A Fresh Geometrical Look at the General S-Procedure

Michel de Lara (1, 2) , Jean-Baptiste Hiriart-Urruty (3)

1 ENPC - École des Ponts ParisTech
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 IMT - Institut de Mathématiques de Toulouse UMR5219

We revisit the S-procedure for general functions with "geometrical glasses". We thus delineate a necessary condition, and almost a sufficient condition, to have the S-procedure valid. Everything is expressed in terms of convexity of augmented sets (convex hulls, conical hulls) of images built from the data functions.

Mots clés: Theorem of alternatives; Separation of convex sets; Convexity of image sets; S-lemma

Domaines: Optimisation et contrôle [math.OC]

Michel de Lara, Jean-Baptiste Hiriart-Urruty. A Fresh Geometrical Look at the General S-Procedure. Optimization Letters, 2021, 16, pp.1129-1135. ⟨10.1007/s11590-021-01789-y⟩. ⟨hal-03137154v2⟩