Dual problems, which we call output and input variance constrainedH2/H∞controls, are considered. In these problems, we seek control-laws that satisfy mixedH2/H∞performance criteria, under multiple variance constraints on either outputs or inputs of time-invariant multivariable systems. The approach taken is to convert the problems into non-linear programming with both equality and inequality constraints. For both problems, the Kuhn-Tucker optimality condition is employed to obtain a first-order necessary condition for a regular point that minimizes an upper bound on the quadratic performance for the givenH∞constraint. A second-order necessary condition and sufficiency for the strict local minimizer of the upper bound are investigated. Efficient algorithms for synthesizing the desired controllers are proposed.