Von Neumann, John
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since it allows for a unified treatment of equilibrium and optimization problems. DEFINITION 1 (variational inequality problem) The finite-dimensional variational inequality problem, VI(F, K), is to determine a vector x* E K C R n, such that
(F(x*)T
- x*}
>_0,
VxEK,
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 R n. [-I We now discuss some basic problem types and their relationships to the variational inequality problem. We also provide examples. Proofs of the theoretical results may be found in [4], [5]. For algorithms for the computation of variational inequalities, see also [1], [5], [8], [7]. We begin with systems of equations, which have been used to formulate certain equilibrium problems. We then discuss optimization problems, both unconstrained and constrained, as well as complementarity problems. We conclude with a fixed point problem and its relationship with the variational inequality problem. P r o b l e m Classes. We here briefly review certain problem classes, which appear frequently in equilibrium modeling, and identify their relationships to the variational inequality problem. S y s t e m s of E q u a t i o n s . Systems of equations are common in equilibrium analysis, expressing, for example, that the demand is equal to the supply of various commodities at the equilibrium price lev-
Variational inequalities els. Let K - R n and let F" R n --+ R n be a given function. A vector x* E R n is said to solve a system of equations if
F(z*) - o . The relationship to a variational inequality problem is stated in the following PROPOSITION 2 Let K - R n and let F" R n --+ R n be a given vector function. Then x* E R n solves the variational inequality problem VI(F, K) if and only if x* solves the system of equations
system of equations"
s(p*) -d(p*) - 0 . Clearly, this expression into the standard nonlinear equation form, if we define the vectors x - p and F(x) - s(p) - d(p). Note, however, that the problem class of nonlinear equations is not sufficiently general to guarantee, for example, that x* >_ 0, which may be desirable in this example in which the vector x refers to prices. [:]
F(z*)-0. E] EXAMPLE 3 (Market equilibrium with equalities only) As an illustration, we now present an example of a system of equations. Consider m consumers, with a typical consumer denoted by j, and n commodities, with a typical commodity denoted by i. We let p denote the n-dimensional column vector of the commodity prices with components"
,;.}. Assume that the demand for a commodity i, di, may, in general, depend upon the prices of all the commodities, that is, Tr~
d, (p) - Z
(p),
j=l
where d~(p) denotes the demand for commodity i by consumer j at the price vector p. Similarly, the supply of a commodity i, si, may, in general, depend upon the prices of all the commodities, that is, m
-
5-].
J(p) ,
O p t i m i z a t i o n P r o b l e m s . Optimization problems, on the other hand, consider explicitly an objective function to be mini
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