For the following problem:
$\text{min:}\ f(x)\\ s.t. \ g(x)\leq t$
Is the above problem equalivant to the following problem?
$\text{min:}\ f(x) + \lambda g(x) \\ s.t. \ \lambda\geq0$
where $t$ and $\lambda$ are variables. It seems equalivant, because if we increase $\lambda$, $t$ tend to be decrease, if we decrease $\lambda$, then $t$ tend to be increase.
The two problem formulations are equivalent for some choice of $\lambda$, at least if $g(x)$ is non-negative; the problem is that in general there's no way to figure out which $\lambda$ will give you the solution corresponding to $g(x) \leq t$.