Uniqueness in the theory of variational inequalities

AbstractFor a given convex set K in Rn, we look for the conditions on the matrix A which ensure uniqueness of the solution x∗ in K to the standard variational inequality: (Ax∗ − b)T)(x − x∗) ⩾ 0 for all x in K. If K is the unit ball, or any smooth compact strictly convex set, then x∗ is unique for every b if and only if A is both invertible and semidefinite. On the unit cube, the condition on A is the same as the one discovered by Samelson, Thrall, and Wesler for K = R+n: all principal minors must have positive determinants. We briefly discuss the generalizations to infinite dimensions