A Newton Collocation Method for Solving Dynamic Bargaining Games

We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.

Solving Systems of Non-Linear Equations by Broyden's Method with Projected Updates

We introduce a modification of Broyden's method for finding a zero of n nonlinear equations in n unknowns when analytic derivatives are not available. The method retains the local Q-superlinear convergence of Broyden's method and has the additional property that if any or all of the equations are linear, it locates a zero of these equations in n+1 or fewer iterations. Limited computational experience suggests that our modification often improves upon Eroyden's method.

Improving Tatonnement Methods for Solving Heterogeneous Agent Models

This paper modifies standard block Gauss-Seidel iterations used by tatonnement methods for solving large scale deterministic heterogeneous agent models. The composite method between first- and second-order tatonnement methods is shown to considerably improve convergence both in terms of speed as well as robustness relative to conventional first-order tatonnement methods. In addition, the relative advantage of the modified algorithm increases in the size and complexity of the economic model. Therefore, the algorithm allows significant reductions in computational time when solving large models. The algorithm is particularly attractive since it is easy to implement - it only augments conventional and intuitive tatonnement iterations with standard numerical methods.

Improving Tatonnement Methods of Solving Heterogeneous Agent Models

This paper modifies standard block Gauss-Seidel iterations used by tatonnement methods for solving large scale deterministic heterogeneous agent models. The composite method between first- and second-order tatonnement methods is shown to considerably improve convergence both in terms of speed as well as robustness relative to conventional first-order tatonnement methods. In addition, the relative advantage of the modified algorithm increases in the size and complexity of the economic model. Therefore, the algorithm allows significant reductions in computational time when solving large models. The algorithm is particularly attractive since it is easy to implement – it only augments conventional and intuitive tatonnement iterations with standard numerical methods.

On the Modifications of a Broyden's Single Parameter Rank-Two Quasi-Newton Method for Unconstrained Minimization

The thesis concerns mainly in finding the numerical solution of non-linear unconstrained problems. We consider a well-known class of optimization methods called the quasi-Newton methods, or variable metric methods. In particular, a class of quasi-Newton method named Broyden's single parameter rank two method is focussed. We also investigate the global convergence properties for some step-length procedures. Immediately from the investigations, a global convergence proof of the Armijo quasi-Newton method is given. Some preliminary modifications and numerical experiments are carried out to gain useful numerical experiences for the improvements of the quasi-Ne"-'ton updates.We then derived two improvement techniques: the first we employ a switching criteria between quasi-Newton Broyden-Fletcher-Goldfrab-Shanno or BFGS and steepest descent direction and in the second we introduce a reduced trace-norm condition BFGS update. The thesis includes results illustrating the numerical performance of the modified methods on a chosen set of test problems. Limitations and some possible extensions are also given to conclude this thesis

Why Broyden's Nonsymmetric Method Terminates on linear equations

Abstract. The family of algorithms introduced by Broyden in 1965 for solving systems of nonlinear equations has been used quite effectively on a variety of problems. In 1979, Gay proved the then surprising result that the algorithms terminate in at most 2n steps on linear problems with n variables. His very clever proof gives no insight into properties of the intermediate iterates, however. In this work we show that Broyden's methods are projection methods, forcing the residuals to lie in a nested set of subspaces of decreasing dimension. (Also cross-referenced as UMIACS-TR-93-23

Computational Experiments with Systems of Nonlinear Equations

Higher Educatio

Nonlinear structural dynamics via Newton and quasi-Newton methods

This paper is an attempt to compare Newton and quasi-Newton methods in nonlinear structural dynamics. After a review of the classical iterative methods, several quasi-Newton updates are presented and tested. Special attention is devoted to the solution of large sparse systems for which two original procedures are described: a substructure correction and a vectorial correction. The numerical examples presented include the dynamic analyses of geometrical, material and combined nonlinearities. All the results are assorted with a complete discussion of the different methods used, of the convergence rates and of the associated computer costs. From the present results, Newton's methods appear to exhibit the best convergence rates when an efficient computational strategy is adopted. Nevertheless computational costs for the solution of large systems can be reduced drastically by using convenient quasi-Newton updates