Differential properties of Euclidean projection onto power cone

In this paper, we study differential properties of Euclidean projection onto the high dimensional power cone $$\begin{aligned} K^{\alpha }_{m,n}=\left\{ (x,z)\in \mathbb {R}^m_+ \times \mathbb {R}^n, \left| \left| {z} \right| \right| \le \prod \nolimits _{i=1}^m x_i^{\alpha _i}\right\} , \end{aligned}$$ K m , n α = ( x , z ) ∈ R + m × R n , z ≤ ∏ i = 1 m x i α i , where $$0>\alpha _i, \sum \nolimits _{i=1}^m \alpha _i=1$$ 0 > α i , ∑ i = 1 m α i = 1 . Projector’s formulas, its directional derivative formulas, its first order Fréchet derivative formulas are considered. Copyright Springer-Verlag Berlin Heidelberg 2015