Optimal Design with Bounded Retardation for Problems with Non-separable Adjoints

Augmenting the one-shot framework by additional constraints

“…The latter fact is represented by the quantity cG = g y 2 arising in formula (13), which measures the partial derivative ∂g/∂y 2 . The other quantity dG can be understood as the influence of y 1 on y 2 since the projection matrix (I − G y 2 ) −1 G y 1 denotes the partial derivative ∂y 2 /∂y 1 for example, the solution y 1 of g(u, y 1 , y 2 ) = 0 is independent of the choice of y 2 , then (13) simplifies to…”

“…It can be used to design so-called one-shot methods [1-3,8,11-13,16, *Email: tbosse@anl.gov 2 T. Bosse 18,22], for the (local) solution of the resulting design optimization problem min (u,y) f (u, y), s.t. y = G(u, y) (1) if the functions f and G are sufficiently smooth 1 and the fixed-point function satisfies the contraction condition…”

“…In the finite-dimensional case, Y = R n and U = R m , the KKT conditions ensure the existence of a unique adjoint variablē y * in the corresponding dual spaceȲ = R m such that 0 = L u (u * , y * ,ȳ * ) = f u (u * , y * ) + G u (u * , y * ) ȳ * 0 = L y (u * , y * ,ȳ * ) = f y (u * , y * ) + (G y (u * , y * ) − I) ȳ * 0 = Lȳ(u * , y * ,ȳ * ) = G(u * , y * ) − y * holds under the stated assumptions. Here, L : U × Y ×Ȳ → R denotes the Lagrangian L(u, y,ȳ) = f (u, y) +ȳ (G(u, y) − y) associated with (1). In combination with the contraction condition (2), the necessary optimality conditions yield an adjoint fixed-point iteration (see also [5,6,13])…”

“…The proposed method extends the multi-step Seidel-one-shot method (5) for the original problem (1) and is based on the first-order optimality conditions for the extended problem (6), which will be given at the beginning of Section 2. In detail, we replace the previous state and adjoint update in (5) by the extended state and adjoint updates (state y 2 → state y 1 ) and (adjoint (ȳ 1 ,ȳ 2 )) that now include the quantities y 1 andȳ 1 , respectively.…”

“…y is close to one. They are based upon the Karush-Kuhn-Tucker (KKT) conditions for the firstorder stationary points (u * , y * ) of the design optimization problem (1). In the finite-dimensional case, Y = R n and U = R m , the KKT conditions ensure the existence of a unique adjoint variablē y * in the corresponding dual spaceȲ = R m such that 0 = L u (u * , y * ,ȳ * ) = f u (u * , y * ) + G u (u * , y * ) ȳ * 0 = L y (u * , y * ,ȳ * ) = f y (u * , y * ) + (G y (u * , y * ) − I) ȳ * 0 = Lȳ(u * , y * ,ȳ * ) = G(u * , y * ) − y * holds under the stated assumptions.…”