Modified Augmented Lagrange Multiplier Methods for Large-Scale Chemical Process Optimization

Chemical process optimization can be described as large-scale nonlinear constrained minimization. The modified augmented Lagrange multiplier methods (MALMM) for large-scale nonlinear constrained minimization are studied in this paper. The Lagrange function contains the penalty terms on equality and inequality constraints and the methods can be applied to solve a series of bound constrained sub-problems instead of a series of unconstrained sub-problems. The steps of the methods are examined in full detail. Numerical experiments are made for a variety of problems, from small to very large-scale, which show the stability and effectiveness of the methods in large-scale problems.

Accelerated Matrix Recovery via Random Projection Based on Inexact Augmented Lagrange Multiplier Method

In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.

A Parameter-Free Approach to Determine the Lagrange Multiplier in the Level Set Method by Using the BESO

A parameter-free approach is proposed to determine the Lagrange multiplier for the constraint of material volume in the level set method.It is inspired by the procedure of determining the threshold of sensitivity number in the BESO method.It first computes the difference between the volume of current design and the upper bound of volume.Then,the Lagrange multiplier is regarded as the threshold of sensitivity number to remove the redundant material.Numerical examples proved that this approach is effective to constrain the volume.More importantly,there is no parameter in the proposed approach,which makes it convenient to use.In addition,the convergence is stable,and there is no big oscillation.

Modified Lagrange Multiplier Method and Generalized Variational Principle in Fluid Mechanics

The Lagrange multiplier method plays an important role in establishing generalized variational principles notonly in tluid mechallics. but also in elasticity. Sometimes, however, one may come across variational crisis(somemultipliers vanish identically). failing to achieve his aim. The crisis is caused by the fact that the Inultipliers are treatedas independent variables in the process of variatioll. but after identification they become functions of the originalindependent variables. To overcome it, a Inodified Lagrange multiplier method or semi-inverse method has beenproposed to deduce generalized varistional principles. Some e-camples are given to illustrate its convenience andeffectiveness of the novel method.

Fully Coupled Fluid-Structure Interaction Model Based on Distributed Lagrange Multiplier/Fictitious Domain Method

This paper, with a finite element method, studies the interaction of a coupled incompressible fluid-rigid structure system with a free surface subjected to external wave excitations. With this fully coupled model, the rigid structure is taken as ＂fictitious＂ fluid with zero strain rate. Both fluid and structure are described by velocity and pressure. The whole domain, including fluid region and structure region, is modeled by the incompressible Navier-Stokes equations which are discretized with fixed Eulerian mesh. However, to keep the structure＇ s rigid body shape and behavior, a rigid body constraint is enforced on the ＂fictitious＂ fluid domain by use of the Distributed Lagrange Multipher/Fictitious Domain （DLM/ FD） method which is originally introduced to solve particulate flow problems by Glowinski et al. For the verification of the model presented herein, a 2D numerical wave tank is established to simulate small amplitude wave propagations, and then numerical results are compared with analytical solutions. Finally, a 2D example of fluid-structure interaction under wave dynamic forces provides convincing evidences for the method excellent solution quality and fidelity.

Characterizing Shadow Price via Lagrange Multiplier for Nonsmooth Problem

In this paper,the relation between the shadow price and the Lagrange multiplier for nonsmooth optimization problem is explored.It is obtained that the Lagrange multipliers are upper bounds of the shadow price for a convex optimization problem and a class of Lipschtzian optimization problems.This result can be used in pricing mechanisms for nonsmooth situation.Several nonsmooth functions involved in this class of Lipschtzian optimizations are listed.Finally,an application to electricity pricing is discussed.

A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

A PRECONDITIONER FOR DOMAIN DECOMPOSITION METHODS WITH MORTAR LAGRANGE MULTIPLIERS

In this paper we consider the domain decomposition methods with mortar element Lagrange multipliers to two-dimensional elliptic problems.We shall construct a kind of simple preconditioners for the corresponding interface equation.It will be shown that condition number of the preconditioned interface matrix is almost optimal.

Local curve fitting based Lagrange multiplier selection for Id-slice in multi-view video coding

The rate and distortion of Id-slice do not fit the globally linear relationship on a logarithmic scale. Lagrange multiplier selection methods based on the globally linear approximate relationship are neither efficient nor optimal for multi-view video coding (MVC). To improve the coding efficiency of MVC, a local curve fitting based Lagrange multiplier selection method is proposed in this paper, where Lagrange multipliers are selected according to the local slopes of the approximate curves. Experi-mental results showed that the proposed method improves the coding efficiency. Up to 2.5 dB gain was achieved at low bitrates.

Improved Faddeev-Jackiw quantization of the electromagnetic field and Lagrange multiplier fields

We use the improved Faddeev-Jackiw quantization method to quantize the electromagnetic field and its Lagrange multiplier fields. The method＇s comparison with the usual Faddeev-Jackiw method and the Dirac method is given. We show that this method is equivalent to the Dirac method and also retains all the merits of the usual Faddeev-Jackiw method. Moreover, it is simpler than the usual one if one needs to obtain new secondary constraints. Therefore, the improved Faddeev-Jackiw method is essential. Meanwhile, we find the new meaning of the Lagrange multipliers and explain the Faddeev-Jackiw generalized brackets concerning the Lagrange multipliers.

A PRECONDITIONER FOR THREE-DIMENSIONAL DOMAIN DECOMPOSITION METHODS WITH LAGRANGE MULTIPLIERS

In this paper we consider domain decomposition methods for three-dimensional elliptic problems with Lagrange multipliers, and construct a kind of simple preconditioner for the corresponding interface equation. It will be shown that condition number of the resulting preconditioned interface matrix is almost optimal.

SUPERCONVERGENCE OF THE NON-CONFORMING DOMAIN DECOMPOSED FINITE ELEMENT METHOD WITH LAGRANGE MULTIPLIERS FOR PARABOLIC PROBLEMS

Abstract. In this paper which is motivated by computation on parallel machine, we showthat the superconvergence results of the finite element method(FEM) with Lagrange mul-tipliers based on domain decomposition method (DDM) with nonmatching grids can becarried over to parabolic problems. The main idea of this paper is to achieve the combina-tion of parallel computational method with the higher accuracy technique by interpolationfinite element postprocessing.

Chebyshev-Lagrange Multipliers Technique for Vibration Analysis of Functionally Graded Material Beams using Various Beam Theories

In this paper,a new technique for analysing functionally graded material(FGM)beams using the Chebyshev polynomials and Lagrange multipliers with various beam theories is presented.By utilizing the inner products and the Chebyshev polynomials’orthogonality properties incorporated with Lagrange multipliers,we can combine the governing equation and boundary conditions to yield the matrix equations with explicit weighting coefficients.Numerical examples are provided for vibration analysis of various beam theories and assumptions.Based on numerical evaluations,it is revealed that the proposed technique can efficiently achieve good agreement with those of the references.

Lagrange multiplier method used in BESⅢ kinematic fitting

In this paper, the kinematic fitting with the Lagrange multiplier method has been studied for BESⅢ experiment. First we introduce the Lagrange multiplier method and implement kinematic constraints. Then we present the performance of the kinematic fitting algorithm. With the kinematic fitting, we can improve the resolution of track parameters and reduce the background.

APPROXIMATE LAGRANGE MULTIPLIER ALGORITHM FOR STOCHASTIC PROGRAMS WITH COMPLETE RECOURSE:NONLINEAR DETERMINISTIC CONSTRAINTS

In this paper we design an approximation method for solving stochastic programs with com-plete recourse and nonlinear deterministic constraints. This method is obtained by combiningapproximation method and Lagrange multiplier algorithm of Bertsekas type. Thus this methodhas the advantages of both the two.

A Relation between Resolvents of Subdifferentials and Metric Projections to Level Sets

An equation concerning with the subdifferential of convex functionals defined in real Banach spaces and the metric projections to level sets is shown. The equation is compared with the resolvents of general monotone operators, and makes the geometric properties of differential equations expressed by subdifferentials clear. Hence, it can be expected to be useful in obtaining the steepest descents defined by the convex functionals in Banach spaces. Also, it gives a similar result to the Lagrange multiplier method under certain conditions.

Study on the tradeoff between interpretability and precision in fuzzy modeling

An approach to identifying fuzzy models considering both interpretability and precision was proposed. Firstly, interpretability issues about fuzzy models were analyzed. Then, a heuristic strategy was used to select input variables by increasing the number of input variables, and the Gustafson-Kessel fuzzy clustering algorithm, combined with the least square method, was used to identify the fuzzy model. Subsequently, an interpretability measure was described by the product of the number of input variables and the number of rules, while precision was weighted by root mean square error, and the selection objective function concerning interpretability and precision was defined. Given the maximum and minimum number of input variables and rules, a set of fuzzy models was constructed. Finally, the optimal fuzzy model was selected by the objective function, and was optimized by a genetic algorithm to achieve a good tradeoff between interpretability and precision. The performance of the proposed method was illustrated by the well-known Box-Jenkins gas furnace benchmark; the results demonstrate its validity.

Multi-phase signal setting and capacity of signalized intersection

A capacity model of multi-phase signalized intersections is derived by a stopping-line method. It is simplified with two normal situations： one situation involves one straight lane and one left-turn lane; the other situation involves two straight lanes and one left-turn lane. The results show that the capacity is mainly relative to signal cycle length, phase length, intersection layout and following time. With regard to the vehicles arrival rates, the optimal model is derived based on each phase＇s remaining time balance, and it is solved by Lagrange multipliers. Therefore, the calculation models of the optimal signal cycle length and phase lengths are derived and simplified. Compared to the existing models, the proposed model is more convenient and practical. Finally, a practical intersection is chosen and its signal cycles and phase lengths are calculated by the proposed model.

Mantle flow beneath the Asian continent and its force to the crust

The mantle unsteady flows, which are in an incompressible and isoviscous spherical shell, are investigated by using algorithms of the parallel Lagrange multiplier dissonant decomposition method (LMDDM) and the parallel Lagrange multiplier discontinuous deformation analyses (LMDDA) in this paper. Some physical fields about mantle flows such as velocity, pressure, temperature, stress and the force to the crust of the Asian continent are calculated on a parallel computer.

A FICTITIOUS DOMAIN METHOD FOR DIRICHLET PROBLEM AND ITS APPLICATIONS TO GENER ALIZED STOKES PROBLEM

This paper discusses a fictitious domain method for the linear Dirichlet problem and its applications to the generalized Stokes problem. This method treats Dirichlet boundary condit ion via a Lagrange multiplier technique and is well suited to the no-slip bound ary condition in viscous flow problems. In order to improve the accuracy of solu tions, meshes are refined according to the a posteriori error estimate. The mini -element discretization is applied to solve the generalized Stokes problem. Fin ally, some numerical results to validate this method are presented for partial d ifferential equations with Dirichlet boundary condition.