Reduced form of the Primal Dual Interior Point residual and the action of its Jacobian. More...

#include <ROL_PrimalDualInteriorPointReducedResidual.hpp>

Detailed Description

Reduced form of the Primal Dual Interior Point residual and the action of its Jacobian.

     The orginal system has the form

\[ \begin{pmatrix} W & J^\ast & -I & I \\ J & 0 & 0 & 0 \\ Z_l & 0 & X-L & 0 \\ -Z_u & 0 & 0 & U-X \end{pmatrix} \begin{pmatrix} d_x \\ d_\lambda \\ d_{z_l} \\ d_{z_u} \end{pmatrix} = -\begin{pmatrix} \nabla f+J^\ast \lambda -z_l + z_u \\ c \\ (x-l)z_l - \mu e \\ (u-x)z_u - \mu e \end{pmatrix} = -\begin{pmatrix} g_x \\ g_\lambda \\ g_{z_l} \\ g_{z_u} \]

    Using the last two equations, we have

\[ d_{z_l} = -(X-L)^{-1} g_{z_l} - (X-L)^{-1}Z_l d_x \]

\[ d_{z_u} = -(U-X)^{-1} g_{z_u} + (U-X)^{-1}Z_u d_x \]

    Substituting into the first equation, we get

\[ [W+(X-L)^{-1}Z_l+(U-X)^{-1}Z_u]d_x + J^\ast d_\lambda = -g_x + (U-X)^{-1}g_{z_u} - (L-X)^{-1}g_{z_l} \]

    This leads to the reduced system

\[ \begin{pmatrix} W+\Sigma_l+\Sigma_u & J^\ast \\ J & 0 \end{pmatrix} \begin{pmatrix} d_x \\ d_\lambda \end{pmatrix} = - \begin{pmatrix} \nabla \varphi_{\mu}(x) + J^\ast \lambda \\ c \end{pmatrix} \]

Where \(\varphi_\mu(x)\) is the barrier penalty objective

The documentation for this class was generated from the following file:

ROL_PrimalDualInteriorPointReducedResidual.hpp