# 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

MathOverflow Asked by user263322 on January 1, 2021

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