2025/12/14 by Dmitry Vorotnikov, Amit Acharya, Vorotnikov, Dmitry +1 · 3 citations
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Distributed Control Multi-Agent Systems #Optimization and Variational Analysis #cs.NA #math.AP #math.NA #math.OC
paper · pdf · doi:10.48550/arxiv.2512.12878
openalex publication_date 2025/12/14 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28
We investigate the influence of base states on the consistency of the dual variational formulation for quadratic systems of PDEs, which are not necessarily conservative (typical examples include the noise-free Nash system with a quadratic Hamiltonian and multiple players). We identify a sufficient condition under which consistency holds over large time intervals. In particular, in the single-player case, there exists a sequence of base states (each exhibiting full consistency) that converges in mean to zero. We also prove existence of variational dual solutions to the noise-free Nash system for arbitrary base states. Furthermore, we propose a scheme based on Hilbertian gradient flows that, starting from an arbitrary base state, generates a sequence of new base states that is expected to converge to a solution of the original PDE.