Projections onto the Intersection of a One-Norm Ball or Sphere and a Two-Norm Ball or Sphere

This paper focuses on designing a unified approach for computing the projection onto the intersection of a one-norm ball or sphere and a two-norm ball or sphere. We show that the solutions of these problems can all be determined by the root of the same piecewise quadratic function. We make use of the special structure of the auxiliary function and propose a novel bisection algorithm with finite termination. We show that the proposed method possesses quadratic time worst-case complexity. The efficiency of the proposed algorithm is demonstrated in numerical experiments, which show the proposed method has linear time complexity in practice.