  We consider a misspecified optimization problem that requires minimizing a
function f(x;q*) over a closed and convex set X where q* is an unknown vector
of parameters that may be learnt by a parallel learning process. In this
context, We examine the development of coupled schemes that generate iterates
{x_k,q_k} as k goes to infinity, then {x_k} converges x*, a minimizer of
f(x;q*) over X and {q_k} converges to q*. In the first part of the paper, we
consider the solution of problems where f is either smooth or nonsmooth under
various convexity assumptions on function f. In addition, rate statements are
also provided to quantify the degradation in rate resulted from learning
process. In the second part of the paper, we consider the solution of
misspecified monotone variational inequality problems to contend with more
general equilibrium problems as well as the possibility of misspecification in
the constraints. We first present a constant steplength misspecified
extragradient scheme and prove its asymptotic convergence. This scheme is
reliant on problem parameters (such as Lipschitz constants)and leads us to
present a misspecified variant of iterative Tikhonov regularization. Numerics
support the asymptotic and rate statements.
