Dear Amir,

Your question is not very clear, if you want more informative answers, You should describe more clearly

As far as know, The global optimality can be proved in the following cases:

1-when you have a convex programming problem

2-the feasible space is closed and bounded,

Thank you

but one more question: how can we prove convexity when our formulation consists of integer or binary variables? Are they similar to formulations with integral variables?

what do you mean by integral variable? do you mean continuous variable?

In order to understand the convexity, you can refer to good books. I like this book by Jasbir Arora

Introduction to optimum design.

Chapter 4 has a section on convex programming problems. Most of real life problems are not convex.

A very good set of software for nonlinear (including convex) optimization problem over integer variables are found here:

http://www.coin-or.org/projects/

Check in particular the three codes under the banner "Optimization deterministic nonlinear discrete", as well as the following published paper describing part of the software:

http://www.sciencedirect.com/science/article/pii/S1572528607000448