I have an optimisation problem with a two-stage cost function, consisting of two parts. For a parameter value less than a threshold (which is an optimal value by itself), only the first part appears in the optimal value function. For the parameter value more than this threshold, on the other hand, both parts appear in the optimal value function.
I tried to under-estimate the problem by under-estimating all non-convex terms in the cost function and constraints; however, the optimal value function of the resulting under-estimated problem is not convex because of the two-stage operation.
I am wondering if there is any way to under-estimate such a problem considering the multi-stage operation in order to have a convex optimal value function of the resulting under-estimated problem.
Thanks Isaac. The problem is that the 'threshold parameter' is not known beforehand and, indeed, must be found by solving the problem.
Therefore, if I use two binary variables, as you suggested, I think I still need to under-estimate the similar problem again in order to solve the problem for the case that e.g. both binary variables are '1'.
More comments would be appreciated.
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