The problem of designing multivariable stabilization controllers that satisfy an optimal condition is considered in this paper. Specifically, it is required to find a robust stabilization controller so that closed-loop stability and asymptotic regulation occur, and also so that a quadratic performance index will be minimized. Necessary and sufficient conditions for a robust optimal output-feedback stabilization are presented. The gradient is considered over a functional defined on matrix spaces. Conditions for an optimal point are derived for this kind of functional. Finally, an algorithm for the robust optimal output-feedback design and an example are given.