Are there any efficient algorithms to solve, multi-linear objective and multi-linear constraint optimization problems? The multilinear functions are sums of bilinear, trilinear (and so on) terms
\begin{align*} \min\quad & f_1(x_1,x_2,x_3,...,x_n)\\ \text{s.t.}\quad &f_2(x_1,x_2,x_3,...,x_n)\leq b\\ &f_3(x_1,x_2,x_3,...,x_n)\leq c\\ &f_i(x_1,x_2,x_3,...,x_n)=a_{i1}x_1x_2...x_n+a_{i2}x_1x_2...x_{n-1}+...+a_{i(n-1)}x_1+a_{in} \end{align*}
Optimization of general multilinear functions is difficult, but for the specific form given in the problem statement, there are only $n$ monomial terms, so we can reparameterize: $$ \begin{align} z_1 &= x_1 \\ z_k &= z_{k - 1} x_k, \quad k = 2 \ldots n \end{align} $$ Then the problem is linear in the $z_k$ and can then be solved with any LP solver.
A general multilinear function has $2^n$ monomial terms, but this is not what the question is asking for.
From the optimal $\mathbf{z}^*$ then we can recover the optimal $\mathbf{x}^*$, unless we have $z^*_k=0$ and $z^*_{k+1}\ne 0$ for some $k$. However, if this happens when $\mathbf{z}^*$ is the unique minimum of the transformed problem, then this implies that the original problem has no finite minimum. Otherwise, if the transformed problem has multiple minima, we can solve a modified LP to find an optimal $\mathbf{z}^*$ which corresponds to a finite $\mathbf{x}^*$.