Using gap functions to devise error bounds for some special
classes of monotone variational inequality is a fruitful venture since it
allows us to obtain error bounds for certain classes of convex
optimization problem. It is to be noted that if we take a Hoffman type
approach to obtain error bounds to the solution set of a convex
programming problem it does not turn out to be fruitful and thus using the
vehicle of variational inequality seems fundamental in this case. We begin
the discussion by introducing several popular gap functions for
variational inequalities like the Auslender gap function and the
Fukushima's regularized gap function and how error bounds can be created
out of them. We then also spent a brief time with gap functions for
variational inequalities with set-valued maps which correspond to the
non-smooth convex optimization problems. We then quickly shift our focus
on the creating error bounds using the dual gap function which is possibly
the only convex gap function known in the literature to the best of our
knowledge. In fact this gap function was never used for creating error
bounds. Error bounds can be used as stopping criteria and this the dual
gap function can be used to solve the variational inequality and also be
used to develop a stopping criteria. We present several recent research on
error bounds using the dual gap function and also provide an application
to quasiconvex optimization.