Consider a optimization problem: $\sum_{i=1}^n(y_i-\beta_0+\beta_1x_i)^2$ subject to $\beta_0+\beta_1x_{i+1}\geq\beta_0+\beta_1x_{i}$ for $i=1,2$.
This can be solved by using (ref:page (282)) the fact that: Given m-inequality constraints there could are $2^m$ possible subsets of inequality constraints which can be converted to equality constraints.
I am NOT getting how the above fact is coming? For example in the above given example which are $2^{m-1}$ (here $m$=2) subsets of inequality constraints that can be converted to equality constraints?
The objective function of your problem is non-linear. So I assume that you are solving your problem via Karush-Kuhn-Tucker (KKT) optimality conditions and you are looking for the active sets of constraints in order to satisfy the complementary conditions. This can be done checking exhaustively all the potential active subsets of the constraint set. If you have $m$ constraints, then you have $2^m$ subsets of constraints that can be active, that is satisfied as equalities. In your case $m=2$ and the active subsets you have to consider are: $$ I_1=\{1\}, I_2=\{2\}, I_3=\{1,2\}, I_4=\{\emptyset\}$$
When you solve the KKT system for $I_h, \ h=1\ldots m$, you can set the Lagrangean multipliers $\lambda_i=0,\ \ \forall i \notin I_h$.