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7.5 Hybrid Filter Trust-region Algorithm region algorithm takes tiny steps and marches considerably slowly once feasibility is attained. The penalty parameter doesn’t allow infeasible moves as aredk, and thus ρk in this case becomes negative which entails sharp reduction in the trust-region radius and therefore, short steps. Hence, instead of developing a hybrid algorithm with the exact penalty algorithm, we develop a filter-based approach. Such an approach is desirable since it can allow taking a step which reduces objective while increasing infeasibility, and therefore can march faster towards the optimum. Moreover, the motivation for developing a filter-based algorithm comes from the difficulty of determining a suitable penalty parameter μ or its updation strategy in the exact penalty function. Filter methods have been extensively studied and applied for nonlinear programming prob- lems. Fletcher et al. [143] provide a brief survey of the literature on filter methods. Filter approach was first proposed by Fletcher in 1996; later described in [74]. The first global convergence proof for these methods was given for an SLP method [75], which was later gen- eralized to SQP methods [76]. Fletcher et al. [72] analyze a trust-region SQP filter method that decomposes the SQP step into a normal step to attain feasibility, and a tangential step which reduced objective function. Nie et al. [135], on the other hand, combine the normal and tangential problem with a penalty parameter and solve them simultaneously in Fletcher’s trust-region SQP filter method. Other studies with filter method include a bundle method for non-smooth optimization [73], and a pattern-search algorithm for derivative-free optimization [16]. Benson et al. [24] and Ulbrich et al. [187] have studied filter methods in the context of interior-point methods for solving NLPs. W ̈achter and Biegler [193, 194] have successfully incorporated filter mechanism in the NLP solver IPOPT [195]. They develop a line-search filter method that avoids convergence to arbitrary stagnation points, as illustrated by the example in [192]. In this work, we use Fletcher’s trust-region filter method [72] with additional modifica- tions for POD-based ROMs. The proposed modifications still enjoy the globally convergent properties of Fletcher’s algorithm (see section 7.5.9). Chapter 7. Trust-region Framework for ROM-based Optimization 163PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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