I am new to optimization theory and I am facing this optimization problem.

\begin{equation} maximize \qquad f(x) = \sum_{i} \log\Big(\frac{\mathbf{c}_i^T\mathbf{x}+N}{\mathbf{d}_i^T\mathbf{x}+N}\Big) \\ s.t.\qquad (\mathbf{a}_k^{T}\mathbf{x}-b) \ge 0\qquad\qquad\forall k\in {1,2,...K} \\ 0 \preceq\mathbf{x}\preceq1 \end{equation}

where x (the optimization parameter) is a column vector and all the other parameters are constants, the inequalities in the last constraint are component wise. This is clearly not a convex optimization problem.

I was wondering if anyone could provide a description of the problem type and the algorithms or solutions available to it.


Looks like a non-convex nonlinear optimization problem. If you use a change of variable $y_i$ for the $i$-th term in the objective function, your problem becomes maximizing $\prod_{i} y_i$ over a set of polyhedral and bilinear constraints. This might have some structure that is exploitable.

I would try to use BARON on this problem or another global optimization solver. If you want some suggestions look at the list on NEOS.


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