Karush-kuhn-tucker conditions for vector optimization

Question

I have a problem to minimize the components of a vector $x$. However I also have contraints such that $x-c_1<0$ and $c_2 - x<0$. I have defined the new problem as $$\underset{x}{\operatorname{argmin}}||x||_2^2 + \mu_1 (c_2 -x)+\mu_2 (x-c_1 )$$ where $\mu$'s are lagrangian multipliers. This is the definition of lagrangian function defined by karush-kuhn-tucker conditions. However cost function is a vector function now. Probably I could not interpret the Lagrangian function definition for my problem. What could you say about this problem? Thank you in advance.