Algorithm 1: The solution of Equation (3). The Newton step is computed with linear conjugate gradient.

Input: $S=(<(x_1,1),\cdots ,(x_n,n)>,V,W,C)$

${\psi }_i=\frac{{\displaystyle \underset{t=1}{\sum ^i}}{v}_t\ast x_t}{i}$

$\omega (i,j)=max(w_i,w_j)$

$q_k\in Q=\left\{\omega (i,j)\right|i>j\}$

u is randomly iniatlized

repeat

$D_{kk}=\mathbf{1}(u^T{q}_k<1)$

$g=u+2C{Q}^{T}D(Q^{T}u-1)$

repeat

Update based on the computation of $s+2C{Q}^{T}D\left(Qs\right)$ for some s.

until Convergence of linear conjugate gradient

$\delta u={(I+2C{Q}^{T}DQ)}^{-1}g$

$u\leftarrow u-\tau \delta u$ ($\tau$ found by line gradient)

until Convergence of Newton

return u