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Considering the first given objective function, the problem is trivial to solve. The decision variables X1 and X2 are positive. Because polynomial with positive coefficients only, the objective function is thus monotonically increasing over the space of feasible solutions. The last constraint over X1 and X2 is inactive because dominated by the previous ones. ...


NP-hardness is an asymptotic result of increasing problem dimensions. It is not a property of one fixed problem instance. So to ask whether your problem class is NP-hard, you would have to explain how the problem would grow with increasing numbers of $X$ variables. As far as solving your specific instance, a number of nonlinear solvers could probably do it, ...


The notion of NP-hardness relates to whether one class of problems can be solved by a solver for another class of problems where the translation overhead is negligible. The problem you presented is member of many classes of problems. However NP-hardness is the property of a class of problems. So it is impossible to answer whether this particular instance is ...


The KKT conditions are necessary conditions for an optimum to your problem, so if you can find all feasible points satisfying them, the one with the best objective function will be your optimum. There is no need to consider the corners of the feasible region explicitly. If any of them is optimal, it will also be a KKT point.

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