I have to minimize $f(x_1,x_2,...,x_k)$ where $f$ is an affine function, subject to constraints where products between these variables are involved.
The number of variables can vary from 10 to 100. What kind of optimization problem this one above refer ? What is the best method ? The most feasible/easiest method ? Is this a difficult problem for a non specialist ?
You are providing very little information. In general, the difficulty of an optimization problem depends on whether we can establish general properties for the objective function and the constraints, or not. These properties depend, in turn and among other things, on the functional forms and on the domains specified.
Your objective function is affine, and so convex... But in order for convexity to even be defined, the domain of the function under examination must be a convex set.
The fact that the constraints are non-linear means that we must check whether they have a similar property - you would want them to be convex too, because then you would have a convex minimization problem, which is a well developed field, with many theoretical results and numerical algorithms (and one of its authoritative textbooks is officially free to download). But non-linearity does not exclude convexity. Here too the domain of the $x's$ must be a convex set, to be able to define convexity, i.e. to be able to even check whether it holds. Is it a convex set?
Also, if your constraints are many, a question related to the feasible set arises: what are the actual values the $k$-dimensional vectors of $x$'s are allowed to take, given the constraints? the "feasible set" is the set of values the remain as candidate minimizers after the constraints are applied to the initial domain.
Moreover, are you looking to solve the problem theoretically, or the various coefficients involved have specific numerical values and you want to obtain a specific numerical solution, and you intend to plug it into a computer?
Perhaps I could expand on this answer (perhaps), if you provided more information, although sometimes, the exact form of the constraints is critical.
