We know that $f(x) \to \min$ subject to $g(x) = t$ and $h(x) \leq m$ can be written as $f(x) + \lambda g(x)\to\min$ subject to $h(x) \leq m$.
How do we get value of lambda so that the two problems are equivalent.
We know that $f(x) \to \min$ subject to $g(x) = t$ and $h(x) \leq m$ can be written as $f(x) + \lambda g(x)\to\min$ subject to $h(x) \leq m$.
How do we get value of lambda so that the two problems are equivalent.
You may formulate the inequality constraint using a log-barrier, as it does not allow $m-h(x)$ to be less than or equal to zero. This is the formulation of the minimization problem :$f(x)+\lambda_{1}(g(x)-t)-\lambda_{2}log(m-h(x))$
Now to find the lambda's first solve a closed form for x by setting the gradient w.r.t x as zero. You will have a closed form for x, containing the lambda's.
Now consider this to be $x^{*}=c(x)$. Now substitute the closed form for $x^{*}$ in the constraint as, $g(x^{*})-t=0$ and solve for $\lambda_{1}$ . Similarly solve for $\lambda_{2}$ by substituting $x^{*}$ in the second constraint.
For your further-reading, reference: you may look up log-barrier methods for inequality constraints.
In a statistical modeling scenario, the \lambda's are estimated by cross-validation. But I am not sure about the domain of your work.