This paper considers the design of a stabilizing static output feedback gain which keeps the linear quadratic (LQ) performance index less than a specified number (we call this an 'LQ suboptimal controller'). Existence of such a controller is shown to be equivalent to the existence of a positive-definite matrix P such that P satisfies two linear matrix inequalities (LMIs) while P-1 satisfies another LMI. All LQ suboptimal controllers are explicitly parameterized by the freedom in the choice of the positive-definite matrix P satisfying the LMIs, and an arbitrary positive scalar and an arbitrary matrix of fixed dimension with a norm bound. A modified version of the min/max algorithm is given to find a positive-definite solution P to the LMIs.236421430