satisfying the first-order necessary optimality conditions, we have that $\bar \mu$ is a stationary point of the cost functional ${\hat{\mathcal{J}}}$. Thus, $\bar \mu$ can be either a local minimum, a saddle point or a local maximum of the cost functional ${\hat{\mathcal{J}}}$ (and obviously the same relationship occurs between $(\bar u,\bar \mu)$ and ${\mathcal{J}}$). We thus consider second-order sufficient optimality conditions in order to characterize local minima of the functional ${\hat{\mathcal{J}}}$, requiring its hessian.

\[prop:hessian\_Jhat\] The hessian of ${\hat{\mathcal{J}}}$, ${\hat{\mathcal{H}}}_\mu := {\mathcal{H}}_\mu {\hat{\mathcal{J}}}: {\mathcal{P}}\to {\mathbb{R}}^{P \times P}$, is determined by its application to a direction $\nu\in \mathbb{R}^P$, given by $$\begin{aligned}
    {\hat{\mathcal{H}}}_\mu(\mu) \cdot \nu
    = \nabla_\mu\Big(&\partial_u {\mathcal{J}}(u_\mu, \mu)[d_{\nu}u_\mu] + l_\mu(d_{\nu}p_\mu) - a_\mu(d_{\nu}u_\mu, p_\mu) - a_\mu(u_\mu, d_{\nu}p_\mu)
    \notag\\
    &+\big(\partial_\mu {\mathcal{J}}(u_\mu, \mu)+ \partial_\mu l_\mu(p_\mu)- \partial_\mu a_\mu(u_\mu, p_\mu)\big)\cdot \nu\Big),
    \notag
    \end{aligned}$$ where $u_\mu, p_\mu \in V$ denote the primal and dual solutions, respectively. For a quadratic ${\mathcal{J}}$ as in the above formula simplifies to $$\begin{aligned}
    {\hat{\mathcal{H}}}_\mu(\mu) \cdot \nu
    = \nabla_\mu\Big(&j_\mu(d_{\nu}u_\mu) + 2k_\mu(d_{\nu}u_\mu, u_\mu) + l_\mu(d_{\nu}p_\mu) - a_\mu(d_{\nu}u_\mu, p_\mu) - a_\mu(u_\mu, d_{\nu}p_\mu)
    \notag\\
    &+\big(\partial_\mu {\mathcal{J}}(u_\mu, \mu)+ \partial_\mu l_\mu(p_\mu)- \partial_\mu a_\mu(u_\mu, p_\mu)\big)\cdot \nu\Big).
    \notag
    \end{aligned}$$

See, e.g., [@HPUU2009] for the first part. The second one follows from Remark \[prop:gateaux\_wrt\_V\].

\[prop:second\_order\] Let Assumption [[\[asmpt:differentiability\]]{}]{} hold true. Suppose that 