I have a optimization problem presented as following: \begin{equation} \mathop {\max }\limits_{({x_1} \to {x_5})} f({x_1},{x_2},{x_3},{x_4},{x_5}) \end{equation} where $f(x_1, x_1, x_3, x_4, x_5)$ is a multi-variable function. It is impossible to solve the aforementioned optimization problem and find the optimal values of $(x_1, x_1, x_3, x_4, x_5)$ at the same time. However, I can prove that the function $f$ is convex w.r.t. $(x_1, x_2)$ when we fix $(x_3, x_4, x_5)$. It means I can find a global solution for the optimization problem in case of fixing $(x_3, x_4, x_5)$. Similarly, I also can prove the convexity of the function $f$ w.r.t. $(x_3, x_4, x_5)$ with fixing $(x_1, x_2)$. Therefore, I intend to use an algorithm to address this problem, described as follows:
However, I do not know whether I can use this algorithm to find the optimal values of the function $f$? Can the algorithm converges? AND If yes, then with what conditions?
Moreover, which reference saying about this algorithm should I refer to?
Thank you!