I'm trying to solve a constrained optimization problem:
Let $x=(x_1,x_2,\cdots,x_n)$
$$\min_{x\in \mathbb{R}^n}\ \textrm{or}\ \max_{x\in \mathbb{R}^n}\quad\Sigma x_i^4 $$
subject to $\Sigma x_i^2=1$ and $\Sigma x_i=\rho$, where $\rho=\sqrt{n}(1-\epsilon)$, $\epsilon$ is a small positive number.
Consider the Lagrange function: $$F(x,\lambda,\mu)=\Sigma x_i^4+\lambda(\Sigma x_i^2-1)+\mu (\Sigma x_i-\rho)$$
$$\frac{\partial}{\partial x_i}F(x,\lambda,\mu)=4 x_i^3+2x_i\lambda+\mu=0,\quad i=1,\cdots,n$$ $$\frac{\partial}{\partial \lambda}F(x,\lambda,\mu)=\Sigma x_i^2-1=0$$ $$\frac{\partial}{\partial \mu}F(x,\lambda,\mu)=\Sigma x_i-\rho=0$$
I'm stuck in this step. I don't know how to use these equations to compute $x_i$ nor directly compute $\Sigma x_i^4$.