By Shaidurov V. V., Timmerman G.
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Extra info for A Cascadic Multigrid Algorithm for Semilinear Indefinite Elliptic Problems
33) to evaluate the inverse of the KKT system, we get x(u) = x + A−1 BT S−1 u, where S = BA−1 BT denotes the Schur complement matrix. Thus x(u) − x = A−1 BT S−1 u, so that p(u) − p(o) = f (x(u)) − f (x) 1 T x(u) − x A x(u) − x 2 1 = ∇f (x)T A−1 BT S−1 u + uT S−1 BA−1 BT S−1 u. 2 = ∇f (x)T x(u) − x + It follows that the gradient of the primal function p at o is given by ∇p(o) = ∇f (x)T A−1 BT S−1 T = S−1 BA−1 ∇f (x). Recalling that ∇f (x) = −BT λ, we get ∇p(o) = −S−1 BA−1 BT λ = −λ. 45) Our analysis shows that if the total diﬀerential of f at x decreases outside ΩE , then this decrease is compensated by the increase of λT (Bx − c).
1 Optimization Problems and Solutions Optimization problems considered in this book are described by a cost (objective) function f deﬁned on a subset D ⊆ Rn and by a constraint set Ω ⊆ D. The elements of the constraint set Ω are called feasible vectors. The main topic of this book is development of eﬃcient algorithms for the solution of quadratic programming (QP) problems with a quadratic cost function f and a constraint set Ω ⊆ Rn described by linear equalities and inequalities. 1) f (x) ≤ f (x), x ∈ Rn , or for a solution x ∈ Ω of the constrained minimization problem f (x) ≤ f (x), x ∈ Ω.
6. 18) are answered by the next proposition. 8. 18) be deﬁned by a symmetric matrix A ∈ Rn×n , a constraint matrix B ∈ Rm×n whose column rank is less than n, and vectors b ∈ Rn , c ∈ ImB. 24) for any d ∈ KerB. 18) if and only if A|KerB is positive semideﬁnite and there is a vector λ ∈ Rm such that Ax − b + BT λ = o. 25) Proof. 18), so that for any d ∈ KerB and α ∈ R 0 ≤ f (x + αd) − f (x) = α(Ax − b)T d + α2 T d Ad. 26) implies dT Ad ≥ 0. Thus A|KerB must be positive semideﬁnite. 26) is determined by the sign of α(Ax − b)T d.
A Cascadic Multigrid Algorithm for Semilinear Indefinite Elliptic Problems by Shaidurov V. V., Timmerman G.