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7.5 Hybrid Filter Trust-region Algorithm Since Section II involves expensive gradient computation, we desire to achieve larger re- duction in the objective and infeasibility in Section I itself and delay switching to Section II. POD subspace augmentation not only allows such a delay, but also improves the performance of normal and tangential subproblems with ZOC by producing more accurate ROM. Note that if Mmax is same as the number of spatial discretization nodes Nx, ROM is essentially as accu- rate as the original DAE system and Section I itself can be used to attain convergence to an optimum. However, we avoid choosing Mmax as high as Nx as we lose all the computational advantage offered by ROMs. Usually an Mmax is chosen which is reasonably high compared to current M but considerably low compared to Nx. In this sense, it is beneficial to resort to Section II with FOC and accurate gradients instead of utilizing more expensive ROMs. Finally, we note that unless Mmax = Nx, we cannot guarantee convergence to an optimal point within Section I itself. Therefore, our filter algorithm never terminates in Section I and at least one iteration of Section II is always executed. 7.5.6 Algorithm Section II: Filter with FOC In Section II of our hybrid filter algorithm, normal subproblem (7.36) and tangential subprob- lem (7.37) are constructed with First-order Corrections for the objective and the constraints. This involves computing exact gradients for each trust-region iteration. Because FOC and the exact gradients can ensure proper descent direction, we do not calculate Cauchy steps for the objective and infeasibility as done in Section I. Moreover, we do not utilize POD basis augmentation strategy for this section as the ROMs, even with few basis functions M, can generate accurate steps with accurate original gradients. As a result, we can work with smaller ROMs which leads to computationally cheaper trust-region iterations compared to Section I. We note that once the algorithm proceeds from Section I to Section II, it never resorts back to Section I in the future course of optimization. Section II is essentially the SQP-filter algorithm proposed by Fletcher et al. [72]. The difference lies in the fact that Fletcher et al. use a quadratic model approximation for their Chapter 7. Trust-region Framework for ROM-based Optimization 170PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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