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7.5 Hybrid Filter Trust-region Algorithm 7.5.3 Trust-region Step Computation Following the strategy proposed by Fletcher et al. [72], we decompose the trust-region sub- problem (7.7) into a normal subproblem and a tangential subproblem. In [72], the normal subproblem computes a step vk which reduces infeasibility of (7.7), while the tangential sub- problem evaluates a step pk which improves the objective and lies in the null space of equality and inequality constraints, thus maintaining feasibility achieved by the normal problem. The overall step sk = vk + pk . In our case, we compute a normal step vk to determine a level δ up to which the infeasibility can be reduced in the given trust-region. Tangential subproblem, then, computes the overall step sk which reduces the objective while maintaining this infeasibility level δ. Formally, to minimize the following maximum constraint violation θR(x) = δ = max 0, max |cRi (x)|, max cRi (x) i∈E i∈I we write the following normal subproblem min δ v,δ s.t. −δ≤cRE,k(xk+v)≤δ cRI,k(xk + v) ≤ δ δ≥0 xL ≤ xk + v ≤ xU ∥v∥∞ ≤ ∆c (7.35) (7.36) In order to ensure a non-zero tangential step, we choose ∆c = 0.6∆k. Once the optimum infeasibility level δ is obtained, it is fixed to δ ̄ and we solve the following tangential subproblem min fR(xk+s) sk s.t. −δ ̄ ≤ cRE,k(xk + s) ≤ δ ̄ cRI,k(xk + s) ≤ δ ̄ xL ≤ xk + s ≤ xU ∥s∥∞ ≤ ∆k (7.37) These subproblems are similar to the ones proposed by Alexandrov et al. [10] in their MAESTRO-AMMO algorithm. However, unlike our case, MAESTRO-AMMO takes the nor- mal step vk and solves the tangential subproblem at xk + vk to obtain a tangential step pk which reduced objective in the null space of the constraints. Hence, their overall step is a Chapter 7. Trust-region Framework for ROM-based Optimization 166PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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