# Is it possible to make a posynomial concave using a change of variables?

Note: this question was already posted on Math.SE but received no answers, so I'm re-posting it here for better reach.

Consider the following posynomial with respect to the variables $$x_1,\dots,x_n$$: \begin{align} f(x_1,\dots,x_n) &= \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{a_{2k}} \cdots x_n^{a_{nk}} \\ &= \sum_{k=1}^K c_k \cdot \prod_{i=1}^n x_i^{a_{ik}} \end{align} Using the change of variables $$\tilde x_i = \log x_i$$, such that $$x_i = e^{\tilde x_i}$$, we get the following expression \begin{align} f(\tilde x_1,\dots,\tilde x_n) &= \sum_{k=1}^K c_k \cdot \prod_{i=1}^n e^{\tilde x_i a_{ik}} \\ &= \sum_{k=1}^K c_k \cdot \exp\left[\sum_{i=1}^n \tilde x_i a_{ik}\right] \\ &= \sum_{k=1}^K \exp(\log(c_k)) \cdot \exp\left[\sum_{i=1}^n \tilde x_i a_{ik}\right] \\ &= \sum_{k=1}^K \exp\left[\sum_{i=1}^n \tilde x_i a_{ik} + \log(c_k)\right] \end{align} This expression is convex with respect to $$\tilde x_1,\dots,\tilde x_n$$ (see here for a proof). My question is: does there exist a similar change of variables such that the resulting expression is concave (not convex) with respect to $$\tilde x_1,\dots,\tilde x_n$$? That is, does there exist functions $$g_1,\dots,g_n$$ such that $$f(\tilde x_1,\dots,\tilde x_n)$$, where $$\tilde x_i = g_i(x_i)$$, is concave with respect to $$\tilde x_1,\dots,\tilde x_n$$?

• I see you implicitly assume $c_k > 0 \;\forall k$ and $x_i > 0 \;\forall i$. Commented Sep 19, 2023 at 7:34

You implicitly assume $$c_k > 0$$ for all $$k$$, and $$x_i > 0$$ for all $$i$$, and I will do the same. Using the change of variables $$\tilde x_i = x_i^M$$, such that $$x_i = \tilde x_i^{1/M}$$, for a big constant $$M$$ at least the size of the largest monomial degree, $$\max_k \left(\sum_{i=1}^n \alpha_{ik}\right)$$, we get the following expression \begin{align} f(\tilde x_1,\dots,\tilde x_n) &= \sum_{k=1}^K c_k \cdot \prod_{i=1}^n {\tilde x_i^{a_{ik}/M}}, \\ \end{align} which is a sum of monomials of degree $$p_k = \sum_i \frac{a_{ik}}{M} \leq 1$$ known to be concave and representable by a power cone in conic optimization; see here for homogenous case $$p_k=1$$, and here for the nonhomogenous case $$p_k<1$$. The expression above is thus concave with respect to $$\tilde x_1,\dots,\tilde x_n$$.
• Just wondering, do you know if there is a similar change of variables such that the function $$h(x,y) = \prod_{i=1}^n x_i^{z_i} \cdot y_i^{1-z_i}$$ where $x,y \in \mathbb R_+^n$ and $h(x,y)$ is a monomial in $(x,y)$, is concave in $(x,y)$? That is, does there exist $\tilde x = g_1(x)$ and $\tilde y = g_2(y)$ such that $h(\tilde x,\tilde y)$ is concave? You can further assume that $z_i \in [0,1]$ and $x_i,y_i \in [0,1]$ if it makes things easier. I can also post this as a separate question if you prefer. Commented Sep 20, 2023 at 23:30
• If $z_i$ are constant isn't this just a special case of the above with K=1 where the monomial degree is (z1 + (1-z1) + z2 + (1-z2) + ...) = n? If in doubt, create a separate question yes. Commented Sep 21, 2023 at 10:54