In geometric programming, can I maximize a posynomial objective function?

Question

A Geometric Programming (GP) problem is given by

$\min\limits_{x\in R_+^n}f_0(x)\\s.t.\;\;f_i(x)\le 1,i=1,\cdots,p\\\;\;\;\;\;\;\;\;g_j(x)=1,j=1,\cdots,q$

$f_i(x),i=0,1,\cdots,p$ is a posynomial function. That is, the GP in general deals with the problem of minimizing a posynomial objective.

Then, whether the GP can handle the problem of maximizing a posynomial objective?

Answer 1

You can reformulate the problem into convex form:

$\min\limits_{y\in R^n}\tilde{f}_0 = log\left(\sum_{k=1}^{K_0}e^{a_{0k}^Ty+b_{0k}}\right)\\s.t.\;\;\tilde{f}_i = log\left(\sum_{k=1}^{K_i}e^{a_{ik}^Ty+b_{ik}}\right)\le 0,i=1,\cdots,p\\\;\;\;\;\;\;\;\;\tilde{g}_j^Ty+h_j=0,j=1,\cdots,q\\ with \\ \;\;\;\;\;\;\;\;y_i=log(x_i), i = 1, \cdots, n\\ \;\;\;\;\;\;\;\;b_{ik}=log(c_{ik}), i = 0, \cdots, p, k = 1, \cdots, K_i\\ \;\;\;\;\;\;\;\;h_{j}=log(d_{j}), j = 0, \cdots, q\\ \;\;\;\;\;\;\;\;f_i = \sum_{k=1}^{K_i} c_{ik} * x_1^{a_{ik1}} * \cdots * x_n^{a_{ikn}}, i = 0, \cdots, p\\ \;\;\;\;\;\;\;\;g_j = d_j * x_1^{\tilde{g}_1} * \cdots * x_n^{\tilde{g}_n}, j = 0, \cdots, q$

then you can basically negate the objective function: $\max \tilde{f}_0 \Leftrightarrow \min -\tilde{f}_0$

Answer 2

I'll admit to not understanding the reformulation above, but from my understanding: no, Posynomials are convex (after the transformation) and thus their maximization is not a convex optimization.