Since the question is generic, my generic answer would be, if the constraint set is non-convex, then there will be more than one optimal solution. By forcing a non-convex problem to be convex, you will obtain one optimal solution, which may not be one of the the actual optimal solutions. So yes, could end up being an approximation.

To convert a non convex problem into other convex it is required to formulate an aproximate to the original problem task. That one can make by means of posinomial approximations to the objective function and/or the set of restricted functions of the problem, what transforms it into a particular case of the Geometric Programming task. I have published 2 related papers, but in Spanish Language.

If you were able to convert your original non-convex optimization problem to a convex problem, this is a big step in the right direction. The problem is not that the solution for this convex model can be approximate, and not necessarily optimal. The problem is to evaluate the measure of closeness of this approximate solution to the optimal solution and this is very valuable as a scientific result. Only finding an approximate solution without its evaluation is not a science.

Thanks alot for the responses.

Continuing with my previous answer: Once you formulate an auxiliary approximate to the original task, you can find optimal solution solution to the original task among the solutions that differ of the optimal of the auxiliary task, in not more than an alfa parameter, supposing that for the auxiliary task some method exits for generating all the alfa oprimal solutions. The above-mentioned had been demonstrated in the frame of the Aproximate Combinatorial method of V. R. Jachaturov (Russian Academy of Sciences)

Unfortunately, I did not attend this, which might provide additional background: https://www.cs.washington.edu/events/colloquia/details?id=2992

Here is a set of publications that might be useful: https://people.eecs.berkeley.edu/~jordan/publications.html