In the study of Duality theory, Nonlinear programming problem, Dual space etc,( particularly on fuzzy system also) we generally see a duality gap appears and this gap varies differently according to the selection of feasible region. So, it is trouble some to get best approximation of the solution of the problem. However, in linear programming problem no duality gap exists .
You simply have to learn mathematical optimisation, like the rest of us. Global optimisation (which excludes metaheuristics, as they are NOT convergent) is a mature field, and there are both good books and software, too. If you are unfamiliar with non-convex optimisation, then read! :-)
Respected Prof. Patriksson, you are right. For a definite solution space the idea is ok. But if the solution space itself covers a dual space like xy less than 1, x and y separately less than 1 and both of them are positive, the objective function is a function of (x,y) then how to proceed? However, the variables x and y may be the functions of some other variables ( belongs to real line) also.
I think, the problem will have to solve in one space ( optimizing x satisfying x
Why the "xy < 1"? It will define set that is not closed ---> you may not have optimal solutions!!
My problem was a part of fuzzy system. I think you never heard about fuzzy system/ rough system and hence you do not have primary knowledge about fuzzy sets. So you need to read Zadeh (1965)' fuzzy sets. I gave a research link in support of my answer.
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