Minimizing product subject to linear constraints I am looking for a solver that allows me to solve an optimization problem of the form
$$\begin{array}{ll} \text{minimize} & x_1 x_2 \cdots x_n\\ \text{subject to} & \color{gray}{\text{(some linear constraints)}}\end{array}$$
I've used Gurobi before. However, I couldn't find the way to include products in the objective function as well as in the constraints.
 A: This is a hard problem (maximizing the product is a bit better one, as sometimes one can take $\log$ of the objective function, and it becomes concave...). Your best shot might be to use the sum of squares approach for polynomial optimization, as implemented e.g. in YALMIP.
A: Recently discovered minizinc

MiniZinc is a medium-level constraint modelling language. It is high-level enough to express most constraint problems easily, but low-level enough that it can be mapped onto existing solvers easily and consistently. It is a subset of the higher-level language Zinc. We hope it will be adopted as a standard by the Constraint Programming community.
  FlatZinc is a low-level solver input language that is the target language for MiniZinc. It is designed to be easy to translate into the form required by a solver. 

There are several backends for the translated problem (MIP, SAT, etc).
Here is how something similar to your question will look like in minizinc:
var int: a;
var int: b;
constraint a + b <= 10;
constraint a>0;
constraint b>0;
solve maximize a*b;
output [ show([a,b]) ];
=======
[5, 5]

