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Math. Meth. Oper. Res. (2006) 64: 5577DOI 10.1007/s00186-006-0070-8
Received: 15 August 2004 / Accepted: 24 August 2005 / Published online: 9 June 2006
Springer-Verlag 2006Abstract We give a concise review and extension of S-procedure that is an
instrumental tool in control theory and robust optimization analysis. We also discuss the approximate S-Lemma as well as its applications in robust optimization.Keywords S-Procedure S-Lemma Robust optimization Control theory1 IntroductionThe purpose of this paper is to give a concise review of recent developments related
to the S-procedure in a historical context as well as to offer a new extension.
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. It was given in 1944 by
Lure and Postnikov without any theoretical justification. Theoretical foundations of
S-procedure were laid in 1971 by Yakubovich and his students (Yakubovich 1971).S-procedure deals with the nonnegativity of a quadratic function on a set described by quadratic functions and provides a powerful tool for proving stability
of nonlinear control systems. For simplicity, if the constraints consist of a single
quadratic function, we refer to it as S-Lemma. If there are at least two quadratic
inequalities in the constraint set, we use the term of S-procedure. YakubovichThe many suggestions and detailed corrections of an anonymous referee are gratefully
acknowledged.K. DerinkuyuLehigh University, Bethlehem, USAM. . Pnar (B)Bilkent University, Ankara, TurkeyE-mail: [email protected] ARTICLEKrsad Derinkuyu Mustafa . PnarOn the S-procedure and some variants56 K. Derinkuyu and M. . Pnar(1971) was the first to prove the S-Lemma and to give a definition of S-procedure.
Recently, Polyak (1998) gave a result related to S-procedure for problems involving
two quadratic functions in the constraint set.Although the S-Lemma was proved in 1971, results on the convexity problems
of quadratic functions were already there since 1918. From Toeplitz (1918) and
Hausdorff (1919) theorem to more recent results, many important contributions to
the field are available. In this period, not only the S-Lemma was improved, but also
a new result was introduced, called the approximate S-Lemma. The approximate
S-Lemma developed by Ben-Tal et al. (2002) establishes a bound for problems with
more than one constraints of quadratic type. Their result also implies the S-Lemma
of...