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7.5.2 Filter 7.5 Hybrid Filter Trust-region Algorithm There are two objectives in a general nonlinear programming problem, minimization of the objective function f(x), and minimization of the constraint violation θ(x), where θ(x) = ∥cE (x)∥ + ∥ max(0, cI (x))∥ A penalty function combines both these goals in one single measure and minimizes ψ(x) = f(x) + μθ(x). In contrast, a filter method considers both of these as separate goals, and interprets the NLP as a bi-objective optimization problem. There is a special emphasis on the second goal since a point has to be feasible in order to be an optimal solution, and thus θ(x∗) should be zero at the optimum x∗. Filter methods borrow the concept of domination from multiobjective optimization and state that a point xk dominates xl whenever θ(xk) ≤ θ(xl) and f(xk) ≤ f(xl) As a result, xl is of no use as xk is better in terms of both feasibility and optimality. A filter method involves storing iterates xk that are not dominated by any other iterates. More precisely, a filter is a list F of pairs (θi,fi) such that either θi ≤θj or fi ≤fj ∀j,i̸=j During optimization, we aim to accept a new iterate xi only if it is not dominated by any other iterate in the filter. Figure 7.5 illustrates the concept by showing (θk,fk) at xk as black dots in the (θ, f )-space. The lines emanating from each (θ, f )-pair (forming filter envelope) indicate that any iterate whose associated (θ,f)-pair occurs in the shaded region in Figure 7.5 is dominated by at least one of these black dots. Iterates which do not lie in the shaded region are acceptable. The contours of the l1 exact penalty function will be straight lines with slope −μ in this plot, indicating that the filter is generally less restrictive than penalty methods. We do not accept new iterate xk+sk if its (θ,f)-pair is quite close to the filter envelope, and thus set a small “margin” around this envelope. Formally, we say that a point x is acceptable for the filter if and only if θ(x)≤(1−γθ)θj or f(x)≤fj −γfθj ∀(θj,fj)∈F (7.33) Chapter 7. Trust-region Framework for ROM-based Optimization 164PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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