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7.5 Hybrid Filter Trust-region Algorithm composite step defined as sk = vk + pk. Unlike MAESTRO-AMMO, we do not take the normal step vk and compute the overall step sk in the tangential subproblem in order to eschew computational expense involved in constructing corrections (ZOC or FOC) for the objective and the constraints at xk +vk. If the normal step is taken, corrections will have to constructed once for the normal subproblem at xk and again for the tangential subproblem at xk +vk. This involves evaluating the values and derivatives of the objective and the constraints of the original problem twice. To circumvent this, we construct ZOC or FOC only once at xk and solve both subproblems with xk as the center of the trust-region. We note that although Fletcher et al. [72] allow obtaining approximate solutions for nor- mal and tangential problem, we solve both problems using IPOPT and compute exact local solutions in this work. 7.5.4 Switching Condition Relying solely on the condition (7.34) can produce a situation when the sequence of iterates {xk} always provide sufficient reduction of the constraint violation alone, and not the objective function. This could result in convergence to a feasible but non-optimal point. In order to prevent this, we check the following switching condition during optimization fR(x)−fR(x +s)≥κθψ (7.38) kkkkkθk def θ(xk) is the current actual constraint violation, different from θR in (7.35), and is defined as k θ(x) = max 0, max |ci(x)|, max ci(x) (7.39) i∈E i∈I Note that θk is used for the filter margin (7.34) instead of θR defined in (7.35). Current iterate k xk is added to the filter if condition (7.38) fails. The role of this switching condition can be interpreted as follows. If it fails, then the current constraint violation θk is significant and we aim to improve on this in the future by inserting xk into the filter. On the other hand, if it R where fk (x) is the ROM-based objective function of the tangential subproblem, and θk = Chapter 7. Trust-region Framework for ROM-based Optimization 167PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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