Can these quadratic constraints be linearized?

Question

I have the following optimization problem:

$\max s$

$\displaystyle \left (\sum_{n=1}^NA_nx^i_n\right )^2 + (1-y_i)M\geq B\sum_{n=1}^N(x^i_n)^2,\forall i=1,2,\ldots S$

$\displaystyle \sum_{i=1}^Sx^i_n=C_n,\forall n\in \mathcal{N}$

$y_i\geq \dfrac{s-i+1}{S},\forall i=1,2,\ldots S$

$y_i\leq s-i+1,\forall i=1,2,\ldots S$

In this optimization problem, $x^i_n\in \mathbb{R^+}$, $s\in \mathbb{N}$, and $y_i\in \{0,1\}$ are decision variables. I have the following questions:

1. Can the quadratic constraints be linearized to transform the problem into an equivalent linear form?
2. If not, what is the best way to solve the problem for exact (not approximate) optimal solution?

Thank you.

Answer 1

The constraints imply that $s\in\{S-1,\dots,2S-1\}$. You can maximize $s$ by performing a bisection search on this set, solving a (convex) quadratically constrained feasibility problem at each stage. If a given $s$ yields an infeasible problem, so do all larger values of $s$.