I have the following normal cone inclusion
$$-(A x + b) \in \mathcal{N}_\mathcal{C}(x) \qquad (1)$$
where $\mathcal{N}_\mathcal{C}$ denotes the normal cone to the convex set $\mathcal{C}$ at the point $x \in \mathcal{C}$. Matrix $A$ is non-symmetric.
Normally if $A$ would be symmetric, the convex optimization problem is
$$\min \frac{1}{2} x^\top A x + b^\top x + I_{\mathcal{C}}(x) \qquad (2)$$ where $I_{\mathcal{C}}(x)$ is the indicator function of the set $\mathcal{C}$. The inclusion (1) is the necessary and sufficient optimality condition of the convex optimization problem (2).
However, what are the implications of $A$ beeing non-symmetric and how does that relate to an optimization function? I guess there does not exist an optimization function?