Quantification and Use of System Coupling in Decomposed Design Optimization Problems

Decomposition-based optimization strategies are used to solve complex engineering design problems that might be otherwise unsolvable. Yet, the associated computational cost can be prohibitively high due to the often large number of separate optimizations needed for coordination of problem solutions. To reduce this cost one may exploit the fact that some systems may be weakly coupled and their interactions can be suspended with little loss in solution accuracy. Suspending such interactions is usually based on the analyst’s experience or experimental observation. This article introduces an explicit measure of coupling strength among interconnected subproblems in a decomposed optimization problem, along with a systematic way for calculating it. The strength measure is then used to suspend weak couplings and thus improve system solution strategies, such as the model coordination method. Examples show that the resulting strategy can decrease the number of required system optimizations significantly. NOMENCLATURE fi objective function associated with system i, fi : Rqi → R ∂ f/∂x gradient vector of f (x) a row vector F objective function representing the supersystem objective, F : RN+∑i=1 ni → R gi inequality constraints associated with system i, gi :Rqi →Rmi ∂g/∂x Jacobian matrix of g with respect to x; it is m × n, if g is an m-vector and x is an n-vector hi equality constraints associated with system i, hi :Rqi →Roi k (subscript only) denotes values at kth iteration li j number of interaction variables associated with system interaction variable yi j mi number of inequality constraints associated with system inequality constraint gi ni number of design variables associated with system design variable xi N total number of systems oi number of equality constraints associated with system equality constraint hi qi total number of design and interaction variables associated with system i, qi , ∑j=1(n j + l ji) dx̂ j/dxi gradient of optimal solution of system j with respect to xi for the optimization problem with xi suspended Rn n-dimensional Euclidean (real) space xi vector of design variables associated with system i, xi ∈ Rni yi j data transfer or interaction variable vector from system i to system j where yi j ∈ Rli j ; yii ∈ Rlii from system i to itself represents system simulation (analysis) models Γi optimization coupling function vector associated with system design variable xi +,×, ||.|| matrix sum, matrix product and Euclidean norm respectively , definition