Equivalent objective functions?

[abhik_giri]

Is Minimize ∑[x(i)+y(i)] the exact equivalent of ∑x(i)+∑y(i) for a LPP?

Some time back, I read somewhere that the formulation ∑x(i)+∑y(i) is more effective than ∑[x(i)+y(i)] because we are minimizing each section of the objective function separately. Same for maximizaton.

Is this correct? Can it be mathematically proved if this is correct or incorrect?

[RobPratt]

Yes, they are equivalent, and neither one is more efficient than the other.  In both cases, the LP solver sees a single vector of objective coefficients and optimizes the x and y parts simultaneously.  The variable names do not influence the solver, which internally considers all problems as having one set of variables indexed from 0 to n - 1.