Let me start by saying I know almost nothing about optimization so please bear with me. Basically, I am wondering whether it is possible to solve a problem with two constraints by solving the problem with each constraint individually and somehow combining the solutions. However, I'm having trouble visualizing the scenario. More specifically:
Suppose $V$ is some Banach space and $f:V \to \mathbb{R}$ is convex and continuous. Furthermore, let $K_1, K_2 \subset V$ be convex and consider the solutions $x_i$ of the optimization problems $$\quad \min_{x \in K_i} f(x).$$
For $K = K_1 \cap K_2$ let $x$ solve $$\min_{x \in K} f(x).$$
- Is $x$ the projection of $x_1$ onto $K_2$?
- Is $x$ a linear combination of $x_1$ and $x_2$?
- Are there any relations between $x$, $x_1$ and $x_2$ at all?
- Does the situation change if we further restrict $f$? For example to be quadratic?
Thank you very much in advance.