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.

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.

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.

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.

THE LAGRANGE DYNAMIC EQUATIONS OF MULTI－RIGIDBODY SYSTEMS WITH EXTERNAL SHOCKS

In the paper, the problems of dynamie response of multi-rigidbody systems with external impulsive forces are diseussed, and a set of equations with form of Lagrange method is presented. These equations are easy to be caleulated by computer.

Secured Health Data Transmission Using Lagrange Interpolation and Artificial Neural Network

In recent decades,the cloud computing contributes a prominent role in health care sector as the patient health records are transferred and collected using cloud computing services.The doctors have switched to cloud computing as it provides multiple advantageous measures including wide storage space and easy availability without any limitations.This necessitates the medical field to be redesigned by cloud technology to preserve information about patient’s critical diseases,electrocardiogram(ECG)reports,and payment details.The proposed work utilizes a hybrid cloud pattern to share Massachusetts Institute of Technology-Beth Israel Hospital(MIT-BIH)resources over the private and public cloud.The stored data are categorized as significant and non-significant by Artificial Neural Networks(ANN).The significant data undergoes encryption by Lagrange key management which automatically generates the key and stores it in the hidden layer.Upon receiving the request from a secondary user,the primary user verifies the authentication of the request and transmits the key via Gmail to the secondary user.Once the key matches the key in the hidden layer,the preserved information will be shared between the users.Due to the enhanced privacy preserving key generation,the proposed work prevents the tracking of keys by malicious users.The outcomes reveal that the introduced work provides improved success rate with reduced computational time.

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.

A New Algorithm for Reducing Dimensionality of L1-CSVM Use Augmented Lagrange Method

Principal component analysis and generalized low rank approximation of matrices are two different dimensionality reduction methods. Two different dimensionality reduction algorithms are applied to the L1-CSVM model based on augmented Lagrange method to explore the variation of running time and accuracy of the model in dimensionality reduction space. The results show that the improved algorithm can greatly reduce the running time and improve the accuracy of the algorithm.

Structure Analysis of Locking Mechanism of Gear-Rack Typed Ship-Lift

Contact nonlinear theory was researched. Contact problem was transformed into optimization problem containing Lagrange multiplier, and unsymmetrical stiffness matrix was transformed into symmetrical stiffness matrix. A finite element analysis （FEA） model defining more than 300 contact pairs for long nut-short screw locking mechanism of a large-scale vertical gear-rack typed ship-lift was built. Using augmented Lagrange method and symmetry algorithm of contact element stiffness, the FEA model was analyzed, and the contact stress of contact interfaces and the von Mises stress of key parts were obtained. The results show that the design of the locking mechanism meets the requirement of engineering, and this method is effective for solving large stole nonlinear contact pairs.

Base force element method of complementary energy principle for large rotation problems

Using the concept of the base forces, a new finite element method （base force element method, BFEM） based on the complementary energy principle is presented for accurate modeling of structures with large displacements and large rotations. First, the complementary energy of an element is described by taking the base forces as state variables, and is then separated into deformation and rotation parts for the case of large deformation. Second, the control equations of the BFEM based on the complementary energy principle are derived using the Lagrange multiplier method. Nonlinear procedure of the BFEM is then developed. Finally, several examples are analyzed to illustrate the reliability and accuracy of the BFEM.

Numerical modeling of large deformation and nonlinear frictional contact of excavation boundary of deep soft rock tunnel

Roadways excavated in soft rocks at great depth are difficult to be maintained due to large deformation of surrounding rocks, which greatly influences the safety and efficiency of deep resources exploitation. During the excavation process of a deep soft rock tunnel, the rock wall may be compacted due to large deformation. In this paper, the technique to address this problem by a two-dimensional (2D) finite element software, large deformation engineering analyses software (LDEAS 1.0), is provided. By using the Lagrange multiplier method, the kinematic constraint of non-penetrating condition and static constraint of Coulomb friction are introduced to the governing equations in the form of incremental displacement. The numerical example demonstrates the efficiency of this technology. Deformations of a transportation tunnel in inclined soft rock strata at the depth of 1 000 m in Qishan coal mine and a tunnel excavated to three different depths are analyzed by two models, i.e. the additive decomposition model and polar decomposition model. It can be found that the deformation of the transportation tunnel is asymmetrical due to the inclination of rock strata. For extremely soft rock, large deformation can converge only for the additive decomposition model. The deformation of surrounding rocks increases with the increase in the tunnel depth for both models. At the same depth, the deformation calculated by the additive decomposition model is smaller than that by the polar decomposition model.

Frictional contact algorithm study on the numerical simulation of large deformations in deep soft rock tunnels

There exist three types of nonlinear problems in large deformation processes of deep softrock engineering, i.e., nonlin- earity caused by material, geometrical and contact boundary. In this paper, the numerical method to tackle the nonlinear eontact and large deformation problem in A Software on Large Deformation Analysis for Soft Rock Engineering at Great Depth was presented. In the software, based on Lagrange multiplier method and Coulomb friction law, kinematic constraints on contact boundaries were introduced in functional function, and the finite element equations was established for two incremental large deformation analyses models, polar decomposition model and additive decomposition model. For every incremental loading step, by searching for the contact points in the potential contact interfaces （the excavation boundaries）, the Lagrange multipliers, i.e., contact forces are cal- culated iteratively by Gauss-Seidel method, and justified through satisfy the inequalities of static constraint on contact boundaries. With the software, large deformation and frictional contact of a transport roadway were analyzed numerically by the two models. The numerical examples demonstrated the efficiency of the method used in the software.

Nonlinear optimal model and solving algorithms for platform planning problem in battlefield

Platform planning is one of the important problems in the command and control(C2) field. Hereto, we analyze the platform planning problem and present nonlinear optimal model aiming at maximizing the task completion qualities. Firstly, we take into account the relation among tasks and build the single task nonlinear optimal model with a set of platform constraints. The Lagrange relaxation method and the pruning strategy are used to solve the model. Secondly, this paper presents optimization-based planning algorithms for efficiently allocating platforms to multiple tasks. To achieve the balance of the resource assignments among tasks, the m-best assignment algorithm and the pair-wise exchange(PWE)method are used to maximize multiple tasks completion qualities.Finally, a series of experiments are designed to verify the superiority and effectiveness of the proposed model and algorithms.

Sedimentation of a single particle between two parallel walls

The sedimentation of a single circular particle between two parallel walls was studied by means of direct numerical simulation (DNS) and experiment. The improved implementation of distributed Lagrange multiplier/fictitious domain method used in our DNS is a promising new way for simulation of particulate flows. The settling behaviors of the particle are presented ranging in Reynolds number from 0 to about 700, which showed that our results for low Reynolds numbers agreed well with that reported before. Nevertheless, for higher Reynolds numbers our results were different from theirs. The long-term mean equilibrium positions in our results were all on the centerline, but not at off-center position as reported before. In order to validate our simulation, experiments were also conducted. The results showed that the sedimenting behavior simulated in this paper agreed well with our experiment result.

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.

Active set truncated-Newton algorithm for simultaneous optimization of distillation column

An active set truncated-Newton algorithm (ASTNA) is proposed to solve the large-scale bound constrained sub-problems. The global convergence of the algorithm is obtained and two groups of numerical experiments are made for the various large-scale problems of varying size. The comparison results between ASTNA and the subspace limited memory quasi-Newton algorithm and between the modified augmented Lagrange multiplier methods combined with ASTNA and the modified barrier function method show the stability and effectiveness of ASTNA for simultaneous optimization of distillation column.

INTERACTIONS BETWEEN TWO SEDIMENTING PARTICLES WITH DIFFERENT SIZES

An improved implementation of Distributed Lagrange multiplier/fictitious domain method was presented and used to simulate the interactions between two circular particles sedimenting in a two_dimensional channel. The simulation results were verified by comparison with experiments. The results show that the interactions between two particles with different sizes can be described as drafting, kissing, tumbling and separating. Only for small diameter ratio, the two particles will interact undergoing repeated DKT (Drafting, Kissing and Tumbling) process. Otherwise, the two particles will separate after their tumbling. The results also show that, during the interaction process, the motion of the small particle is strongly affected while the large particle is affected slightly.

Research on dynamic model of rotors with bearing misalignment

Based on the structural and mechanics analysis of aero-engines rotor system, the dynamic model of the flexible rotor system with multi-supports are presented in order to solve the bearing misalignment problem of rotor system in aero-engines. The motion equations are derived through Lagrange method. The relationship between structural and mechanics characteristics parameters are built up. Finally, the dynamic influence of bearing misalignment on rotor system are divided into three kinds： additional rotor bending rigidity, additional bearing misalignment excitation force and additional imbalance. The equations suggest that additional imbalance excitation force activates the nonlinearity on rotor system and an additional 2x excitation force might appear.

Novel Method to Handle Inequality Constraints for Nonlinear Programming

By redefining the multiplier associated with inequality constraint as a positive definite function of the originally-defined multiplier, say, u2_i, i=1, 2, ..., m, nonnegative constraints imposed on inequality constraints in Karush-Kuhn-Tucker necessary conditions are removed. For constructing the Lagrange neural network and Lagrange multiplier method, it is no longer necessary to convert inequality constraints into equality constraints by slack variables in order to reuse those results dedicated to equality constraints, and they can be similarly proved with minor modification. Utilizing this technique, a new type of Lagrange neural network and a new type of Lagrange multiplier method are devised, which both handle inequality constraints directly. Also, their stability and convergence are analyzed rigorously.

Estimating the bound for the generalized Lorenz system

A chaotic system is bounded, and its trajectory is confined to a certain region which is called the chaotic attractor. No matter how unstable the interior of the system is, the trajectory never exceeds the chaotic attractor. In the present paper, the sphere bound of the generalized Lorenz system is given, based on the Lyapunov function and the Lagrange multiplier method. Furthermore, we show the actual parameters and perform numerical simulations.