Consider a continuous function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that $\forall y \in \mathbb{R}^m$ $f(\cdot,y)$ is convex, and $\forall x \in \mathbb{R}^n$ $f(x,\cdot)$ is convex as well.
Consider the convex optimization problem $$ \min_x c^\top x \quad \text{subject to: } \ f(x,y^i) \leq 0 \ \ \forall i = 1,...,N,$$ and assume that $x^\star$ is the unique optimizer.
Let $y^1, ..., y^m$ be all the active constraints, i.e. $f(x^\star,y^i) = 0$ iff $i \in \{1,2,...,m\}$.
We know (KKT) that there exist $\lambda_1, ..., \lambda_m \geq 0$ such that $$ c^\top + \sum_{i=1}^m \lambda_i \frac{\partial}{\partial x} f( x^\star, y^i ) = 0.$$
Trying to exploit convexity of $f(x,\cdot)$, I am wondering then if there exists $y^\star \in \mathbb{R}^m$ and $\lambda \geq 0$ such that $$ c^\top + \lambda \frac{\partial}{\partial x} f( x^\star, y^\star ) = 0. $$