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7.5 Hybrid Filter Trust-region Algorithm holds, then the reduction in the objective function predicted by ROM is more significant than the current θk, and the algorithm should promote descent in the objective. In this case, it is important that a sufficient decrease is also realized in the objective function of the original optimization problem. In other words, ρk = aredk = f(xk)−f(xk +sk) ≥η1 (7.40) predk fR(xk) − fR(xk + sk) kk should hold together with condition (7.38). In the parlance of filter methods, a step generated in such a case is called an “f-type step”. With an f-type step, xk is not added to the filter. If an iterate xk is feasible (θk = 0), equation (7.38) becomes fR(xk) − fR(xk + sk) ≥ 0. kk Consequently, the filter mechanism is irrelevant if all iterates are feasible, and the algorithm reduces to a classical unconstrained trust-region method. Moreover, condition (7.38) ensures that no feasible iterate is ever included in the filter. This is vital to avoid convergence to a feasible but suboptimal point, and crucial in allowing finite termination of the feasibility restoration phase dicussed further in section 7.5.7. 7.5.5 Algorithm Section I: Filter with ZOC As mentioned before, our interest here is to develop a hybrid filter algorithm which utilizes benefits of both ZOC and FOC. Since ZOC is cheap to construct and can predict descent in objective or infeasibility even without accurate gradients, the algorithm begins in Section I with normal subproblem (7.36) and tangential subproblem (7.37) defined with ZOC, and proceeds until no further improvement in the objective or the infeasibility measure is obtained. After this, the algorithm moves to Section II where subproblems are constructed with FOC (discussed in the next section). In Section I, instead of haphazardly solving normal and tangential subproblems, we verify if the ROM-based objective function and constraints constructed with ZOC can indeed provide a descent in either objective or infeasibility or both. Normal and tangential subproblems are solved only if we can guarantee descent. Since we have no information about accurate gradients Chapter 7. Trust-region Framework for ROM-based Optimization 168PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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