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Denote $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I want to minimize a function $f:N\rightarrow \mathbb{R}$. It is very easy to minimize $f$ coordinate by coordinate so one natural algorithm is to iterate on a mapping $T$ for which the $i$th element is defined as $$(Tn)_i=\arg\min_{\tilde{n}_i\in N_i} f\left( \left\{ n_1,\dots,\tilde{n}_i,\dots,n_I\right\}\right)$$ until we have convergence. This is essentially a coordinate descent algorithm but in a discrete space.

My question is: under what conditions does this approach yield the true global minimum of $f$? For instance, is $f$ strictly convex a sufficient condition for this procedure to work? Also, if anybody has a reference on the topic that would be highly appreciated.

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