Fréchet Discrete Gradient and Hessian Operators on Infinite-Dimensional Spaces

Moreschini A., GÖKSU G., Parisini T.

7th IFAC Conference on Analysis and Control of Nonlinear Dynamics and Chaos, ACNDC 2024, London, İngiltere, 5 - 07 Haziran 2024, cilt.58, ss.78-83 identifier

  • Yayın Türü: Bildiri / Tam Metin Bildiri
  • Cilt numarası: 58
  • Doi Numarası: 10.1016/j.ifacol.2024.07.067
  • Basıldığı Şehir: London
  • Basıldığı Ülke: İngiltere
  • Sayfa Sayıları: ss.78-83
  • Anahtar Kelimeler: Discrete gradients, Fréchet derivative, Geometric integration on Banach spaces, Infinite-dimensional convex optimization, Infinite-dimensional spaces, Structural preservation
  • Yıldız Teknik Üniversitesi Adresli: Evet

Özet

Benefiting from the notion of Fréchet derivatives, we define Fréchet discrete operators, such as gradient and Hessian, on infinite-dimensional spaces. The Fréchet discrete gradient expands upon the concept of the discrete gradient of Gonzalez (1996) for finite-dimensional spaces. The Fréchet discrete Hessian elevates the property to second-order representations of the Fréchet derivative. By leveraging these operators, we offer an initial exploration of discrete gradient methods for convex optimization in infinite-dimensional spaces. Under mild conditions on the objective functional, we establish the convergence of any sequence generated by the proposed Fréchet discrete gradient method, regardless of the choice of the finite learning rate.