Design and Operation of Pressure Swing Adsorption Processes

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Design and Operation of Pressure Swing Adsorption Processes ( design-and-operation-pressure-swing-adsorption-processes )

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Additive: Φ􏳧Rk (x) = ΦRk (x) + (Φ(xk) − ΦRk (xk)) + (∇Φ(xk) − ∇ΦRk (xk))T (x − xk) (7.5) (7.6) Multiplicative: Φ􏳧Rk (x) = ΦRk (x) Φ(xk) + 􏱴􏱲R R􏱳T􏱵 7.2 Optimization Problem et al. [83] propose enforcing assumptions (A2) and (A3) by using scaled/corrected objective and constraints in the trust-region subproblem which can be derived by using local corrections corresponding to the current iterate k. However, these corrections require computation of the gradient of the original objective function and the Jacobian of original constraints. In particular, they propose two types of local corrections, additive and multiplicative, which can be defined in the following manner ΦRk (xk) where Φ(x) : f(x), ci(x) Φk (xk)∇Φ(xk) − Φ(xk)∇Φk (xk) (x − xk) ΦRk (xk)2 ΦRk (x) : fkR(x), cRi,k(x) i ∈ {E, I} Multiplicative correction can become ill-conditioned and may require additional modification when ΦRk (xk) gets close to zero, especially for the equality constraints and active inequality constraints. Hence, we prefer to use the additive correction for our work. For both corrections, it is obvious that at the current iterate xk f􏳧R(x ) = f(x ), 􏳧cR (x ) = c (x ), and kk k i,kkik ∇ f􏳧R(x )=∇ f(x ), ∇ 􏳧cR (x )=∇ c(x ), i∈{E,I} xkkxk xi,kkxik Therefore, we re-define the trust-region subproblem (7.2) in terms of the corrected objective and constraints as below min f􏳧R(xk+s) sk s.t. 􏳧cRE,k(xk + s) = 0 􏳧cRI,k(xk + s) ≤ 0 xL ≤ xk + s ≤ xU ∥s∥∞ ≤ ∆k (7.7) Chapter 7. Trust-region Framework for ROM-based Optimization 140

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