Lagrangean Technique Research Articles

Repeated moral hazard and recursive Lagrangeans

This paper shows how to solve dynamic agency models by extending recursive Lagrangean techniques à laMarcet and Marimon (2011) to problems with hidden actions. The method has many advantages with respect to the promised utilities approach (Abreu et al., 1990): it is a significant improvement in terms of simplicity, tractability and computational speed. Solutions can be easily computed for hidden actions models with several endogenous state variables and several agents, while the promised utilities approach becomes extremely difficult and computationally intensive even with just one state variable or two agents.

Reliability Optimization Model for Redundant Systems with Multiple Constraints

The Reliability Optimization Models for Redundant Systems with multiple constraints for one mathematical function is established by applying Lagrangean approach, the related case problem is presented to find the component reliabilities (rj), the number of components in each stage (xj), stage reliability (Rj) and the System Reliability (Rs). The purpose of this Paper is to optimize a class of Integrated Reliability Model for Redundant Systems with weight as additional constraint apart from basic cost constraint. By taking the mathematical function ▪ the optimum component reliability, stage reliability, the number of components in each stage and the system reliability are determined after taking the pre-determined values of Cost and Weight. In this work, an attempt is made to develop an integrated reliability redundant model for a Series – Parallel configuration subject to the multiple constraints. Generally reliability is treated as the function of Cost but in any given practical situation apart from Cost other constraint like Weight will have hidden impact on the reliability of the system. In this model the Lagrangean technique is implemented to determine the Cost and Weight as constraints. The model has yielded very encouraging results and it can be applied to any type of system, simple or complex. The advantage of this model is very flexible and requires little processing time.

Constrained multi-objective trajectory planning of parallel kinematic machines

This paper presents a new approach to multi-objective dynamic trajectory planning of parallel kinematic machines (PKM) under task, workspace and manipulator constraints. The robot kinematic and dynamic model, (including actuators) is first developed. Then the proposed trajectory planning system is introduced. It minimizes electrical and kinetic energy, robot traveling time separating two sampling periods, and maximizes a measure of manipulability allowing singularity avoidance. Several technological constraints such as actuator, link length and workspace limitations, and some task requirements, such as passing through imposed poses are simultaneously satisfied. The discrete augmented Lagrangean technique is used to solve the resulting strong nonlinear constrained optimal control problem. A decoupled formulation is proposed in order to cope with some difficulties arising from dynamic parameters computation. A systematic implementation procedure is provided along with some numerical issues. Simulation results proving the effectiveness of the proposed approach are given and discussed.

Lagrangean approach to Hamiltonian gauge symmetries and the Dirac conjecture

Using well-known Lagrangean techniques for uncovering the gauge symmetries of a Lagrangean, we derive the transformation laws for the phase space variables corresponding to local symmetries of the Hamilton equations of motion. These transformation laws are shown to coincide with those derived by Hamiltonian methods based on the Dirac conjecture. The connection between the Lagrangean and Hamiltonian approach is illustrated for first class systems involving one primary constraint.

Maximizing votes under imperfect information

This paper presents an alternative, sympathy voting, to the usual public choice approaches. It deals with voters whose decision depends both on an economic component and a sympathy component. The politician is perfectly informed of the economic component, but not of the sympathy component. The paper compares vote maximization under sympathy voting to the maximization of votes which come from pure homines oeconomici. Sympathy voting is defined in such a way that pure economic voting is its limiting case. The latter property can be used to apply the classical Lagrangean technique to maximizing votes under economic voting. The approach is applied to two typical examples of political choice: the property tax/local public expenditure decision, and public pricing of local public utilities.

A heuristic lagrangean algorithm for the capacitated plant location problem

Lagrangean techniques have been widely applied to the uncapacitated plant location problem, and in some cases they have proven to be successfull even when capacitated problems with additional constraints are taken into account. In our paper we study the application of these techniques to the capacitated plant location problem when the model considered is a pure integer one. Several lagrangean decompositions are considered and for some of them heuristic algorithms have been designed to solve the resulting lagrangean subproblems, the heuristics consisting of a two phase procedure. The first (location phase) defines a set of multipliers from the analysis of the dual LP relaxation, and makes a choice of the plants considering the resulting subproblems as a particular case of the general assignment problems. Several heuristics have been studied for this second phase, based either on a decomposition of knapsack type subproblems through a definition of a set of penalties, or of looking into the duality gap and trying to reduce it. Computational experience is reported.

A Lagrangean approach to the limit theorem on the core of an economy

It is shown that an allocation which remains in the core under replication gives rise to an utility allocation with the following property: The utility level of no class can be increased without decreasing the utility level of at least one other class even if the number of consumers is allowed to vary continuously. An examination of the first order conditions the corresponding maximum problem with constraints leads to the conclusion that the allocation can be sustained by a price vector as a competitive equilibrium. This is the limit theorem on the core, proved in this paper with a simple Lagrangean technique.

Inverse Optimization: An Application to the Capacitated Plant Location Problem

Lagrangean techniques have had wide application to the optimization of discrete optimization problems. Inverse optimization refers to the fact that each time a Lagrangean calculation is made for a specific problem with a given resources vector, an optimal solution is obtained for a related problem with a suitably adjusted resources vector. This property is studied in depth for the capacitated plant location problem and new parametric methods for that problem are suggested. Computational experience is reported.

The Economic Significance of Auxiliary Functions in Optimal Control

IN THE OPTIMIZATION of an econometric model using a Lagrangean technique, it is sometimes enlightening to regard a Lagrange multiplier as a marginal rate of change of one quantity with respect to another. Having determined optimal values for the independent or decision variables, one can infer from the values, of the multipliers how the optimal value of the objective function will change with respect to certain changes in the model. In particular, it may be possible to infer from the multipliers the benefit to be gained by altering certain of the constraints imposed on the decision variables. (See Beranek [3, (135 et. seq.)] for an example of this.) The multipliers can be interpreted in certain model as. prices on raw materials. (See for example Zangwill [13, (62-68)] for a discussion of this.) There are at least two senses in which the multipliers may be regarded as rates of change of the optimal value of the objective function, and these are pointed out in the next section for a class of nonlinear programming problems which includes linear problems as a special case. Optimal control problems of the type examined by Pontryagin [12] involve auxiliary functions which are infinite dimensional counterparts to the Lagrange multipliers encountered in nonlinear programming. These functions, too, have interpretations as rates of change of one quantity with respect to another. Although some of these interpretations have been recognized (Dorfman [5], Intrilligator [6], and Lee and Markus [8, (347 et. seq.)] for continuous time. problems, Benavie [2] for the discrete time case), there are many that have apparently escaped notice. In the third section we illustrate some of these interpretations using variations of a growth model analyzed by Arrow [1] and subsequently by Dorfman [5]. This final section contains four theorems on the interpretatior of auxiliary functions which extend the results given in the examples.