Consider the following problem:
$\mathop{\max}\limits_{x}\;\;f_0(x)\\ s.t.\;\;f_i(x)\le0,i=1,2,\cdots,m\\ \;\;\;\;\;\;\;\;h_j(x)=0,j=1,2,\cdots,n$
where $f_i(x),i=0,1,\cdots,m$ is convex function, and $h_j(x)$ is linear function. Then, whether this problem is a convex optimization problem?
This is not a convex optimization problem.
The flaw is with the objective function where we are suppose to minimize a convex function.
Maximizing a convex function doesn't have properties such as local optimal is a global optimal.