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and utilize the following l1 exact penalty function [71, 140] R R R R 7.3 Penalty-based Trust-region Algorithm ψk (xk +s) = fk (xk +s)+μ ci,k(xk +s) +μ max 0,ci,k(xk +s) (7.10) i∈E i∈I to reformulate the trust-region subproblem (7.7) into the following bound-constrained problem min ψkR(xk+s) s s.t. xL ≤xk +s≤xU ∥s∥∞ ≤ ∆k (7.11) Here μ is the penalty parameter, which must be chosen sufficiently large. Note that the bound constraints xL ≤ xk + s ≤ xU are either ignored if the box trust-region lies completely within the polytope defined by them, or help to redefine the box trust-region if it intersects with the polytope. A penalty based formulation enables us to minimize the objective function while controlling constraint violations by penalizing them. The penalty function is exact in the sense that for a sufficiently high μ, the local solution of (7.7) is equivalent to the local minimizer of (7.11). To evaluate the actual reduction obtained in the original objective function in (7.1), we define the corresponding exact penalty function as ψ(xk +s)=f(xk +s)+μ |ci(xk +s)|+μ max(0,ci(xk +s)) (7.12) i∈E i∈I Since the penalty functions are non-differentiable, we adopt the following smoothing approxi- mation [20]. A value of 0.01 is used for ε in the following equations. max(0, f(x)) = 0.5 f(x) + f(x)2 + ε2 (7.13a) |f (x)| = max(0, f (x)) + max(0, −f (x)) = f (x)2 + ε2 (7.13b) One of the main issues with penalty functions is to find a reasonable value for the penalty Chapter 7. Trust-region Framework for ROM-based Optimization 142PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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