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7.5 Hybrid Filter Trust-region Algorithm R. Reinitialize ∆k, λ∗ and corresponding Mk. Construct ROM with updated Mk and go to Step 6. 5. ROM construction: Compute POD basis functions using the snapshots obtained at xk. Choose λ∗, obtain corresponding Mk and construct a reduced-order model. 6. Step computation (a) Solve normal subproblem (7.36) and tangential subproblem (7.37) with FOC. (b) Ifθk =0,vk =0,andpk =0,orθk =0and∆k ≤∆min,STOP. (c) If θk ≥ θmax, or if ∆k ≤ ∆min, add xk to the filter and go to the restoration phase in Step 7. (d) 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 6. (e) Compute ρk from (7.40). (f) If (7.38) holds and ρk < η1, set xk+1 = xk and ∆k+1 = γ1∆k. Increment k by 1 and repeat Step 6. (g) If (7.38) fails, add xk to the filter. (h) Set xk+1 = xk + sk. If (7.38) fails, ∆k+1 = ∆k, else update ∆k+1 as in 3(h). Increment k by 1 and go to Step 5. 7. Restoration with FOC: Solve normal subproblem (7.36) with FOC 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 5, else STOP at an infeasible point. Clearly, algorithm begins in Section I and always terminates in Section II. The choice of the constants in the algorithm depends on the optimization problem and the scaling mechanism used for the decision variables. In this work, we choose the following values: η1 =0.05, η2 =0.5, η3 =1, γ1 =0.25, γ2 =0.5, γ3 =1.5, γf =γθ =0.01, κθ =0.01, ψ=0.09, αf =αθ =0.02 Chapter 7. Trust-region Framework for ROM-based Optimization 174PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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