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7.3 Penalty-based Trust-region Algorithm and constraints as the penalty parameter can put a lot of weight on constraint violation and its gradient, thus allowing tiny reduction in the objective with each trust-region iteration. Moreover, with feasible equalities and active inequalities, smoothing parameter ε (see Equation (7.13)) in the penalty function can substantially skew the results and can make the algorithm terminate prematurely. In this work we use the following values for the constants in the algorithm η1 =0.05, η2 =0.5, η3 =1, γ1 =0.25, γ2 =0.5, γ3 =1.5 Note that the value for η1 is very close to zero. This implies that we take a step even if the reduction in ψ(x) is quite small. This is essential for POD ROM-based optimization because computation of ψ(xk +sk) in ρk involves evaluation of new snapshots from the original DAEs. Hence, it is always beneficial to take a step, even if ρk is small, and use these new snapshots to update our ROM which we expect to perform better in the next iteration. Also, the choice of η2 = 0.5 and η3 = 1 shows that most of the time during the algorithm we wish to keep the trust- region radius constant instead of increasing it frequently. With POD ROM-based optimization we prefer not to be greedy and limit the growth of the trust-region for longer duration because of the oscillatory behavior of the ROM for large trust-regions, as observed in the previous chapter. With oscillations that result from large extrapolation, ROM can take the algorithm in a direction which may not be a descent direction, which can cause ρk to become negative and lead to drastic reductions in ∆k for subsequent iterations. The oscillatory behavior can be monitored by checking whether the state variables are within the defined bounds or not. Such bounds may be used as safeguards to avoid oscillatory behavior. 7.3.3 Convergence and Optimality Since Algorithm I becomes a basic trust-region algorithm because of the smoothing approxi- mation (7.13), it enjoys the following convergence property. Theorem 7.3.1. (See Theorem 6.4.6 in [54]) Suppose that (AF1)–(AF3), (A1)–(A4), and Chapter 7. Trust-region Framework for ROM-based Optimization 145PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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