Smoothing techniques in Euclidean Jordan algebras

We extend the powerful smoothing techniques of Yu. Nesterov to the framework of Euclidean Jordan algebras. This study allows us to design a new scheme for minimizing the largest eigenvalue of an affine function on a Euclidean Jordan algebra. We prove that its complexity is in the order of O(1/ ), where is the absolute tolerance on the value of the objective. Particularizing our result, we propose a new algorithm to minimize a sum of Euclidean norms and we perform its complete complexity analysis.