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7.3 Penalty-based Trust-region Algorithm It is worth noting that in these correction schemes, the gradient of the original objective function and the Jacobian of the constraints is computed only once at the center of the trust- region for a single trust-region iteration. Optimization within a trust-region is performed using the cheap gradient of the objective and Jacobian of the constraints of the reduced-order model, thus offering computational advantage. However, ∇f(xk), ∇cE,k(xk), and ∇cI,k(xk) will have to be evaluated for each trust-region iteration if corrected functions fR(x), cR k E,k are used. In this work, we define two kinds of additive correction schemes Zero-order Correction (ZOC) ΦRk (x) = ΦRk (x) + (Φ(xk) − ΦRk (xk)) First-order Correction (FOC) ΦRk (x) = ΦRk (x) + (Φ(xk) − ΦRk (xk)) + (∇Φ(xk) − ∇ΦRk (xk))T (x − xk) where Φ(x) : f(x), ci(x) ΦRk (x) : fkR(x), cRi,k(x) i ∈ {E, I} We can obtain Zero-order Correction cheaply as it doesn’t require gradient evaluation for the original objective and the constraints. However, with ZOC, only assumption (A2) is satisfied while assumption (A3) is not guaranteed. On the other hand, First-order Correction ensures both assumption (A2) and (A3) are satisfied, although it is expensive to construct. 7.3 Penalty-based Trust-region Algorithm 7.3.1 Penalty Formulation Note that if Problem (7.7) is constructed at an infeasible point, the trust-region box may be too small to satisfy the constraints in (7.7). Thus, handling feasibility needs special treatment in a trust-region framework. Few trust-region algorithms are designed to deal with general equality and inequality constraints. To attain feasiblity, first we explore a penalty-based formulation, (x), and cR (x) I,k (7.8) (7.9) Chapter 7. Trust-region Framework for ROM-based Optimization 141PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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