The S-procedure or S-lemma is a mathematical result that gives conditions under which a particular quadratic inequality is a consequence of another quadratic inequality. The S-procedure was developed independently in a number of different contexts and has applications in control theory, linear algebra and mathematical optimization.
Let F<sub>1</sub> and F<sub>2</sub> be symmetric matrices, g<sub>1</sub> and g<sub>2</sub> be vectors and h<sub>1</sub> and h<sub>2</sub> be real numbers. Assume that there is some x<sub>0</sub> such that the strict inequality holds. Then the implication
holds if and only if there exists some nonnegative number û such that