I’m working on a channel maximization problem which is MAX -log2(1 + p / d *d). We know that the -log2(1 + 1 / x) is concave with respect to x. So I let x = d * d, then the initial problem became MAX -log2(1 + 1 / x), constr(x <= d * d).
Now, objective function is a convex optimization problem because it is a concave problem of maximization. However, due to the constraints, the feasible solution set is not convex, so the whole problem does not meet the criteria of convex optimization.
Because x * x is convex with respect to x, at any given point xc, there will always be x * x > = 2 * xc * x - xc * xc. In this step, the right-hand side of the original constraint equation can be transformed into a linear expression. So the original problem can be turned into MAX -log2(1 + 1 / x), constr(x <= 2 * xc * x - xc * xc). Because of the first order Taylor expansion at xc.
My question is, after such a series of transformations, how different is the optimal solution between the transformed problem and the original problem? What are the reasons for this difference?