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I can suggest a shortest path approach, but you'll have to decide if it's computationally feasible (which depends on how hard $g()$ is to compute and how large $N$ is, among other things) and better than brute force. First, let $X$ be the domain of $x$ and let $y_n=f(x_n)$. We can rewrite the problem as minimizing $\sum_{n=1}^N y_n$ subject to $g(y_n, y_{n+1}... 3 There are broadly 2 approaches for obtaining feasible solutions to your problem, namely (1) Local search, and (ii) exact approaches (i.e. using solvers). If you prefer to solve using exact approaches i.e. model the problem using a solver, then your best bet may be to model the problem using a Constraint Programming solver. At least from what I gather (so ... 2 It sounds like your Benders master problem is to minimize$\sigma = \sum_{d,k} \sigma_{d,k}$, where your independent subproblems are indexed by$(d,k)$. For each fixed master solution$x^*$, let$\phi(x^*)_{d,k}$be the optimal objective value (or any lower bound, perhaps obtained from the LP relaxation) of subproblem$(d,k)$, and let$M\$ be a valid lower ...