I have a non-linear optimization problem formulation in which my objective is convex with a convex equality constraint. However, I parametrized my equality constraint and plugged this parametric form into the objective.
Now, in the new formulation, the objective is constrained in the sense that my parametrization limits the objective to a certain set (the convex constraint set) but there is no explicitly defined constraint.
Strictly speaking, is this a constrained or unconstrained optimization problem? I would appreciate it if you could answer with a reference of citation to a mathematic/scientific paper. This fundamental point will go into my thesis and a reference would be nice to have.
Thank you
Your optimization problem is now $$\min_{x \in X} f(x)$$ where $X \neq \mathbb{R}^n$. There are two ways to look at this. The first way is to say it's constrained optimization by writing it as: $$\min_{x \in \mathbb{R}^n} \{ f(x) : x \in X \}$$ This problem has advantages in terms of solvability when you can explicitly write $X$ in terms of convex functions.
The second way is to define a new function $g$, where $g(x)=f(x)$ when $x\in X$, $g(x)=\infty$ elsewhere. Note that $g$ is convex. Now it is clearly an unconstrained problem: $$\min_{x \in \mathbb{R}^n} g(x)$$ A disadvantage is that you don't have a differentiable objective function.