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Characterizations of the Solution Sets of Convex Programs and Variational Inequality Problems1 Abstract:For a convex program in a normed vector space with the objective function admitting the Gateaux derivative at an optimal solution, we show that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the Gateaux derivative. For a general continuous convex program, a feasible point is an optimal solution iff it lies in a hyperplane with a normal vector belonging to the subdifferential of the objective function at this point. In several cases, the solution set of a variational inequality problem is shown to coincide with the solution set of a convex program with its dual gap function as objective function, while the mapping involved can be used to express the above normal vectors. KeyWords:Convex programs, Gateaux derivatives,variational inequalities, dual gap function, pseudomonotonicity. 1. Introduction For a convex program min{:C}, where C is a convex set in Rn andis a convex function on Rn，Mangasarian has proved in Ref.1 that its solution set can be characterized completely by a constant gradient of , whenis twice continuously differentiable on some open convex set containing C, and by the subdifferential off, whenis continuous and the relative interior of the solution set is nonempty. According to the nature of the solution set of a convex program, one can understand with more depth several important optimization problem models(see Ref. 2). It is worthy to extend the above result to a more general convex program. In the nonsmooth case, Burke and Ferris (Ref. 2) have presented another more specific characterization of the solution set, while Jeyakumar (Ref. 3) has extended the Mangasarian result to the case whereis a lower semicontinuous convex function on a Banach space with a generalized interior condition. Even for a pseudolinear program, the Managasarian result has an interesting extension; see Jeyakumar and Y