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7.3 Penalty-based Trust-region Algorithm parameter μ. Since the Lagrange multipliers, and thus the lower bound on μ are not known a priori, choice of a value for μ is not intuitive. Too high a value for μ can cause the algorithm to suffer from poor performance and ill-conditioning. Usually μ is adjusted by some update criterion as the algorithm proceeds, and an acceptable step is decided thereafter. Such an approach can provide considerable flexibility in choosing larger steps. However, in this work, we do not propose any update mechanism for μ and work with a constant value which is decided before the algorithm begins. 7.3.2 Trust-region Algorithm Algorithm I: Exact penalty trust-region algorithm Choose0<η1 ≤η2 <1≤η3,0<γ1 ≤γ2 <1<γ3,penaltyμ>0,aninitialtrust-region radius ∆0, minimum radius ∆min, and an initial iterate x0. Compute ψ(x0) and set k = 0. 1. Compute POD basis functions and construct a reduced-order model using the snapshots obtained for current iterate xk. 2. Compute a step sk from (7.11). Problem (7.11) can also be solved “approximately” such that a sufficient decrease condition (7.14) or (7.15) is satisfied. 3. Compute ψ(xk + sk) and define the ratio ρk = aredk = ψ(xk)−ψ(xk +sk) predk ψkR(xk) − ψkR(xk + sk) If ρk < η1, set xk+1 = xk and ∆k+1 = γ1∆k. If ∆k+1 ≤ ∆min, STOP, otherwise increment k by 1 and go to Step 2. 4. Set xk+1 = xk + sk, 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. Chapter 7. Trust-region Framework for ROM-based Optimization 143PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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