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7.5 Hybrid Filter Trust-region Algorithm As in the case of exact penalty trust-region algorithm, we choose a small value for η1 to allow taking a step even if the reduction in f (x) is quite small. Since computation of f (xk + sk ) in ρk involves evaluation of new snapshots from the original DAEs, which can be used to update ROM at xk +sk, it is always beneficial to move to xk +sk and expect the new ROM to predict a better descent step. Also, as in the penalty trust-region case, we choose η2 = 0.5 and η3 = 1 to maintain the trust-region for longer duration because of the oscillatory behavior of the ROM for large trust-regions. One peculiar feature of the algorithm is step 3(e) in Section I. Even though predk < 0, this step allows us to move from xk to xk + sk because of the aredk being positive. Such a sce- nario is possible especially with ROM-based trust-region subproblems without exact gradient information. In particular, we encounter this situation when the normal and the tangential sub- problems focus more on reducing infeasibility, leading to an increase in the objective function fR which causes predk to become negative. However, such an iterate can actually decrease k both infeasibility and objective for the original optimization problem, leading to a positive aredk. Inaccurate gradients in the ROM-based tangential subproblem with ZOC entails predk to become negative. Consequently, ρk becomes negative (< η1). If we move from 3(e) to 3(f), a step will be denied which is undesirable as aredk > 0. Hence, we jump from 3(e) to 3(g). Note that a counterpart of 3(e) is missing in Section II since such a scenario cannot occur as a consequence of the availability of exact gradient information in this section. Another important feature of the algorithm is that in both sections, trust-region radius is updated only when (7.38) holds. If (7.38) fails, the main effect of the current iteration is not to reduce objective (which makes ρk essentially irrelevant), but rather to reduce constraint violation (which is taken care of by inserting xk to the filter in steps 3(g) and 6(g)). In this case, we impose no further restriction on ∆k+1 and keep it same as ∆k because reducing ∆k+1 might cause steps towards infeasibility that are too small, or an unnecessary call for the restoration phase. If, on the other hand, (7.38) holds, iteration’s emphasis is on reducing the objective and ∆k+1 is updated in the conventional way. Chapter 7. Trust-region Framework for ROM-based Optimization 175PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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