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7.5 Hybrid Filter Trust-region Algorithm trust-region (tangential) subproblems, while we use a POD-based ROM as a surrogate model. However, we ensure all the assumptions made by Fletcher et al. are satisfied, which renders our algorithm similar to the SQP-filter algorithm from convergence theory point of view (see further details in section 7.5.9). 7.5.7 Feasibility Restoration Phase The algorithm switches to a feasibility restoration phase when it is not able to obtain an “admissible” step size. The step is not admissible: • when the next iterate xk + sk obtained after solving the normal and the tangential subproblem is not acceptable for the filter and xk, or • when xk + sk is acceptable for the filter and xk, and satisfies the switching condition (7.38) as well, but fails to provide sufficient reduction in the original objective function f(x), and generates ρk < η1. Restoration phase is invoked when either the current trust-region radius ∆k goes below ∆min, when the current infeasibility level θR goes beyond a maximum limit θmax, or when in Section k I, τf ≤ 0 together with τθ > 0, i.e., when ROM-based subproblems with ZOC can only decrease infeasibility but not the objective function. The purpose of the restoration phase is to decrease the current constraint violation and generate a new iterate xk+sk which is acceptable for the current filter and xk. In our algorithm, restoration phase involves solving the normal subproblem using the basic trust-region algorithm [54] until such an iterate is obtained. Consequently, restoration phase can follow its own trust- region update rules separate from the ones used in the filter trust-region algorithm. In the terminology of filter methods, such an infeasibility reducing step is known as “θ-type step” or “h-type step”. We define distinct restoration phases for Section I and Section II. Restoration phase in Section I solves the normal subproblem with ZOC, while that in Section II solves it with FOC. Moreover, restoration phase in Section I also involves POD subspace augmentation Chapter 7. Trust-region Framework for ROM-based Optimization 171PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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