$\mu-\lambda<0$ at points where $\lambda=0$, given the definition of $K$.
The existence and uniqueness of $(u,\lambda)\in V\times K$ of this saddle
-point problem has been stated in Glowinski, Lions and Tr{é}moli{è}res: Numerical Analysis of Variational
+point problem has been stated in Glowinski, Lions and Trémolières: Numerical Analysis of Variational
Inequalities, North-Holland, 1981.
<h3>The primal-dual active set algorithm</h3>
-The algorithm for the primal-dual active set method works as follows:
+The algorithm for the primal-dual active set method works as follows (NOTE: $B = B^T$):
- [(0)] Initialize $\mathcal{A}_k$ and $\mathcal{F}_k$, such that
$\mathcal{S}=\mathcal{A}_k\cup\mathcal{F}_k$ and
- [(1)] Find the primal-dual pair $(U^k,\Lambda^k)$ that satisfies
@f{align*}
AU^k + B\Lambda^k &= F,\\
- [BU^k]_i &= G\quad&&\forall i\in\mathcal{A}_k,\\
+ [BU^k]_i &= G_i\quad&&\forall i\in\mathcal{A}_k,\\
\Lambda_i^k &= 0\quad&&\forall i\in\mathcal{F}_k.
@f}
Note that the second and third conditions imply that exactly $|S|$ unknowns