Under what conditions the globally optimal solution and solution obtained by step-by-step optimization are the same?

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More specifically, I have a problem where it is hard to find the optimal solution w.r.t all the parameters together, so I solve it w.r.t one parameter at a time. In each step, I either use the optimal value of other parameters obtained from the previous step or just keep them as variables to be optimized later. I can not prove the convexity / concavity of the problem but can show the objective function to be monotone w.r.t all the parameters. Thanks in Advance.