# Primal-dual splitting method ###### tags: `11th-joint-workshop` * L. Condat, “[A primal–dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms](https://hal.archives-ouvertes.fr/hal-00609728v5/document),” Journal of Optimization Theory and Applications, vol. 158, 08 2013. $$ \begin{aligned} x^{[n+1]} &= \mathrm{prox}_{\sigma_1 g} \left[ x^{[n]} - \sigma_1 \left(\nabla f(x^{[n]}) + \Phi^* z^{[n]}\right) \right], \\ z^{[n+1]} &= \mathrm{prox}_{\sigma_2 h^*} \left[ z^{[n]} + \sigma_2\Phi \left(2x^{[n+1]} - x^{[n]}\right) \right]. \end{aligned} $$ where 1. $\Phi = D_t E_{\mathrm{iPC}} E_{\mathrm{PC}}\mathscr{F}^w$ ## Convergence condition of the algorithm Arguments $\sigma_1$ and $\sigma_2$ should satisfy the following inequality $$ \frac{1}{\sigma_1} - \sigma_2 \| \Phi \|_{\mathrm{op}}^2 \geq \frac{1}{2}. $$ So we need to estimate the operator norm of $\Phi$. > But finding the exact solution is somehow not an easy job, so we estimate the upper bound of $\|\Phi\|_{\mathrm{op}}$ instead. > [color=pink] $$ \| D_t E_{\mathrm{iPC}} E_{\mathrm{PC}}\mathscr{F}^w \|_{\mathrm{op}} \leq \| D_t \|_{\mathrm{op}} \cdot \sup_{i, j} | E_{\mathrm{iPC}} (i, j) | \cdot \sup_{i, j} | E_{\mathrm{PC}} (i, j) | \cdot \| \mathscr{F} \|_{\mathrm{op}} \cdot \sup_{i} | w (i) | $$ > 1. $D_t: \mathbb{R}^{n'\times N}\to \mathbb{R}^{n'\times (N-1)}$ > 2. $E_{\mathrm{PC}} (i, j)$ and $E_{\mathrm{iPC}} (i, j)$ are piecewise scalar multiplications > 3. $\mathscr{F}$ is the specified Vandermonde matrix. > 4. $w$ is a window function (Hann).