Consider the linear complementarity problem with $\mathbf{M}$ a $P$-matrix and a vector $\mathbf{q}$ with unknown $\mathbf{z}$:
Find $\mathbf{z}$ such that $\mathbf{M}\mathbf{z}+\mathbf{q} \geq \mathbf{0}, \mathbf{z} \geq \mathbf{0}, \mathbf{z}^\top (\mathbf{M}\mathbf{z}+\mathbf{q}) = \mathbf{0}$.
Let's call the solution to this LCP $\mathbf{z} = \mathbf{LC}(\mathbf{M},\mathbf{q})$. Given that $\mathbf{M}$ is a $P$-matrix, we know that $\mathbf{z}$ exists for any $\mathbf{q}$.
Now, consider a vector function $\mathbf{q}(t)$ that has a continuous derivative $\dot{\mathbf{q}}(t)$ and $\mathbf{z}(t) = \mathbf{LC}(\mathbf{M},\mathbf{q}(t))$. Assume there exist an instant $t_0$ where, for some $i$, we have: \begin{equation} z_i(t_0) = \lambda_i(t_0) = 0 \end{equation} where \begin{equation} \lambda_i(t_0) = q_i(t_0) + \sum\limits_{j} M_{ij}z_j(t_0) \end{equation}
Problem: Define the derivative $\dot{\mathbf{z}}(t_0^+)$.
My procedure:
$\dot{\mathbf{z}}(t_0^+)$ exists since $\mathbf{z}(t) = \mathbf{LC}(\mathbf{M},\mathbf{q}(t))$ exists everywhere and is piecewise-linear and continuous.
$t_0$ marks a point of discontinuity in $\dot{\mathbf{z}}(t)$, i.e. we expect $z_i(t_0)$ to switch between satisfying $z(t) = 0, \lambda(t) > 0$ and $z(t) > 0, \lambda(t) = 0$
We know that the elements of $\mathbf{z}(t_0)$ can be in one of three states \begin{equation} \begin{cases} z_i(t_0) = 0 & \lambda_i(t_0) > 0, \quad \forall i \in \mathcal{I}_1\\ \lambda_i(t_0) = 0 & z_i(t_0) > 0, \quad \forall i \in \mathcal{I}_2 \\ \lambda_i(t_0) = 0 & z_i(t_0) = 0, \quad \forall i \in \mathcal{I}_3 \\ \end{cases} \end{equation} where the problem requires $\mathcal{I}_3 \neq \emptyset$. Also, the union of $\mathcal{I}_1$, $\mathcal{I}_2$, $\mathcal{I}_3$ forms the complete set of all indices in $\mathbf{z}(t)$. Via differentiation of the above w.r.t $t$, we get \begin{equation} \begin{cases} \dot{z}_i(t_0^+) = 0 & \lambda_i(t_0^+) > 0, \quad \forall i \in \mathcal{I}_1\\ \dot{\lambda}_i(t_0^+) = 0 & z_i(t_0^+) > 0, \quad \forall i \in \mathcal{I}_2 \\ \dot{\lambda}_i(t_0^+) = 0 & \text{ and/or } \dot{z}_i(t_0^+) = 0, \quad \forall i \in \mathcal{I}_3 \\ \end{cases} \end{equation} The issue is with the last condition. However, since I know that we are at the moment of switch and both $\lambda_i(t_0^+) \leq 0$ and $z_i(t_0^+) \leq 0$ must be satisfied, I can say confidently \begin{equation} \dot{\lambda}_i(t_0^+) \geq 0 , \dot{z}_i(t_0^+) \geq 0, \dot{z}_i(t_0^+)\dot{\lambda}_i(t_0^+) = 0, \quad \forall i \in \mathcal{I}_3 \end{equation} which results in a new LCP problem (by plugging the necessary values in $\lambda_i(t_0^+)$) now acting on the derivative of the elements $i \in \mathcal{I}_3$.
Here I am at a bit of a loss:
A. Can I guarantee that the LCP resulting in 3 has a solution? the matrix of the LCP is indeed formulated from the minors of $\mathbf{M}$ but I am unsure if the matrix formed in 3 is a $P$-matrix as well.
B. I claim the following: since 1 is valid (!!!if it is!!!), the LCP formed in 3 leads to the formation of a problem with a unique solution and the matrix formed is therefore a $Q$-matrix.