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max f(x) s.t. c(x)=0 a ≤ x ≤ b ∇f(x∗)+∇c(x)λ+μl −μu =0 c(x∗)=0, μl,μu ≥0 μl(x∗ − a) = 0 μu(x∗ − b) = 0 By shifting the decision variables to the beginning of vector x and defining a null space basis matrix that satisfies ZT [I 0]T = I and ZT ∇c(x∗) = 0, allows KKT conditions to be written as: ZT ∇f(x∗) − μu = 0 : for decision variables at upper bound (μl = 0) ZT ∇f(x∗) + μl = 0 : for decision variables at lower bound (μu = 0) Sinceμu,μl ≥0,KKTconditionssimplifyto(ZT∇f(x∗))i ≤0ifadecisionvariableiisatits lower bound, and (ZT ∇f(x∗))i ≥ 0 if a decision variable i is at its upper bound. We indeed 6.5 Case Study - Hydrogen PSA in AMPL. Thus, we infer that the ROM is behaving reasonably accurately at the optimum too. Since increase in the hydrogen recovery is same, ROM is not depicting much error at the optimum. Moreover, optimum is feasible since the hydrogen purity constraint is also satisfied by the rigorous model. To futher consolidate our inference, we plot gas-phase mole fraction profiles of methane in Figure 6.7 for all steps. Mole fraction profiles are generated from the rigorous model simulation at the optimum and compared with profiles obtained from the ROM after optimization in AMPL. The profiles match reasonably accurately and show that the behavior of the ROM is close to the rigorous model at the optimum. It is worth noting that all decision variables are at their bounds after optimization. There- fore, to verify that (locally) optimal values for the ROM are also optimal for the rigorous model, we evaluate KKT conditions of the rigorous model at this point, x∗. A general rigorous model based optimization problem with its KKT conditions is given by: Chapter 6. Reduced-order Modeling for Optimization 126PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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