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where fkR(xk + s) is the objective function and cRE,k(xk + s), and cRI,k(xk + s) are the equality and inequality constraints, respectively, computed from the reduced set of state variables of the reduced-order model. For this subproblem also, DAEs of the ROM are solved outside Problem (7.2) and the solution of the unknown temporal coefficients in the POD expansion is then used to obtain fkR(xk + s), cRE,k(xk + s), and cRI,k(xk + s). The last inequality in Problem (7.2) is the trust-region constraint which limits the step size within the current trust-region radius ∆k. In this work, we prefer to use an infinity norm for the trust-region constraint, i.e., we use a box-type (l∞) trust-region to restrict the step size of the decision variables. It should be noted that the dimension of x, f(x), cE(x), and cI(x) remains same for both original optimization problem as well as ROM-based trust-region subproblem. In other words, the number of decision variables and constraints remain same for both problems. Computa- tional advantage is achieved in terms of the smaller number of DAEs of the reduced-order model which leads to cheap calculation of the gradients of the objective function and the constraints in the trust-region subproblem (7.2). 7.2.2 Correction (scaling) for Objective and Constraints To develop a robust and globally convergent trust-region algorithm involving approximate models, the following assumptions should hold [54] (AF1) Functions f(x), cE(x), and cI(x) are twice-continuously differentiable on Rn. (AF2) The function f(x) is bounded below for all x ∈ Rn. 7.2 Optimization Problem trust-region subproblem for kth iteration as min fkR(xk+s) s s.t. cRE,k(xk + s) = 0 cRI,k(xk + s) ≤ 0 xL ≤ xk + s ≤ xU ∥s∥∞ ≤ ∆k (7.2) Chapter 7. Trust-region Framework for ROM-based Optimization 138PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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