Langrange Multipliers Research Articles

Semi-Smooth Newton Method for Solving 2D Contact Problems with Tresca and Coulomb Friction

The contribution deals with contact problems for two elastic bodies with friction. After the description of the problem we present its discretization based on linear or bilinear finite elements. The semi--smooth Newton method is used to find the solution, from which we derive active sets algorithms. Finally, we arrive at the globally convergent dual implementation of the algorithms in terms of the Langrange multipliers for the Tresca problem. Numerical experiments conclude the paper.

Generalized entropy optimization problems and the existence of their solutions

In the present study we have formulated a generalization of entropy optimization problems (GEOP), proposed sufficient conditions for the existence of solution. We have suggested also a new method based on a priori evaluations and Newton's methods for calculation of Langrange multipliers. Mentioned method allows calculating Langrange multipliers by starting from arbitrary initial point for Newton's approximations of constructed auxiliary equation. The solution of auxiliary equation is chosen as initial point for second constructed auxiliary equation. The recurring mentioned process for finite time leads to achieve an initial point for Newton's approximations of given equation and allows to find its unknown solution.

Feedback control for object manipulation by a pair of soft tip fingers

This paper discusses the problem of stable grasping and object manipulation by a pair of robot fingers when fingertips are covered with soft compressible material and fingers are allowed to incline their last link against the object surface. The area contact between the fingertips and the rigid object surface leads to nonholonomic constraints even for the planar case; however, the variational principle can be applied and the equation of motion is derived as a set of nonlinear differential equations with extra terms of Langrange multipliers incorporating the constraints. The proposed feedback controller is a linear combination of simple feedback control signals each designed for realizing grasp stabilization, regulation of object rotation and regulation of object position respectively. The controller is shown to achieve asymptotic convergence to the desired state at a stable grasping configuration. Simulation results are presented confirming the theoretical findings.

A new transmission fast-decoupled state estimation with equality constraints

In this paper, a new fast decoupled state estimator with equality constraints is proposed. The Langrange multipliers were utilized to deal with the zero injections. The proposed method is based on the equivalent-current-measurement and rectangular coordinates. The conventional assumptions on voltage magnitudes, angles and r/ x ratios are no longer necessary by the new method in decoupling the network. The method uses a compact constant-symmetric gain matrix which can be decoupled into two ‘identical’ sub-gain matrices. The update and factorization of the sub-gain matrix needs to be carried out only once. Tests have shown that the proposed low-storage method is efficient, accurate and robust in solving the equality-constrained state estimation.

On regularity and stability in semi-infinite optimization

For stationary solutions and Langrange multipliers of a semi-infinite program withC 1 data, we study some stability behaviour which is closely related to (metric) regularity of the constraint system. The multiplier set mapping considered here has its images in a finite-dimensional space. In this framework, regularity is a necessary and sufficient condition to have bounded sets of multipliers.

Gauge fixing and abelianization in simple BRST quantization

In a previous paper [Nucl. Phys. B395 (1993) 647] it was shown that the BRST charge Q for any gauge model with a Lie algebra symmetry may be decomposed as Q = δ + δ †, δ 2 = δ †2 = 0 , [δ,δ †] + = 0 provided dynamical Langrange multipliers are used, but without introducing other matter variables in δ than the gauge generators in Q. In this paper further decompositions are derived, but now by means of gauge fixing operators. As in the previous paper it is shown that δ = c †aφ a , where c a are new ghost and φ a are non-hermitian variables satisfying the gauge algebra. However, in distinction to the previous paper also solutions of the form δ = c †aA a , where the A a satisfy an abelian algebra, are derived (abelianization). By means of a bigrading the BRST condition reduces to δ|ph〉 = δ †|ph〉 = 0 on inner product spaces whose general solutions are expressed in terms of the solutions to a proper Dirac quantization. Thus, the procedure provides for inner products for the solutions of a Dirac quantization.

A general multi-configuration Hartree-Fock program

A general multi-configuration Hartree-Fock program is described. Unlike earlier versions, the present code includes a rotation analysis to determine whether the energy is invariant with respect to a rotation of an orbital basis and off-diagonal Langrange multipliers may be set to zero; if not, orbitals are rotated for a stationary energy. Also a special data structure has been introduced which eliminates numerous evaluations of the same radial integral in the calculation of the potential and exchange functions as well as the evaluation of the interaction matrix.

A surrogate and Lagrangian approach to constrained network problems

This paper presents the use of surrogate constraints and Lagrange multipliers to generate advanced starting solutions to constrained network problems. The surrogate constraint approach is used to generate a singly constrained network problem which is solved using the algorithm of Glover, Karney, Klingman and Russell [13]. In addition, we test the use of the Lagrangian function to generate advanced starting solutions. In the Lagrangian approach, the subproblems are capacitated network problems which can be solved using very efficient algorithms. The surrogate constraint approach is implemented using the multiplier update procedure of Held, Wolfe and Crowder [16]. The procedure is modified to include a search in a single direction to prevent periodic regression of the solution. We also introduce a reoptimization procedure which allows the solution from thekth subproblem to be used as the starting point for the next surrogate problem for which it is infeasible once the new surrogate constraint is adjoined. The algorithms are tested under a variety of conditions including: large-scale problems, number and structure of the non-network constraints, and the density of the non-network constraint coefficients. The testing clearly demonstrates that both the surrogate constraint and Langrange multipliers generate advanced starting solutions which greatly improve the computational effort required to generate an optimal solution to the constrained network problem. The testing demonstrates that the extra effort required to solve the singly constrained network subproblems of the surrogate constraints approach yields an improved advanced starting point as compared to the Lagrangian approach. It is further demonstrated that both of the relaxation approaches are much more computationally efficient than solving the problem from the beginning with a linear programming algorithm.

Topological Analysis of Multibody Systems for Recursive Dynamics Formulations

Abstract Graph theory is used to characterize the topology of multibody systems for high speed parallel computer dynamic simulation. Decoupled loops are defined for use in the recursive dynamics formulations, wherein Langrange multipliers can be eliminated concurrently within each decoupled loop. A row-scanning process for the nondirected adjacency matrix of a graph is described, to generate an identification array that facilitates compact storage and defines system connectivities. A decision tree search algorithm is employed to generate all spanning trees associated with a closed loop graph. For each spanning tree generated from the decision tree search algorithm, a precedence array is defined. A path matrix for the spanning tree and an outdegree of junction nodes are defined to assist in parallel implementation of recursive dynamics algorithms and are then constructed from the precedence array. Finally, an optimal tree search algorithm that identifies a spanning tree that leads to best solution sequences for parallel implementation of recursive dynamics formulations is presented.

Solution to the lot size multi-item deterministic inventory problems with budget and/or space constraints using a microcomputer

This paper presents solutions for the multi-item deterministic inventory problem with constraints. Langrange multipliers are used to find mathematical solutions to the problems. A microcomputer program in BASIC has been developed for this purpose and due to the complexity of the problem solved by hand. The use of the program is illustrated by solving numerical examples of these types of inventory problems.

Use of regularization operators together with Lagrange multipliers in numerical deconvolution of fluorescence decay curves

Abstract A method for deconvolution of fluorescence decay curves is proposed in which no assumption is made about the decay law. It is based on the Langrange multipliers formalism and on recent theories about the regularization of ill-posed problems and keeps errors under control by means of a statistically significant norm. The decay is obtained by an iterative least square Gauss-Seidel algorithm. The method is then compared with other non- a priori deconvolution techniques by its application to some real experiments. Special attention is paid to the decay of emission anistropy.

Analysis of optimality criteria and gradient projection methods for optimal structural design

Abstract A general mathematical model for optimal design of structural and mechanical systems is defined. The finite element techniques for analysis of these systems are incorporated into the model. Optimality conditions for the model are derived. It is shown that an analytical solution of the optimality conditions is impossible except for trivial design problems. Numerical methods for solving these optimality conditions are presented. A direct method based on the gradient projection concept is derived. Numerical aspects for the method are discussed. These include step size selection, calculation of Langrange multipliers, identification of dependent design sensitivity vectors of constraint functions, and convergence criterion. Both direct and indirect approaches require total derivatives of cost and various constraint functions with respect to design variables. This is accomplished by integrating into both the approaches the state space design sensitivity analysis that has proven to be very general and efficient. It is shown that other major calculations of the two approaches are essentially the same. Thus there is potential for continuous transition between various algorithms for structural optimization. Optimal solutions for two design examples are obtained by hybrid methods to show the potential for further development of these methods.