Variational Inequalities

Background Equilibrium is a central concept in numerous disciplines including economics, management science/operations research, and engineering. Methodologies that have been applied to the formulation , qualitative analysis, and computation of equilibria have included • systems of equations, • optimization theory, • complementarity theory, and • fixed point theory. Variational inequality theory is a powerful unifying methodology for the study of equilibrium problems. 1 Variational inequality theory was introduced by Hart-man and Stampacchia (1966) as a tool for the study of partial differential equations with applications principally drawn from mechanics. Such variational inequalities were infinite-dimensional rather than finite-dimensional as we will be studying here. The breakthrough in finite-dimensional theory occurred in 1980 when Dafermos recognized that the traffic network equilibrium conditions as stated by Smith (1979) had a structure of a variational inequality. This unveiled this methodology for the study of problems in economics, management science/operations research , and also in engineering, with a focus on transportation. 2 To-date problems which have been formulated and studied as variational inequality problems include: • traffic network equilibrium problems • spatial price equilibrium problems • oligopolistic market equilibrium problems • financial equilibrium problems • migration equilibrium problems, as well as • environmental network problems, and • knowledge network problems. Variational inequality theory provides us with a tool for: formulating a variety of equilibrium problems; qualitatively analyzing the problems in terms of existence and uniqueness of solutions, stability and sensitivity analysis, and providing us with algorithms with accompanying convergence analysis for computational purposes. It contains, as special cases, such well-known problems in mathematical programming as: systems of nonlin-ear equations, optimization problems, complementarity problems, and is also related to fixed point problems. The finite-dimensional variational inequality problem, VI(F, K), is to determine a vector x * ∈ K ⊂ R n , such that F (x *) T · (x − x *) ≥ 0, ∀x ∈ K, or, equivalently, F (x *) T , x − x * ≥ 0, ∀x ∈ K (1) where F is a given continuous function from K to R n , K is a given closed convex set, and ·, ·· denotes the inner product in n dimensional Euclidean space. In geometric terms, the variational inequality (1) states that F (x *) T is " orthogonal " to the feasible set K at the point x *. This formulation, as shall be demonstrated, is particularly convenient because it allows …