Fix some equations

exanauts · Apr 20, 2024 · ef09b73 · ef09b73
1 parent 1aaf467
commit ef09b73
Showing 1 changed file with 10 additions and 10 deletions.
diff --git a/tex/sections/ipm.tex b/tex/sections/ipm.tex
@@ -143,12 +143,12 @@ \subsection{Solving the KKT conditions with the interior-point method}
   \setlength\arraycolsep{5pt}
   \tag{$K_3$}
   \begin{bmatrix}
-    W_k & 0 & G_k^\top           & H_k^\top           & -I           & \phantom{-}0 \\
-    0 & 0 & 0\phantom{^\top} & I\phantom{^\top} & \phantom{-}0 & -I           \\
-    G_k & 0 & 0\phantom{^\top} & 0\phantom{^\top} & \phantom{-}0 & \phantom{-}0 \\
-    H_k & I & 0\phantom{^\top} & 0\phantom{^\top} & \phantom{-}0 & \phantom{-}0 \\
-    U_k & 0 & 0\phantom{^\top} & 0\phantom{^\top} & \phantom{-}X_k & \phantom{-}0 \\
-    0 & V_k & 0\phantom{^\top} & 0\phantom{^\top} & \phantom{-}0 & \phantom{-}S_k
+    W_k & 0   & G_k^\top         & H_k^\top         & -I             & \phantom{-}0 \\
+    0   & 0   & 0\phantom{^\top} & I\phantom{^\top} & \phantom{-}0   & -I           \\
+    G_k & 0   & 0\phantom{^\top} & 0\phantom{^\top} & \phantom{-}0   & \phantom{-}0 \\
+    H_k & I   & 0\phantom{^\top} & 0\phantom{^\top} & \phantom{-}0   & \phantom{-}0 \\
+    U_k & 0   & 0\phantom{^\top} & 0\phantom{^\top} & \phantom{-}X_k & \phantom{-}0 \\
+    0   & V_k & 0\phantom{^\top} & 0\phantom{^\top} & \phantom{-}0   & \phantom{-}S_k
   \end{bmatrix}
   \begin{bmatrix}
     d_x \\
@@ -206,13 +206,13 @@ \subsection{Solving the KKT conditions with the interior-point method}
 \end{equation}
 with the diagonal matrices $D_x := X^{-1} U$ and $D_s := S^{-1} V$.
 The vectors forming the right-hand-sides are given respectively by
-$r_1 := \nabla f(x) + \nabla g(x)^\top y + \nabla h(x)^\top z + \mu X^{-1} e$,
-$r_2 := z + \mu S^{-1} e$,
+$r_1 := \nabla f(x) + \nabla g(x)^\top y + \nabla h(x)^\top z - \mu X^{-1} e$,
+$r_2 := z - \mu S^{-1} e$,
 $r_3 := g(x)$,
 $r_4 := h(x) + s$.
 Once \eqref{eq:kkt:augmented} solved, we recover the updates on bound multipliers with
-$d_u = - X^{-1}(U d_x + X u - \mu e)$ and
-$d_v = - S^{-1}(V d_s + S v - \mu e)$.
+$d_u = - X^{-1}(U d_x - \mu e) - u$ and
+$d_v = - S^{-1}(V d_s - \mu e) - v$.
 
 Note that we have added additional regularization terms $\delta_w \geq 0 $
 and $\delta_c \geq 0$ in \eqref{eq:kkt:augmented}, to ensure the