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, être résolue avec une précision arbitraire en utilisant une hiérarchie de programmes semi-définis. On trouve plus d'informations sur les applications de la programmation semi-définie en optimisation polynomiale dans les livres

, pour plus d'informations sur la programmation semi-définie et ses applications. En pratique, on résout des programmes semi-définis en utilisant des méthodes de points intérieurs. Ces méthodes ont été généralisées de la programmation linéaire à la programmation semidéfinie par Alizadeh, Grâce à sa puissance expressive, la programmation semi-définie a des applications dans la théorie du contrôle [BEGFB94], en information quantique

, Il y a beaucoup des questions ouvertes sur les spectraèdres et la programmation semi-définie. Par exemple, Nemirovski [Nem07] a demandé de caractériser les spectraèdres projetés. Helton et Nie [HN09] ont conjecturé que tout ensemble semi-algébrique convexe est une projection d'un spectraèdre, Nous renvoyons à [dK02, Ren01, GM12] pour plus d'informations sur les méthodes de points intérieurs

, De plus, on sait qu'elle est vraie en dimension 2

L. Cependant, . Conjecture-À-Été-récemment-réfutée-par, and . Scheiderer, qui a montré que le cône des formes semi-définies positives n'est pas une projection d'un spectraèdre, sauf pour quelques cas particuliers [Sch18b]. Son article contient une liste exhaustive de références. La conjecture généralisée de Lax est une autre question ouverte. Elle demande si tout cône d'hyperbolicité est un spectraèdre. La réponse est positive pour plusieurs classes de ces cônes

, Nous renvoyons aux travaux cités pour plus d'informations. La géométrie des spectraèdres a été étudié, Néanmoins, quelques généralisations de la conjecture sont fausses

, Une solution exacte au programme semi-défini est un nombre algébrique et le degré de son polynôme minimal peut être élevé [NRS10]. De plus, les méthodes mentionnées ci-dessus dépendent d'hypothèses supplémentaires sur la structure d'un spectraèdre sous-jacent. Nous renvoyons à [Ram97, dKV16, LMT15] pour plus d'informations et à [LP18] pour une classe de programmes semi-définis de petite taille qui sont difficiles pour les logiciels contemporains. Les méthodes de pointe qui sont capables de résoudre les programmes semidéfinis généraux sont fondées sur les méthodes de points critiques, Cependant, il y a plusieurs questions ouvertes dans cette domaine. Par exemple, on ne comprend pas bien la structure faciale des spectraèdres et des cônes d'hyperbolicité. Dans cette thèse, nous nous intéressons à la complexité théorique de la programmation semidéfinie. Dans le modèle de machine du Turing