This paper studies the existence of a (Lipschitz) continuous (single-valued) solution function of parametric variational inequalities under functional and constraint perturbations. At the most elementary level, this issue can be explained from classical parametric linear programming and its resolution by the parametric simplex method, which computes a solution trajectory of the problem when the objective coefficients and the right-hand sides of the constraints are parameterized by a single scalar parameter. The computed optimal solution vector (and not the optimal objective value) is a continuous piecewise affine function in the parameter when the objective coefficients are kept constant, whereas the computed solution vector can be discontinuous when the right-hand constraint coefficients are kept fixed and there is a basis change at a critical value of the parameter in the objective. We investigate this issue more broadly first in the context of an affine variational inequality (AVI) and obtain results that go beyond those pertaining to the lower semicontinuity of the solution map with joint vector perturbations; the latter property is closely tied to a
stability theory of a parametric AVI and in particular to Robinson's seminal concept of strong regularity. Extensions to nonlinear variational inequalities {are} also investigated without requiring solution uniqueness (and therefore applicable to non-strongly regular problems). The role of solution uniqueness in this issue of continuous single-valued solution selection is further clarified.
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