Convex programming when the problem has an underlying combinatorial structure that's a DAG

I have a nonlinear convex objective function to minimize. The function is defined on a set of variables: $\{ x_1,x_2, \ldots ,x_p \},$ where each $x_i$ is a number associated with a path in the DAG.

I'm wondering if there's previous work available for such convex programming problems when the problem has an underlying combinatorial structure such as a graph?

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If $x_i$ would be associated not with paths but nodes of undirected graph it would looks similar to result of "functional lifting" method of convexificaton, for example Tom Goldstein et al "Global minimization of markov random fields with applications to optical flow"