This paper presents a tractable method of solving a non-convex, nonlinear optimization problem formulated for robust static attitude determination based on a least squares approach with nonlinear constraints. Considering infinity-norm bounded uncertainties, this robust min-max problem is approximated with a minimization problem, although the objective function and constraints are still nonlinear. We propose an additional regularization term to improve the robust performance. We then use semidefinite relaxation to convert the approximate nonlinear optimization problem into a tractable semidefinite program with a linear objective and linear matrix inequality constraints. We show how to extract the solution of the nonlinear optimization problem from the solution of the semidefinite relaxation. Numerical simulations suggest that the gap between the considered problem and its relaxation is zero.