is it possible for Gurobi to solve a optmization problem with a cubic convex objective function? I only found, that it can solve "Quadratically Constrained Program (QCP)". They didn't mentioned anything for the obj function. Does it mean quadratically constrained quadratic programming? Below is an example on how my problem looks like.
\begin{align} & \min & \sum_{a \in A} |x_a|^3 + \sum_{v \in V} y_u^3 &&\\ %% & \text{subject to} & x_a \cdot |x_a|= d && \forall a \in A \\ & & y_u \geq 0 && \forall u \in V \end{align}
Thank you in advance!
To build a cubic, first note that (the rotated scalar Lorentz cone) $w^2 \leq u v$ can be written as the standard SOCP constraint $\left| \begin{pmatrix}2w\\u- v\end{pmatrix} \right|_2 \leq u + v$
We now note that $s^3 \leq t$, with non-negative $s$, can be written as $s^4 \leq ts$, which we expand as $w^2 \leq ts$ and $s^2 \leq w$. Hence, SOCP-representable as those two are SOCP-representable.
Unfortunately, you have the constraint $x_a \cdot |x_a| = d$ which is nonconvex and thus not SOCP-representable.
EDIT: ...unless $d$ is fixed, as then you simply have a linear constraint $x_a = \text{sign}(d)\sqrt{|d|}$.