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7.5 Hybrid Filter Trust-region Algorithm and evaluation of the Cauchy step τθ to determine if a descent direction exists for infeasibility with current ROM incorporating Mk basis functions for the current iterate k. We note that the iterates generated during the restoration phase are never added to the filter as it can also lead to the addition of a feasible point to the filter which is detrimental for the algorithm. But we also note that whenever restoration phase is invoked at an iterate xk, it is augmented to the filter to avoid a visit to this point again in the future. Since a feasible iterate is never included in the filter, feasibility restoration phase always either generates a successful iterate, or converges to a local minimizer with some measure of infeasibility, indicating that the original optimization problem might be infeasible. 7.5.8 Trust-region Algorithm Algorithm II: Hybrid filter trust-region algorithm Choose0<η1 ≤η2 <1≤η3,0<γ1 ≤γ2 <1<γ3,γf,γθ ∈(0,1),κθ ∈(0,1),β∈(0,1), ψ > 1/(1 + β), αf , αθ ∈ (0, 1), an initial trust-region radius ∆0, minimum radius ∆min, and an initial iterate x0. Set k = 0. Section I: Filter with ZOC 1. ROM construction: Compute POD basis functions using the snapshots obtained at xk. Choose λ∗, obtain corresponding Mk and construct a reduced-order model. 2. Cauchy step computation (a) Compute sCf , sCθ , τf, and τθ. (b) If τf > 0, go to Step 3. (c) If τθ > 0, add xk to the filter and go to the restoration phase in Step 4. (d) If Mk < Mmax, decrease λ∗ to update Mk and ROM, and repeat Step 2, else, go to Step R. 3. Step computation Chapter 7. Trust-region Framework for ROM-based Optimization 172PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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