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7.5 Hybrid Filter Trust-region Algorithm f(x) xk Chapter 7. Trust-region Framework for ROM-based Optimization 165 (x) Figure 7.5: A filter with four (θ,f)-pairs for some γf,γθ ∈ (0,1). In Figure 7.5, this margin corresponds to the thin dotted line. In practical implementation, γf , γθ are typically chosen to be small. As the algorithm proceeds, we keep adding (θk,fk)-pairs to the filter for the acceptable iterates xk. However, it is important to note that (θ,f)-pairs are not added to a filter for all the acceptable iterates. We observe that θ(x) in (7.33) dominates to a certain point, especially when infeasibility is large. However, as θ(x) → 0, the method must focus on descent of f(x). For this case, no filter point is added (see further details in section 7.5.4). Since xk may not be in the filter, we move from xk to xk +sk only if xk +sk is “acceptable for the filter and xk”, i.e., if the following condition holds θ(xk+sk)≤(1−γθ)θj or f(xk+sk)≤fj−γfθj ∀(θj,fj)∈F∪(θk,fk) (7.34) Maintaining a list of (θ,f)-pairs in a filter avoids what is known as cycling. Cycling results between two points that alternately improve one of the measures, θ or f, but worsen the other one at the same time. The filter avoids cycling because for a movement from xk to xk+1 that improves θ but not f, the filter is augmented with xk, and thus xk becomes unreachable during optimization. Hence, cycling cannot occur with filter methods [74, 193].PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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