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ψkR(xk) − ψkR(xk + sk) ≥ κp∥∇ψ(xk)∥ min ∆k, k (7.14) 7.3 Penalty-based Trust-region Algorithm The algorithm repeatedly solves the ROM-based trust region subproblem (7.11) with the Zero- order Correction (ZOC) or the First-order Correction (FOC) for the objective and the con- straints until the trust-region radius shrinks to a value less than ∆min, and no further improve- ment is obtained. In order to estimate the quality of the next iterate, we compare the actual reduction in the true objective aredk, to the predicted reduction predk. This requires compu- tation of a new solution for the DAEs of the original problem in order to evaluate ψ(xk + sk). If the trial step sk yields a satisfactory decrease, and if ρk ≥ η1, it is accepted and we update the ROM for next iteration with the help of these new snapshots just calculated. Otherwise, the size of the trust-region is reduced and Step 2 is repeated with a smaller trust-region. In Step 2, “approximately” means that a solution sk can be obtained in any manner suitable to the application as long as it satisfies the following sufficient decrease condition, also known as the fraction of Cauchy decrease condition [54] βk where 0 < κp < 1, while 1 < βk < ∞ is any bounded sequence of numbers (note that ∇ψ(xk) can be computed because of the smoothing approximation (7.13)). However, at the beginning of the algorithm, when not close to the optimum, equation (7.14) can be replaced by the following gradient free sufficient decrease condition [186] ψkR(xk) − ψkR(xk + sk) ≥ κp min[ν1, ν2∆k] (7.15) for some 0 < κp < 1, 0 < ν2 ≤ 1, ν1 > 0. Although a step sk can be computed approximately, in this work we find an exact local optimum for Problem (7.11) using IPOPT for each iteration. One of the key features of this algorithm is that once we achieve feasiblity during the course of the algorithm, we stop using the penalty formulation, constraint relaxation is removed and they are transferred back to the trust-region subproblem. In other words, Problem (7.11) is converted back to Problem (7.7). This is especially important when FOC is used for objective ∥∇ψ(x )∥ Chapter 7. Trust-region Framework for ROM-based Optimization 144PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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