I am trying to solve the problem:

$\max_{\boldsymbol{s}\in\mathbb{R}^{n}} \frac{\boldsymbol{a}^{T}\boldsymbol{s}+\alpha}{\boldsymbol{b}^{T}\boldsymbol{s}+\beta}\\ \text{s.t} \;\;0\leq s_{i}\leq1,\;\;i=1\cdots n$

with $\boldsymbol{a},\boldsymbol{b}$ being positive value vectors.

I know that the objection is a pseudoconcave function (proposition 2.1 in "On the Global Optimization of Sums of Linear Fractional Functions over a Convex Set" by H. P. BENSON), and further more, I know that any local maxima of the problem is also a global maxima (proposition 2.2 in the same paper)

However, I can't seem to find an algorithm to solve this. In the mentioned paper it says that any convex algorithm would do, but cvx would not accept this objective as it is not convex. I also tried a branch and bound algorithm (this Matlab implementation: https://www.mathworks.com/matlabcentral/fileexchange/36247-function-for-global-minimization-of-a-concave-function)

but the algorithm seems to converge to a point that is not the global maxima, which makes me wonder if indeed any local maxima is the global maxima?

hence my questions are: Is the function indeed pseudoconcave and what would be a suitable simple algorithm to fins the global maxima?

Thanks Shahar

I am trying to solve the problem:

$\max_{\boldsymbol{s}\in\mathbb{R}^{n}} \frac{\sqrt{\boldsymbol{a}^{T}\boldsymbol{s}+\alpha}}{\boldsymbol{b}^{T}\boldsymbol{s}+\beta}\\ \text{s.t} \;\;0\leq s_{i}\leq1,\;\;i=1\cdots n$

with $\boldsymbol{a},\boldsymbol{b}$ being positive value vectors.

I know that the objection is a pseudoconcave function (proposition 2.1 in "On the Global Optimization of Sums of Linear Fractional Functions over a Convex Set" by H. P. BENSON), and further more, I know that any local maxima of the problem is also a global maxima (proposition 2.2 in the same paper)

However, I can't seem to find an algorithm to solve this. In the mentioned paper it says that any convex algorithm would do, but cvx would not accept this objective as it is not convex. I also tried a branch and bound algorithm (this Matlab implementation: https://www.mathworks.com/matlabcentral/fileexchange/36247-function-for-global-minimization-of-a-concave-function)

but the algorithm seems to converge to a point that is not the global maxima, which makes me wonder if indeed any local maxima is the global maxima?

hence my questions are: Is the function indeed pseudoconcave and what would be a suitable simple algorithm to fins the global maxima?

Thanks Shahar