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criticality measure. Theorem 7.5.2. (See Theorem 12.1.6 in [54]) Suppose that (AF1), (A2), and (A3) hold and xk belongs to a nonempty, closed and convex feasible region. Then χ(xk) defined by (7.45) is a first-order criticality measure, in the sense that it is a nonnegative, continuous function of xk,andlimk→∞χk =0ifandonlyifxk →x∗. In other words, we can always compute a Cauchy descent direction if χ(xk) > 0, and χ(xk) vanishes only when xk is a first-order critical point. Therefore, as the trust-region gets smaller, the linear part of the objective and the constraints dominate and thus, a Cauchy step can always be taken to ensure FCD condition (7.43) is satisfied. Also, because of FOC in the Section II of Algorithm II, the Cauchy step of the tangential problem (7.37) coincides with that of problem (7.37) with the original objective and constraints. Hence, Section II satifies Theorem 7.5.1, and thus converges to the first-order critical point if the restoration procedure terminates successfully. Moreover, Algorithm II never terminates in Section I and always reaches Section II. Hence, Algorithm II is globally convergent and always converges to the exact local optimum of the original optimization problem. In order to verify optimality of the termination point of Algorithm II, we conduct a perturbation analysis as done with the exact penalty trust-region algorithm. 7.6 PSA Case Study Revisited We demonstrate Algorithm II for the 2-bed 4-step PSA case study for post combustion CO2 capture, and utilize it to solve the optimization problem (7.31) with same five decision variables. The algorithm begins at the same initial guess as shown in Table 7.5. At this initial guess, ROM is constructed with a threshold error tolerance λ∗ of 0.05, similar to the exact penalty function case study, which yields M = 2, 4, 1, and 3 for pressurization, adsorption, depressurization, and desorption steps, respectively. For Section II, gradients are evaluated using perturbation. Table B.3 in Appendix B lists the trust-region iterations for the tangential subproblem, Chapter 7. Trust-region Framework for ROM-based Optimization 178 7.6 PSA Case Study RevisitedPDF Image | Design and Operation of Pressure Swing Adsorption Processes
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