Alternating optimization (AO) has been widely used to solve the (non-convex) problem with several types of variables. People can decouple the original optimization problem into several sub-problems, with each sub-problem contains one type of variable. Solving each sub-problem separately and then optimize different variables alternately. In general, it refers to the methodology named "divided-and-conquer".
Assume that each sub-problem can be optimally solved, how to know whether the obtained solution by AO is global optimal or not?
Yaru -
I'm not familiar with this algorithm, so just a generalized guess, but I suspect that if there are local optimum values, that that is what you would find. To know if it is also the global optimum would require more information, but perhaps you could try different starting points and see if you get different results. You may not prove you have a global optimum, but you would either prove you do not, or you would have better reason to believe that you do.
But what about the other variables? Perhaps a slightly suboptimal value for one would be associated with slightly suboptimal values for other variables, which might be better overall. It sounds like a bigger operations research problem.
Best wishes - Jim
Unless the problem is convex, a local method cannot guarantee global optimality. This is because first- or second-order characterizations of a global minimizer do not exist in the general case.
Now, if you use global methods (https://en.wikipedia.org/wiki/Deterministic_global_optimization), you will explore the search space exhaustively (and hopefully implicitly) and you'll be able to prove the global optimality of the solution. This can get pretty difficult to solve, though.
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