On the s-procedure and some variants

In this thesis, we deal with the S-procedure that corresponds to verifying that the minimum of a quadratic function over constraints consisting of quadratic functions is positive. S-procedure is an instrumental tool in control theory and robust optimization analysis. It is also used in linear matrix inequality (or semi definite programming) reformulations and analysis of quadratic programming. We improve an error bound in the Approximate S-Lemma used in establishing levels of conservatism results for approximate robust counterparts. Moreover we extend the S-procedure and obtain some general results in this field. Finally, we get a bound similar to Nesterov’s bound for trust region subproblem, which consists in minimizing an indefinite quadratic function subject to a norm-1 constraint by using the Approximate S-Lemma.