While reading many research papers I have following observations.
1- First they prove that the problem is convex optimization problem. (They do this by showing that the objective function is convex and the constraints results in convex sets)
2- After proving that, some of the papers say that Slater condition is valid and hence KKT conditions are necessary and sufficient for finding the solution.
3- They say that the dual variables $\lambda, \mu$ are greater than or equal to zero.
I am not sure on which bases they say that the dual variables are greater than or equal to zero. What is its reason? Is it because the problem is convex optimization problem or because Slater condition is satisfied? Any help in this regard will be much appreciated. Thanks in advance.
In any nonlinear programing the dual variables associated to the inequality constraints must satisfy some sign restriction. For example all dual variable associated to the $\leq $-type inequalities have to be nonnegative. It is natural since if you switch inequalities , problem changes so it's dual has to change in some way.