Let $\lambda_1, \lambda_2 \geq 0$ and consider the problem :
$$\begin{array}{ll} \text{minimize} & x^2(1-2\lambda_2)+\lambda_1x+3y^3+\lambda_1y\\ \text{subject to} & 2x+y=2\\ & x,y \geq 0\end{array}$$
I saw that the curve $~2x+y=2~$ with $x,y \geq 0$ has two extreme points : $(1,0)$ and $(0,2)$.
But $f(x,y)=x^2(1-2\lambda_2)+\lambda_1x+3y^3+\lambda_1y ~~$ is convex only if $~\lambda_2 \leq 1/2$.
I am stuck here. I don't know what way to take to solve this problem.
