My problem Maximize$$\min_{i} \{\ c_i \cdot \prod_{j \in A(i)} {x_{j}} \prod_{j \in B(i)} {y_{j}} \} $$Subject  to\begin{align}&\sum_{j \in C(k)} x_{j} = 1,\ \forall k \\&l \leq x_{j} \leq u,\ \forall j\\&\sum_{j \in D(k)} y_{j} = 1+p,\ \forall k\\&y_{j} \in \{1,\ p\},\ \forall j\end{align}where $c_i, l, u, p$ are positive constants less than 1, and sets $A, B, C, D$ are also given. What I have tried I have tried to use ncpol2sdpa to relax the polynomial programming into a semi-definite programming and call the sdpa solver to solve it. The objective is replaced with a new variable $ \ f$, and the following constraints are added$$\ f \leq c_i \cdot \prod x_{j} \prod y_{j}, \forall i\, .$$Each discrete variable $y_i$ is replaced by $y_i = p \cdot (1-z_i) + z_i$, where $z_i = \{0,\ 1\}$ or equivalently, $z_i^2-z_i=0$.However even for a small problem (5 $x$'s and 8 $y$'s), it took hours for ncpol2sdpa to relax and sdpa to solve the relaxed problem. The level of relaxation is set to be 3.I also tried to relax the constraint on $y$ to be continues, but still quite slow.I wonder if I did anything wrong with both tools? Or is there a better method/solver for this problem?
Minimizing the products of variables
optimization;nonlinear programming;nonconvex