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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 Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
,
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
and
mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
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Statement of the S-procedure

Let F1 and F2 be symmetric matrices, g1 and g2 be vectors and h1 and h2 be real numbers. Assume that there is some x0 such that the strict inequality x_0^T F_1 x_0 + 2g_1^T x_0 + h_1 < 0 holds. Then the implication ::x^T F_1 x + 2g_1^T x + h_1 \le 0 \Longrightarrow x^T F_2 x + 2g_2^T x + h_2 \le 0 holds if and only if there exists some nonnegative number λ such that :: \lambda \begin F_1 & g_1 \\ g_1^T & h_1 \end - \begin F_2 & g_2 \\ g_2^T & h_2 \end is positive semidefinite.


References

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See also

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Linear matrix inequality In convex optimization, a linear matrix inequality (LMI) is an expression of the form : \operatorname(y):=A_0+y_1A_1+y_2A_2+\cdots+y_m A_m\succeq 0\, where * y= _i\,,~i\!=\!1,\dots, m/math> is a real vector, * A_0, A_1, A_2,\dots,A_m are n\times n ...
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Finsler's lemma Finsler's lemma is a mathematical result named after Paul Finsler. It states equivalent ways to express the positive definiteness of a quadratic form ''Q'' constrained by a linear form ''L''. Since it is equivalent to another lemmas used in optimi ...
Control theory Linear algebra Mathematical optimization