Set Of Lagrange Multipliers Research Articles

Partitioned formulation of internal fluid–structure interaction problems by localized Lagrange multipliers

A partitioned formulation of compressible internal fluid–structure interaction problems is presented by employing a displacement model for both the fluid and structure. Partitioning is effected by a localized version of the method of Lagrange multipliers, which assigns two independent sets of Lagrange multipliers to the structural and fluid interfaces. Two major features of the present formulation include: an interface compliance normalization that helps capture the predominant physics of interaction phenomena when the interfaces are characterized by two radically different rigidities, and a novel transformation of the displacement model into a fluid-pressure model that is suitable for both transient and vibration analyses. The present formulation first solves for the interface Lagrange multipliers, which are subsequently used to solve for the structural displacements and the fluid displacement or pressure by employing two independent analysis modules.

FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method

AbstractThe FETI method and its two‐level extension (FETI‐2) are two numerically scalable domain decomposition methods with Lagrange multipliers for the iterative solution of second‐order solid mechanics and fourth‐order beam, plate and shell structural problems, respectively.The FETI‐2 method distinguishes itself from the basic or one‐level FETI method by a second set of Lagrange multipliers that are introduced at the subdomain cross‐points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations. In this paper, we present a dual–primal formulation of the FETI‐2 concept that eliminates the need for that second set of Lagrange multipliers, and unifies all previously developed one‐level and two‐level FETI algorithms into a single dual–primal FETI‐DP method. We show that this new FETI‐DP method is numerically scalable for both second‐order and fourth‐order problems. We also show that it is more robust and more computationally efficient than existing FETI solvers, particularly when the number of subdomains and/or processors is very large. Copyright © 2001 John Wiley & Sons, Ltd.

系統評価のための停止ランキングに基づく電源起動停止計画手法

This paper presents a practical algorithm for solving Unit Commitment (UC) problems with network constraints represented in Optimal Power Flow (OPF). The proposed algorithm is based on Successive Approximation Dynamic Programming (SADP) according to a Unit Decommitment Ranking (UDR) which is a priority list for unit decommitment procedure. The UDR can be determined by Dynamic Programming (DP) according to the information of OPF consisting of a set of Lagrange multipliers (dual variables). Since the set of Lagrange multipliers represents the relationship between the optimal value of objective function and OPF constraints, the proposed algorithm can determine power system resource scheduling in consideration of network constraints, such as line flow and voltage constraints. The feasibility and effectiveness of the proposed algorithm are demonstrated on a modified IEEE Reliability Test System. © 2001 Scripta Technica, Electr Eng Jpn, 138(2): 1–13, 2002

A parallel solver based on the dual Schur decomposition of general finite element matrices

A parallel solver based on domain decomposition is presented for the solution of large algebraic systems arising in the finite element discretization of mechanical problems. It is hybrid in the sense that it combines a direct factorization of the local subdomain problems with an iterative treatment of the interface system by a parallel GMRES algorithm. An important feature of the proposed solver is the use of a set of Lagrange multipliers to enforce continuity of the finite element unknowns at the interface. A projection step and a preconditioner are proposed to control the conditioning of the interface matrix. The decomposition of the finite element mesh is formulated as a graph partitioning problem. A two-step approach is used where an initial decomposition is optimized by non-deterministic heuristics to increase the quality of the decomposition. Parallel simulations of a Navier–Stokes flow problem carried out on a Convex Exemplar SPP system with 16 processors show that the use of optimized decompositions and the preconditioning step are keys to obtaining high parallel efficiencies. Typical parallel efficiencies range above 80%. © 1998 John Wiley & Sons, Ltd.

A parallel solver based on the dual Schur decomposition of general finite element matrices

A parallel solver based on domain decomposition is presented for the solution of large algebraic systems arising in the finite element discretization of mechanical problems. It is hybrid in the sense that it combines a direct factorization of the local subdomain problems with an iterative treatment of the interface system by a parallel GMRES algorithm. An important feature of the proposed solver is the use of a set of Lagrange multipliers to enforce continuity of the finite element unknowns at the interface. A projection step and a preconditioner are proposed to control the conditioning of the interface matrix. The decomposition of the finite element mesh is formulated as a graph partitioning problem. A two-step approach is used where an initial decomposition is optimized by non-deterministic heuristics to increase the quality of the decomposition. Parallel simulations of a Navier–Stokes flow problem carried out on a Convex Exemplar SPP system with 16 processors show that the use of optimized decompositions and the preconditioning step are keys to obtaining high parallel efficiencies. Typical parallel efficiencies range above 80%. © 1998 John Wiley & Sons, Ltd.

Optimization Problems with Perturbations: A Guided Tour

This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems. These methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty. We give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.

The Newton modified barrier method for QP problems

The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues, to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a “hot start”. Starting at such a “hot start”, at mostO(In Inɛ-1) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, ɛ>0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that onlyO(Inɛ-1)O(In Inɛ-1) Newton steps are required to reach an ɛ-approximation to the solution from any “hot start”. In order to reach the “hot start”, one has to perform\(\mathcal{O}(\sqrt m {\text{ }}In{\text{ }}C)\) Newton steps, wherem characterizes the size of the problem andC>0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.

A simplified approach to optimally controlled quantum dynamics

A new formalism for the optimal control of quantum mechanical physical observables is presented. This approach is based on an analogous classical control technique reported previously [J. Botina, H. Rabitz, and N. Rahman, J. Chem. Phys. 102, 226 (1995)]. Quantum Lagrange multiplier functions are used to preserve a chosen subset of the observable dynamics of interest. As a result, a corresponding small set of Lagrange multipliers needs to be calculated and they are only a function of time. This is a considerable simplification over traditional quantum optimal control theory [S. Shi and H. Rabitz, Comp. Phys. Comm. 63, 71 (1991)]. The success of the new approach is based on taking advantage of the multiplicity of solutions to virtually any problem of quantum control to meet a physical objective. A family of such simplified formulations is introduced and numerically tested. Results are presented for these algorithms and compared with previous reported work on a model problem for selective unimolecular reaction induced by an external optical electric field.

A new approach to formulation of the governing equations of constrained multibody systems

In this paper a new method to formulate automatically the governing equations of constrained multibody systems is presented based on the property of constraints. In this method, the equations of motion for an unconstrained multibody system are converted into those of the associated constrained multibody system. Instead of introducing a set of Lagrange multipliers, the constraint forces are evaluated before integrating the governing equations; thus the process of finding an orthogonal-complement matrix of the constraint Jacobian matrix is avoided, which saves much computational time. Compared with other methods, more information about the dynamics of the multibody system is obtained by integrating the same number of differential equations. Additionally, a new stabilization technique is given in this paper.

Optimality criterion for maximum fundamental frequency of free vibrations of frames including axial and bending effects

An optimality criterion for maximum multiple fundamental frequency of free vibrations for structures of prescribed weights is presented. The criterion includes both axial and bending effects and can be used for analysis of truss, beam and frame structures. The error norm based on the criterion is proposed and used to verify trial designs against the optimum. The accompanying iterative procedure reduces this error norm to zero and drives a trial design to the optimum. The modality of the design at the optimum and the corresponding set of Lagrange multipliers are determined automatically.

Scheduling of hydrothermal power systems

The authors present a method for scheduling hydrothermal power systems based on the Lagrangian relaxation technique. By using Lagrange multipliers to relax system-wide demand and reserve requirements, the problem is decomposed and converted into a two-level optimization problem. Given the sets of Lagrange multipliers, a hydro unit subproblem is solved by a merit order allocation method, and a thermal unit subproblem is solved by using dynamic programming without discretizing generation levels. A subgradient algorithm is used to update the Lagrange multipliers. Numerical results based on Northeast Utilities data show that this algorithm is efficient, and near-optimal solutions are obtained. Compared with previous work where thermal units were scheduled by using the Lagrangian relaxation technique and hydro units by heuristics, the new coordinated hydro and thermal scheduling generates lower total costs and requires less computation time. >

Optimization of structures having general cross-sectional relationships using an optimality criterion method

An optimality criterion-based method for locating the optimal designs of structures with members having general cross-sectional relationships is presented. The method exploits the concept of one most active constraint to avoid the need of calculating a large set of Lagrange multipliers for the active constraints and also eliminates the need to decide as to whether or not a particular constraint should be considered active. The method also avoids the scaling procedure normally used in other optimality criterion techniques and can deal with multiple load conditions and design-variables linkage. Several example structures, designed under stress, displacement, minimum and maximum area constraints and design-variables linkage are presented. The results demonstrate the simplicity, efficiency and reliability of the method even when used on small microcomputers. Although this research is based on the concept of one most active constraint, several constraints may be active at the optimal designs.

Load‐Transfer Problems for Transversely Isotropic Elastic Media

This paper presents a unified solution procedure to analyze the load transfer from a partially embedded long (finite) cylindrical elastic bar into a transversely isotropic elastic half‐space. The bar may be subjected to a torque, an axial force, a horizontal force, or a bending moment at the top end. The solution approach is different from that proposed by Muki and Sternberg (1970). It is based on the interaction between a one‐dimensional elastic bar and a three‐dimensional elastic half‐space with a cylindrical cavity. The displacement compatibility is achieved along the contact surface between the bar and the half‐space. A variational technique coupled with a boundary integral equation scheme based on a set of exact Green's functions is used in the analysis. The boundary conditions at the top end of the bar are incorporated into the variational formulation through a set of Lagrange multipliers. Selected numerical results are presented for elastic bars embedded in some representative transversely isotropic media.

An optimality criterion method for structures having general cross-sectional relationships

An optimality criterion-based method is presented which is capable of locating the optimal designs of structures with members having general cross-sectional relationships among the area, section modulus and second moment of area. The method exploits the concept of one most critical constraint to avoid the need to calculate a large set of Lagrange multipliers for the active constraints and also eliminates the need for a decision as to whether or not a particular constraint should be considered active. The method also avoids the scaling procedure normally used in other optimality criterion techniques and can deal with multiple load conditions and design-variables linkage. Several example structures, designed under stress, displacement, minimum and maximum area constraints, and design-variables linkage demonstrate the simplicity, efficiency and reliability of the method, even on small microcomputers. Although this research uses the concept of one most critical constraint, several constraints may be active at the optimal designs.

Dynamic Response of Elastic Plates on Viscoelastic Half Space

The time‐harmonic vertical vibratios of an elastic annular plate resting in smooth contact with a homogeneous isotropic viscoelastic half space is considered. The plate is subjected to a vertical distributed loading, or it may be excited by specified displacements or stress resultants applied along the plate edges. The response of the plate is goverried by classical thin‐plate theory and its vertical displacement is represented by an admissible function containing a set of generalized coordinates. A representation for contact stresses is obtained through the solution of a flexibility equation based on an exact displacement Green's function of the half space. The equation of motion of the plate in terms of generalized coordinates are established through the application of Lagrange's equation of motion. The plate edge boundary conditions are incorporated into the plate Lagrangian function as constraint terms through a set of Lagrange multipliers. Selected numerical results for displacement and contact stresses of annular elastic plates are presented.

An optimality criterion method for structural optimization problems

This work presents the application of an optimality criterion method which exploits the concept of one most critical constraint. The method eliminates the need to calculate a large set of Lagrange multipliers for the active constraints, and also eliminates the need for a decision as to whether or not a particular constraint should be considered active. The method can treat multiple load conditions under both stress and displacement constraints. A study of the effect of using different starting designs and how to deal with passive members has been made. Also some suggestions which have been found to improve the results obtained are presented.

Optimality criterion techniques applied to frames having general cross-sectional relationships

An optimality criterion-based method developed in this research is capable of locating the optimal design of WF steel frames with members having general cross-sectional relationships among the area, moment of inertia, and section modulus. The method uses the idea of one most critical constraint to avoid the calculation of large sets of Lagrange multipliers, and also avoids the scaling procedure normally used in other optimality criterion techniques. Several example structures, designed under stress, displacement, and minimum area constraints, demonstrate the efficiency, reliability, and simplicity of the method. Although the method of this research uses the idea of one most critical constraint, the optimal designs presented in this paper have several active constraints.

Consistent relations in the method of reducibility in the cluster variation method

A redundancy is noted in the set of Lagrange multipliers—effective fields and effective interactions—used in the expressions of the reduced density matrices given by Morita(4) in some problems with less symmetry(6). The expressions of the reduced density matrices without this redundancy are given. An argument showing that a certain expression of the free energy in terms of the Lagrange multipliers is stationary with respect to the variations of these is presented.

The lagrange multiplier set and the generalized gradient set of the marginal function of a differentiable program in a Banach space

We prove that, under the usual constraint qualification and a stability assumption, the generalized gradient set of the marginal function of a differentiable program in a Banach space contains the Lagrange multiplier set. From there, we deduce a sufficient condition in order that, in finite-dimensional spaces, the Lagrange multiplier set be equal to the generalized gradient set of the marginal function.

A CONDITION OF JACOBI TYPE FOR A QUADRATIC FORM TO BE NONNEGATIVE ON A FINITE-FACED CONE

The author considers the problem of nonnegativity and positive definiteness of a Legendre quadratic form on a family of cones in Hilbert space that are the intersection of a fixed finite-faced cone with a monotonically increasing family of subspaces. This problem arises in the investigation to second order of an extremal in the Lagrange-Mayer-Bolza problem of classical variational calculus in the presence of finitely many additional constraints in the form of end and integral inequalities when there is only one set of Lagrange multipliers for the given extremal. It is shown that a complete answer can be given in terms of solutions of the corresponding Euler-Jacobi equation. Bibliography: 5 titles.