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min f (x) x s.t. cE(x)=0 cI(x) ≤ 0 xL ≤x≤xU (7.1) where x ∈ Rn are the decision variables bounded between lower and upper bounds xL and xU , respectively, and the objective function f (x) : Rn → R, equality constraints cE (x) : Rn → RNE , and inequality constraints cI (x) : Rn → RNI are assumed to be sufficiently smooth and at least twice differentiable functions. We note that Problem (7.1) is written in the reduced-space of the original DAE-constrained optimization problem. DAEs are integrated outside Problem (7.1), and the solution profiles are then used to compute the objective function and the constraints. At kth iteration during the course of optimization, the ROM constructed for a particular xk is used to build the model function for the trust-region subproblem. We define a ROM-based Chapter 7. Trust-region Framework for ROM-based Optimization 137 7.2 Optimization Problem a merit function and a trust-region SQP formulation. However, they use squared slack vari- ables for inequalities in the optimization problem which can be easily shown to fail the linear independence constraint qualification (LICQ), thus making the corresponding KKT system inconsistent. In this chapter, we develop a trust-region framework for solving constrained optimization problems using reduced-order models. In particular, we explore penalty and filter based ap- proaches to manage infeasibility and utilize a few concepts from Fahl’s TRPOD algorithm, Alexandrov’s scaling (correction) scheme for the objective function and the constraints, and MAESTRO-AMMO algorithm. The algorithms developed are demonstrated for a case study of a two-bed four step isothermal PSA process for post-combustion CO2 capture. 7.2 Optimization Problem 7.2.1 Trust-region Subproblem In this work, the original optimization problem is represented by a nonlinear program of the following formPDF Image | Design and Operation of Pressure Swing Adsorption Processes
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