any good way to approximate this non-convex function with convex function?

Question

There is a non-convex constraint in my optimization problem, which is given by $\displaystyle -xy\log\left(1+\frac{z}{xy}\right)$. Obviously, it is neither convex or concave. Is there any good convex approximation method to approximate this function? Thank you for your help.

Answer 1

I don't know if this is particularly helpful in your situation, but you might consider the 'double' Legendre transform.

What I mean is the following: Let $f$ represent your function. I assume $f: D \to \mathbb{R}$, where $D \subset \mathbb{R}^3$ is some suitable domain.

The Legendre transform of $f$, denoted by $f^{*}$, is the function defined by

\begin{equation} f^{*}(y) = \sup_{x \in D}(\langle y, x\rangle - f(x)) \end{equation} where $y$ is any point where the $\sup_{x \in D}(\langle y, x\rangle - f(x)) < \infty$. (The angled brackets $\langle\cdot,\cdot \rangle$ represent the inner-product on $\mathbb{R}^3$.)

We can define $f^{**}$ as the Legendre transform of $f^*$. It has the following properties:

• $f^{**}$ is convex.
• $f^{**} \leq f$.
• If $f$ is finite, but not necessarily convex, then $f^{**}$ is its convex envelope (that is, the largest convex function which is smaller than $f$).