A decomposition method to deflate the curse of dimensionality for dynamic stochastic models

Computing equilibria in dynamic economies is still quite challenging even though the noticeable increase in computing power, storage capacity and new approaches in the literature on computational economics. The solvability of many economic models suffers from the curse of dimensionality, which limits the planning horizon practitioners can afford for mapping a real problem into a numerically solvable dynamic model. Consequently, many standard algorithms are computationally burdensome. We propose a decomposition procedure to deflate the dimensionality problem by splitting it into manageable pieces and coordinating their solution. There are two main computational advantages in the use of decomposition methods. First, the subproblems are, by definition, smaller than the original problem and therefore much faster to solve. Second, the subproblems could have special properties such as convexity and sparsity that enable the use of efficient algorithms to solve them. Previous decomposition algorithms break into three groups: Danzting-Wolfe decomposition, Benders decomposition and augmented Lagrangian relaxation procedure. Both Danzting-Wolfe decomposition (see Danzting and Wolfe 1960) and Benders decomposition (see Benders 1962 and Geoffrion 1972) are efficient schemes to deal with convex programming problems. Extension to nonconvex problems is the augmented Lagrangian relaxation (see Rockafellar and Wets 1991), based on an estimate of the Lagrange multipliers to decompose the problem into a set of subproblems, which solutions are used to update the current estimate of the Lagrange multipliers. Applications of decomposition methods to general equilibrium computation are originally due to Mansur and Whalley (1982). They apply Danzting and Wolfe's decomposition procedure to Scarf's (1982) algorithm to improve the speed of convergence. In contrast, the current paper considers the extension of Lagrangian decomposition methods to compute dynamic economic models in their original form. Our results show the computational gain achieved through its way to break the problem into smaller subproblems and its robustness against misspecifications.