algebra. It is actively used in artificial intelligence algorithms including principal component analysis (PCA). Therefore, it is desired that instructors motivate students who first encounter this method in college calculus. In this paper, we provide an integrated perspective for instructors to teach the method of Lagrange multipliers effectively. First, we provide visualization materials and Python-based code, helping to understand the principle of this method. Second, we give a full explanation on the relation between Lagrange multiplier and eigenvalues of a matrix. Third, we give the proof of the first-order optimality condition, which is a fundamental of the method of Lagrange multipliers, and briefly introduce the generalized version of it in optimization. Finally, we give an example of PCA analysis on a real data. These materials can be utilized in class for teaching of the method of Lagrange multipliers.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.2
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• pp.267-281
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• 2016

In this paper, we consider multi-degree reduction of $B{\acute{e}}zier$ curves with continuity of any (r, s) order with respect to $L_2$ norm. With help of matrix theory about generalized inverses we can use Lagrange multipliers to obtain the degree reduction matrix in a very simple form as well as the degree reduced control points. Also error analysis comparing with the least squares degree reduction without constraints is given. The advantage of our method is that the relationship between the optimal multi-degree reductions with and without constraints of continuity can be derived explicitly.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.45 no.4
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• pp.467-473
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• 1996

This Paper presents a new algorithm to solve Dynamic Economic Dispatch problem. Proposed algorithm is composed of two computational modules; one is dispatch, the other adjusting module. In the dispatch module based on the traditional Static Economic Dispatch method, the power dispatch of each unit is calculated. And in case the results of dispatch module violate ramp rate constraints, Lagrange multipliers are adjusted in the adjusting module. Tests and computer results on test systems are given to show the efficiency of the proposed algorithm. (author). 11 refs., 6 figs., 4 tabs.

• Journal of the Korean Institute of Illuminating and Electrical Installation Engineers
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• v.19 no.1
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• pp.59-63
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• 2005

In this article, we apply the standard method of Lagrange multipliers to examine the algorithm in a recent IEEE publication which calculates the optimal generation for minimizing the system loss using loss sensitivities derived by angle reference transposition, and show that the two algorithms are mathematically the same.

• Structural Engineering and Mechanics
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• v.26 no.6
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• pp.673-685
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• 2007

Reinforced concrete beams can be strengthened in a structural retrofit process by attaching steel plates to their sides by bolting. Whilst bolting produces a confident degree of shear connection under conditions of either static or seismic overload, the plates are susceptible to local buckling. The aim of this paper is to investigate the local buckling of unilaterally-restrained plates with point supports in a generic fashion, but with particular emphasis on the provision of the restraints by bolts, and on the geometric configuration of these bolts on the buckling loads. A numerical procedure, which is based on the Rayleigh-Ritz method in conjunction with the technique of Lagrange multipliers, is developed to study the unilateral local buckling of rectangular plates bolted to the concrete with various arrangements of the pattern of bolting. A sufficient number of separable polynomials are used to define the flexural buckling displacements, while the restraint condition is modelled as a tensionless foundation using a penalty function approach to this form of mathematical contact problem. The additional constraint provided by the bolts is also modelled using Lagrange multipliers, providing an efficacious method of numerical analysis. Local buckling coefficients are determined for a range of bolting configurations, and these are compared with those developed elsewhere with simplifying assumptions. The interaction of the actions in bolted plates during buckling is also considered.

• Journal of the Korean Mathematical Society
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• v.38 no.1
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• pp.1-23
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• 2001

This paper is concerned with an optimal shape control problem for the stationary Navier-Stokes system. A two-dimensional channel flow of an incompressible, viscous fluid is examined to determine the shape of a bump on a part of the boundary that minimizes the viscous drag. by introducing an artificial compressibility term to relax the incompressibility constraints, we take the penalty method. The existence of optima solutions for the penalized problem will be shown. Next, by employing Lagrange multipliers method and the material derivatives, we derive the shape gradient for the minimization problem of the shape functional which represents the viscous drag.

• Proceedings of the KIEE Conference
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• summer
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• pp.15-17
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• 2004

The flexible AC transmissions system (FACTS) is the underpinning concept upon which are based promising means to avoid effectively power flow bottlenecks and ways to extend the loadability of existing power transmission networks. This paper proposes a method by which the optimal locations of the FACTS to be installed in power system under cost function. The optimal solution of this type of problem requires large scale nonlinear optimisation techniques. We used Lagrange multipliers to solve a nonlinear equation with equality and ineaquality constraints. Case studies on the standard IEEE 14 bus system show that the method can be implemented successfully and that it is effective for determining the optimal location of the FACTS

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.1
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• pp.17-26
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• 2014

This paper presents development of an improved domain decomposition method for large scale structural problem that aims to provide high computational efficiency. In the previous researches, we developed the domain decomposition algorithm based on augmented Lagrangian formulation and proved numerical efficiency under both serial and parallel computing environment. In this paper, new computational analysis by the proposed domain decomposition method is performed. For this purpose, reduction in computational time achieved by the proposed algorithm is compared with that obtained by the dual-primal FETI method under serial computing condition. It is found that the proposed methods significantly accelerate the computational speed for a linear structural problem.

• Journal of the Korean Data and Information Science Society
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• v.14 no.2
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• pp.337-343
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• 2003

In this paper we present the prediction interval estimation method using bootstrap method for least squares support vector machine(LS-SVM) regression, which allows us to perform even nonlinear regression by constructing a linear regression function in a high dimensional feature space. The bootstrap method is applied to generate the bootstrap sample for estimation of the covariance of the regression parameters consisting of the optimal bias and Lagrange multipliers. Experimental results are then presented which indicate the performance of this algorithm.