Automated Algorithm Design via Nevanlinna-Pick Interpolation

Abstract: The synthesis of optimization algorithms typically follows a design-first-analyze-later approach, which often obscures fundamental performance limitations and hinders the systematic design of algorithms operating at the achievable theoretical boundaries. Recently, a framework based on frequency-domain techniques from robust control theory has emerged as a powerful tool for automating algorithm synthesis. By integrating the design and analysis stages, this framework enables the identification of fundamental performance limits. In this paper, we build on this framework and extend it to address algorithms for solving strongly convex problems with equality constraints. As a result, we obtain a new class of algorithms that offers sharp trade-off between number of matrix multiplication per iteration and convergence rate.