Convergence of solving an OP $ minlimits_{x, alpha} sum_i alpha_i f_i(x) + g(alpha)$

Consider the optimization problem
$$ minlimits_{x, alpha} sum_i alpha_i f_i(x) + g(alpha). $$
I solved the above problem by alternating fixing $x$ solve for $alpha$ and vice versa. When $x$ is fixed, the corresponding optimization with variables $alpha$ is convex and has a closed form solution. When $alpha$ is fixed, the problem is non-convex and it is solved by a gradient descent algorithm (called this problem OP2).

My question is does the above alternative scheme converge (a local optimum) given than OP2 converges to a local optimum. If it is not do you know in which conditions does alternative scheme converge?

Thanks