Year: 2023
Type: article
Abstract: Optimization problems are a staple of today’s scientific and technical landscape. However, at present, solvers of such problems are almost exclusively run on digital hardware. Using Turing machines as a mathematical model for any type of digital hardware, in this paper, we analyze fundamental limitations of this conceptual approach of solving optimization problems. Since in most applications, the optimizer itself is of significantly more interest than the optimal value of the corresponding function, we will focus on computability of the optimizer. In fact, we will show that in various situations the optimizer is unattainable on Turing machines and consequently on digital computers. Moreover, even worse, there does not exist a Turing machine, which approximates the optimizer itself up to a certain constant error. We prove such results for a variety of well-known problems from very different areas, including artificial intelligence, financial mathematics, and information theory, often deriving the even stronger result that such problems are not Banach-Mazur computable, also not even in an approximate sense.
Authors: Yunseok Lee, Holger Boche, Gitta Kutyniok
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Hardware-Plattform und Computing-Modell für neuromorphes Rechnen - NeuroCM -, G1253059744, G1289050001, G2125983667, G3444685969, Verbundprojekt: Digitale Transformation und Souveränität künftiger Kommunikationsnetze - 6G-life -, Teilvorhaben: Grundlagen und Systemlösungen für Post Shannon Kommunikation, In-Network Computing, eingebettete Sicherheit und Resilienz, G4792813178, G5273600302, G7472567491
OpenAlex ID: W4390187812