I have an confusion with the concept of "degrees of difficulty" of a generalized geometric program, I would like to know the degree of difficulty of the following optimization problem:
$$\left\{\begin{array}{lll} {\displaystyle \inf_{(\beta,\lambda)\in\mathbb{R}^{2},s\in\mathbb{R}^{N}} } & {\displaystyle \lambda \varepsilon +\frac{1}{N}\sum_{i=1}^{N}s_{i}} &\\ \mbox{subject to} &\beta^{2}+4a\lambda\beta+4a^{2}\lambda-4\lambda s_{i}+4s_{i}\leq 0 & \forall i\leq N \\ &\lambda >1& \\ & \beta\geq 0.& \end{array}\right.$$ where $a$ and $\varepsilon$ are fixed real numbers.
Also, I would appreciate it if someone indicated a correct definition of "degrees of difficulty" or that I indicated a book where this concept is well explained.
Remark: The concept of "degrees of difficulty" is defined in the paper in the link, but I do not understand the examples.
Note: The document in the link can be accessed at https://doi.org/10.1007/BF01580667
Geometric programs require all quantities to be positive, and they do not need to be constrained as such in the model itself. So assuming that is the case, we have this standard form: \begin{array}{lll} \text{minimize} & \lambda\epsilon + (1/N) \sum_i s_i \\ \text{subject to} & \tfrac{1}{4} \beta^{2}\lambda^{-1}s^{-1}+a\beta s^{-1} +a^{2}s^{-1}+\lambda^{-1}\leq 1 & \forall i=1,2,\dots, N \\ & \lambda^{-1} \leq 1 \end{array} The paper you linked to defines the degree of difficulty as $DD=NTERMS-(NVAR+1)$. The number of variables is $N+1$, the number of terms is $N+1$ (for the objective) plus $4N$ (for the main constraints) plus $1$ (for the lower bound on $\lambda)$, so $DD=4N$.
I actually had never encountered the degree-of-difficulty idea before, but it makes some sense to me. Each term in a geometric program is roughly equivalent to a row in a linear program.