I have formulated a problem where I need to minimize the sum of $N$ functions, with only pairwise dependence between the functions (any single constraint involves only two functions having adjacent indices). \begin{align} [\hat{x}_{1}~\hat{x}_{2}~\cdots~\hat{x}_{N}] &= \text{min}~\sum\limits_{n=1}^{N} f_{n}(x_{n}) \\ g(f_{1}(x_{1}),f_{2}(x_{2})) &\leq k \\ g(f_{2}(x_{2}),f_{3}(x_{3})) &\leq k \\ &\vdots \\ g(f_{N-1}(x_{N-1}),f_{N}(x_{N})) &\leq k \\ \end{align}
The functions $f(x)$ and $g(x)$ are highly non-linear and non-convex, in addition to the decision variables being integer-valued. I am wondering if there is a method for decomposing this problem into smaller sub-problems and iteratively solving them, using a divide-and-conquer approach? Essentially, my question is whether this 'pairwise dependence' can somehow be exploited to make this problem easier to solve? Fortunately, the sample space for the decision variables is small (about 5000 values), and I can actually apply a brute-force search for the minimum, provided the sub-problems are small enough.
I am aware of non-convex solvers such as Baron etc. but I'd like to know whether I can instead reduce the overall objective to a sum of 'smaller' objectives that can each be solved by a brute-force approach and where I am certain that the global minimum has been found.