  This paper considers min-max optimization for a class of robust nonlinear
least squares problems. We show via an example that solving the first order
optimality conditions defined by gradients of the objective function can lead
to incorrect solutions of min-max problems. We give an explicit formula for the
value function of the inner maximization problem. Using the formula, we show
that finding a first order $\epsilon$-approximate necessary minimax point of
the min-max problem needs at most $O(|\log\epsilon| +\epsilon^{-2}) $
evaluations of the function value and gradients of the objective function.
Moreover, we establish error bounds from any solution of the nonlinear least
squares problem to the solution set of the robust nonlinear least squares
problem.
