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7.5 Hybrid Filter Trust-region Algorithm (a) Solve normal subproblem (7.36) and tangential subproblem (7.37) with ZOC. (b) If θk ≥ θmax, or if ∆k ≤ ∆min, add xk to the filter and go to the restoration phase in Step 4. (c) If xk + sk is not acceptable for the filter and xk , i.e., (7.34) fails, set xk+1 = xk and ∆k+1 = γ1∆k. Increment k by 1 and repeat Step 3. (d) Compute ρk from (7.40). (e) Ifpredk <0andaredk >0,goto3(g). (f) If (7.38) holds and ρk < η1, set xk+1 = xk and ∆k+1 = γ1∆k. Increment k by 1 and repeat Step 3. (g) If (7.38) fails, add xk to the filter. (h) Set xk+1 = xk + sk. If (7.38) fails, ∆k+1 = ∆k, else γ2∆k if ρk ∈[η1,η2), ∆k+1 = ∆k if ρk ∈ [η2,η3), γ3∆k if ρk≥η3 Increment k by 1 and go to Step 1. 4. Restoration with ZOC (a) With Mk, solve normal subproblem (7.36) with ZOC using a basic trust-region algorithm until a point acceptable for the filter and xk is found. If found, increment k by 1 and go to Step 1, else continue. (b) IfMk ≥Mmax,gotoStepR. (c) Decrease λ∗ to update Mk and ROM. Compute τθ. i. If τθ > 0, repeat Step 4. ii. If Mk ≥ Mmax, go to Step R, else repeat 4(c). Section II: Filter with FOC Chapter 7. Trust-region Framework for ROM-based Optimization 173PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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