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7.5 Hybrid Filter Trust-region Algorithm lim xkj = x∗ (7.44) j→∞ and x∗ is a first-order critical point for problem (7.1). As discussed before in the case of Algorithm I, assumptions (AF1)–(AF3), (A1), and (A4) are assumed to be true in this work. Moreover, with the First-order Correction (FOC), we ensure that assumptions (A2) and (A3) are satisfied. Also, since xL ≤ x ≤ xU, (AD) is current constraint violation θki = θ(xki ), defined by (7.39), is sufficiently small. For Algorithm II, (AR) is satisfied by the construction of the normal subproblem (7.36) and by assuming that the gradients of the constraints are linearly independent. Since it is solved using a basic trust-region algorithm with exact gradients due to FOC, existence of a nonnegative step vk together with a fraction of Cauchy decrease can always be ensured unless θki = 0. also guaranteed. Assumption (AR) requires existence of a normal step especially when the def For FCD condition, χk is a first-order criticality measure. Based on the tangential problem def (7.37), we define χk = χ(xk) in the following manner χk= |min ∇fR(xk)Td| dk s.t. −δ ̄ ≤ cRi,k(xk) + ∇cRi,k(xk)T d ≤ δ ̄ i ∈ {E} i ∈ {I} (7.45) cRi,k(xk) + ∇cRi,k(xk)T d ≤ δ ̄ ∥d∥ ≤ 1 where δ ̄ is the optimum infeasibility level obtained from the following normal subproblem. min δ q,δ s.t. −δ ≤ cRi,k(xk) + ∇cRi,k(xk)T q ≤ δ cRi,k(xk) + ∇cRi,k(xk)T q ≤ δ ∥q∥≤1−θ, δ≥0 i ∈ {E} i ∈ {I} (7.46) Here θ > 0 ensures (7.45) remains feasible. We note that χk can be defined in terms of fR(xk), k cRE,k(xk), and cRI,k(xk), and their gradients since they match the original objective function and constraints, and their gradients at xk because of the FOC. In order to use χk in (7.45) for the FCD condition, we need to show that it is a first-order criticality measure. Since the constraint set of (7.45) is linear, and thus convex, the following theorem ensures that χk is a first-order Chapter 7. Trust-region Framework for ROM-based Optimization 177PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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