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(AF3) The second derivatives of f(x), cE(x), and cI(x) are uniformly bounded for all x ∈ Rn. (A1) For every iteration k, fkR is twice differentiable on Bk, where Bk ={x∈Rn | ∥x−xk∥∞ ≤∆k}, ∆k >0 (A2) The values of the objective and the constraints for the original optimization problem and the trust-region subproblem coincide at the current iterate, i.e., for all k, f(xk) = fkR(xk) cE(xk) = cRE,k(xk) cI(xk) = cRI,k(xk) (7.3) (A3) The gradient of the objective and the Jacobian of the constraints for both the problems coincide, i.e., for all k, ∇xf(xk) = ∇xfkR(xk) ∇xci(xk) = ∇xcRi,k(xk) i ∈ {E,I} (7.4) (A4) The second derivatives of fkR(xk), cRE,k(xk), and cRI,k(xk) remain bounded within the trust-region Bk for all k. In this work, it is assumed that (AF1)–(AF3) are true, and assumptions (A1) and (A4) hold by construction of the ROM. However, it cannot be guaranteed if assumptions (A2) and (A3) (also called first-order consistency conditions) will be true in general. Depending on the accuracy of the reduced-order model, values of the objective and the constraints for the original problem and the ROM-based trust-region subproblem may reasonably match; however, their gradients will differ since the POD basis set is obtained from the snapshots containing state information but no gradient information. One way to ensure reasonable gradient matching is to develop a separate ROM for the sensitivity equations of the original system, and solve it with the ROM for the state variables. Fahl et al. [66] applied such an approach with the adjoint sensitivity equations of the system and generated a separate ROM for the adjoint variables, different from the ROM for the state variables. However, they reported that such an approach can lead to inconsistent gradients in Problem (7.2), and thus to an algorithm which is not globally convergent. To circumvent the dependency on the accuracy of ROM, and to ensure globally-convergent behavior of the trust-region algorithm, Alexandrov et al. [8, 10], Eldred at al. [64], and Giunta Chapter 7. Trust-region Framework for ROM-based Optimization 139 7.2 Optimization ProblemPDF Image | Design and Operation of Pressure Swing Adsorption Processes
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