I know that sometimes a multi-variable function is convex with respect to each variable, for example $f(x,y)=x^2y^2$, but not convex in the whole. In such cases, can we use nested search methods to find the global optimum? For example, in the case of $f(x,y)=x^2y^2$, a loop can iterate over $x$ based on golden sction search and in each iteration with a given $x$, a nested loop can iterate over $y$ based on golden section search to find the optimal $y$. Is it guaranteed that this method will find the optimum value?
Thanks in advance.