Hi, you can try to use integer programming codes such as CPLEX for example.

Hi, if the problem has only two variables you can compute the branch and bound algorithm by hand. The algorithm you can find it in any book of MIP or article in the internet. But also, you can use any tool like CPLEX or Gurobi for solve the problem if you are not intrest in learn how the problem is solved.

Hi, you can use either Gomory's cutting plane method or branch and bound techniques.

The method for solving this problem depends on the properties of functions F, f, g (convex, concave, other properties).

If properties are not known - the only method is looking through all values of variables.

If the functions are convex, it should be possible to solve the problem efficiently. See these surveys:

https://arxiv.org/pdf/0906.5171.pdf

https://link.springer.com/article/10.1007/s10479-012-1272-5

If the functions are non-convex, things will be a bit harder. If you need an exact solution, you could use a method like branch-and-reduce. See this survey:

S. Burer & A.N. Letchford (2012) Non-convex mixed-integer nonlinear programming: a survey. Surveys in Oper. Res. and Mgmt. Sci., 17(2), 97-106.

you can use CPLEX, Matlab or GAMS

CPLEX can only handle linear or quadratic MIPs, so it is not suited for the problem at hand.

I can only underline what was already said: the methods to handle the problem strongly depend on convexity/concavity properties of F, f and g.

In order for the problem to be a convex one, F has to be convex, but f and g have to be monotone, not convex. Example: F(a,b)=a-b, f(x)=x^2, g(y)=y^2

If it is convex problem then most likely Mosek can solve it. You will have to reformulate on conic form though.

Alternatively you can use cvxpy.org with Mosek.

is F convex? if yes then there are efficient methods as Adam N. Letchford mentioned, otherwise just brute force it.

More information is needed as to the types of the functions in the objective function.

Soumen Atta , you can use Gromory's Cutting Plane method for this integer programming problem. It is better described at the Wikipedia page: https://en.wikipedia.org/wiki/Cutting-plane_method . They can be also be used to solve general, not necessarily differentiable convex optimization problems.