I would like to know if you could either give me a hint to answering the following problem or a reference (article, book, etc) to have a better idea on how to address it
the prove for LASSO estimator is the solution to the optimization problem
$$\text{min}_B \left[(y-X\beta)^T(y-X\beta)\right]$$
$$\text{subject to: }\sum_{j=1}^p|\beta_j| < t$$
For some real positive $t$
$\newcommand{\Minimize}{\operatorname*{Minimize}}\newcommand{\R}{\mathbb{R}}$The question is not very clear, but I presume that the OP is asking whether
\begin{align} \mathbb{P}_t{}:{}&\Minimize_{b\in\R^m}\tfrac{1}{2} \|y - A'b\|^2 \\ &\text{subject to: } \|b\|_1 \leq t, \end{align} for some $t>0$, is equivalent to
\begin{align} \mathbb{P}'_\lambda:\Minimize_{b\in\R^m}\tfrac{1}{2} \|y - A'b\|^2 + \lambda \|b\|_1, \end{align} for some $\lambda \geq 0$.
Matrix $A$ has dimensions $n\times m$ and $y\in\R^n$.
The two problems are equivalent in the sense that for given $\lambda \geq 0$, there exists a $t\geq 0$ so that the solutions of $\mathbb{P}'_\lambda$ are solutions of $\mathbb{P}_t$. Conversely, for every $t\geq 0$, there exists a $\lambda \geq 0$ so that the solutions of $\mathbb{P}_t$ are solutions of $\mathbb{P}'_\lambda$.
This is discussed in detail and proven in Theorem 2.1 in: M.R. Osborne, B. Presnell and B.A. Turlach, On the LASSO and its Dual, Journal of Computational and Graphical Statistics 9, 2000.
