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Hello Janis,

currently (v9.0) we have no automated mechanism that enables the multi-variate PW linear objectives.  One option is to build one yourself using, say, triangulation of the 3D surface corresponding to the function graph (only for completeness, I am adding this link, https://en.wikipedia.org/wiki/Surface_triangulation ). Here you would keep track of which "supporting triangle" you are in using binary inductors.

However, starting with the present version we also support general bilinear expressions, and p-w linear constraints (https://www.gurobi.com/documentation/9.0/refman/constraints.html) so this could be another route.  Namely, you can introduce an auxiliary pair of variables, (t', x'), and connect those to non-linear expressions using p-w approximation, while the objective would have a bilinear term x' * t'.

Personally I would start with the later approach as it is less labour expensive, to see if this would suffice.

Hope this helps.