NP-Hardness of optimizing the sum of Rational Linear Functions over an Asymptotic-Linear-Program

We convert, within polynomial-time and sequential processing, an NP-Complete Problem into a realvariable problem of minimizing a sum of Rational Linear Functions constrained by an Asymptotic-Linear-Program. The coefficients and constants in the real-variable problem are 0, 1, -1, K, or -K, where K is the time parameter that tends to positive infinity. The number of variables, constraints, and rational linear functions in the objective, of the real-variable problem is bounded by a polynomial function of the size of the NP-Complete Problem. The NPComplete Problem has a feasible solution, if-and-only-if, the real-variable problem has a feasible optimal objective equal to zero. We thus show the strong NP-hardness of this real-variable optimization problem.