Are minimizing $f(x)$ and $k\cdot f(x)$ the same?

Question

Suppose I want to minimize an objective function $f(x)$ with respect to some constraints $g(x) = b$. Is it true that minimizing $f(x)$ and $k\cdot f(x)$ is equivalent in the constrained optimization when $k$ is a positive constant?

I feel it is true because the Lagrangian can be rewritten as follows $$\begin{align}&f(x) + \lambda^\top(g(x) - b)\\&k[f(x) + \lambda^{\top*}(g(x) - b)]\end{align}$$ where $\lambda^* = \lambda/k$

Answer 1

Yes, minimising one would also minimise the other. This simply follows from the fact that, since $k > 0$, if $x \leq y$ then $kx \leq ky$, and if $x < y$ then $kx < ky$.