Suppose we want to solve the following optimization problem:
\begin{equation*} \begin{aligned} & \underset{x,y,z}{\text{minimize}} && x(a-y) \\ & \text{subject to} && (1-x)f(z)+xf(y)=b \end{aligned} \end{equation*}
where $0 \leq x \leq 1$, $0 \leq z \leq a \leq y \leq 1$, and $a$ and $b$ are some constants in $[0,1]$. $x$, $y$, and $z$ are variables to be optimized. Moreover, $f(\cdot)$ is defined on $[0,1]$, and is convex-$\cap$ and decreasing on $[0,1]$.
But I am not sure if this problem is solvable by some standard convex optimization problem. Anyone has some idea? Thanks!
Convex optimization problems require affine equality constraints, co convexity of $f$ doesn't really help you here directly. Moreover, you do not have a convex objective function.
For more on relaxing non-affine equality constraints, see Boyd & Vandenberghe Exercise 4.6 (the book is availiable as a free pdf on Stephen Boyd's site).