Variational inequalities in the modelling and computation of spatial economic equilibria: Structural reformulations and the method of multipliers

Variational inequalities have been used to study problems involving partial differential equations with unilateral constraints, such as free-boundary problems. They have also gained much recent interest in the field of operations research, particularly in the study of competitive equilibrium problems. The main focus of this work is to develop efficient algorithms for the computation of large-scale economic equilibria under weaker conditions than those considered previously. The prototype that we use in the analysis is the spatial market equilibrium system with direct price functions. We take advantage of the special structure of the variational inequalities, hence reformulate the problems, via a dual approach of Mosco and a linear algebra argument, as multivalued equations involving two maximal monotone operators. We then apply a relaxed proximal point method with variable parameters to the new formulation. In finite dimensions, we prove that the splitting sequences so generated are convergent to the equilibrium and the Lagrange multipliers, respectively. We also develop variational inequality formulations for migration networks and spatial market systems with goaling constraints. Based on the given economic equilibrium conditions, we establish the corresponding variational inequality formulations. In the second case, we provide direct equivalence proof that is motivated by the governing economic conditions. Essentially, we establish that the economic conditions are the dual forms of the corresponding variational inequalities. By applying the theory of variational inequalities, we then study the qualitative properties of these spatial equilibrium systems. In particular, we show the existence and uniqueness of the equilibrium in each case, assuming some monotonicity conditions that can be interpreted economically. We then apply the above numerical scheme to the variational inequality formulations of spatial equilibrium systems. As a result, we obtain a class of methods of multipliers for the computation of the studied economic equilibria. The methods so derived have an important feature that they require only monotonicity instead of strong monotonicity of supply price functions and demand price functions. They still require strong monotonicity of transaction cost functions. Finally, since they are splitting algorithms, they are suitable for decomposing large-scale problems. With a sequence of penalty parameters being set properly, each split part can then be computed sequentially or parallelly