# Log Fractional optimization problem

Let $$\mathbf{x}$$ be a vector of $$N$$ variables. Then, how can I solve the following optimization problem? \begin{align} \max_\mathbf{x}&\quad \sum_{n} \log(1+\frac{x_n}{\alpha+\sum_{m}\beta_m^{(n)}x_m})\\ \text{subject to}&\quad\mathbf{A}\mathbf{x}\leq \mathbf{p}. \end{align}

Constraints are linear. How about objective function? Is it quasiconcave?

• The objective is neither concave not convex. For example, let $n=2, x1 = x2 = \alpha = \beta_1 = 1, \beta_2 = 3$, then Hessian s indefinite (one positive eigenvalue, one negative eigenvalue). Either use a DC (difference of concave or convex) approach in which the non-convex optimization term is iteratively handled (which at best produces a local optimum), or use a non-convex solver (local or global). Apr 23, 2020 at 23:41

## 1 Answer

Here is an attempt for a special case. Let me write your problem as the following: \begin{align} \max_\mathbf{x}&\quad \sum_{n} \log\left(1+\frac{x_n}{f_n(x)}\right)\\ \text{subject to}&\quad\mathbf{A}\mathbf{x}\leq \mathbf{p}. \end{align}. Assume the following: (i) $$x> 0$$, (ii)the coefficients of $$f_n(x)$$ are all positive $$\forall ~n$$, and (iii) all elements of $$A$$ and $$p$$ are positive.

Firstly, using AM-GM inequality: $$1+\frac{x_n}{f_n(x)} \geq 2\sqrt{\frac{x_n}{f_n(x)}}.$$ This leads to the relaxed problem (barring constants added/multiplied): \begin{align} \max_\mathbf{x}&\quad \sum_{n} \log\left(\frac{x_n}{f_n(x)}\right)\\ \text{subject to}&\quad\mathbf{A}\mathbf{x}\leq \mathbf{p}, ~\mathbf{x}>0. \end{align}. Now, introduce variables $$t_n> 0, \forall n$$, such that: $$\frac{x_n}{f_n(x)} \geq \frac{1}{t_n} \Rightarrow x^{-1}_nt_n^{-1}f_n(x) \leq 1.$$ And then the relaxed problem is equivalent to (removing $$\log$$ as its a monotonic): \begin{align} \min_\mathbf{x}&\quad \prod_{n} t_n\\ \text{subject to}&\quad\mathbf{A}\mathbf{x}\leq \mathbf{p},~\mathbf{x}> 0\\ &~~~~~\mathbf{t}>0,~x^{-1}_nt_n^{-1}f_n(x) \leq 1~ \forall n. \end{align}. Note that the cost function and constraints are all posynomials, with the assumptions made. Thus, this is in the form of a Geometric Program (see https://en.wikipedia.org/wiki/Geometric_programming), which can be converted to a convex program. In fact, softwares like CVXPY will readily solve a problem in this form.

As far as relaxation gap is concerned, that still needs some thought. Anyway, hope this helps.