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Additional info for A branch-reduce-cut algorithm for the global optimization of probabilistically constrained linear programs

Example text

Proof. 17) any x ∈ Rn can be written in the form x = y + z, y ∈ KerB, z ∈ ImBT . 33), we get xT A x = yT Ay + 2yT Az + zT Az + Bz 2 ≥ μ y 2 − 2 A y z + (λmin + σ 2min ) z 2 μ, − A y = y , z . 10 Penalized Matrices 23 We shall complete the proof by showing that the matrix H = μ, − A − A , λmin + σ 2min is positive definite for sufficiently large values of . To this end, it is enough to show that the eigenvalues λ1 , λ2 of H are positive for sufficiently large . First observe that H is symmetric, so that λ1 , λ2 are real.

9. 18). For u ∈ Rm , let us consider also the perturbed problem min f (x) Bx=c+u as in Fig. 9. Its solution x(u) and the corresponding vector of Lagrange multipliers λ(u) are fully determined by the KKT conditions A BT B O x(u) b = , λ(u) c+u so that x(u) A BT = B O λ(u) −1 b A BT = B O c+u −1 b A BT + B O c −1 o . 4 Equality Constrained Problems 47 ∇f (x)T d(u) = −(BT λ)T d(u) = −λT Bd(u) = −λT u. It follows that −[λ]i can be used to approximate the change of the optimal cost due to the violation of the ith constraint by [u]i .

E ∇f (x) x(u) = x + d(u) Bx = c + u x BT λ Bx = c f (x) = d Fig. 9. 18). For u ∈ Rm , let us consider also the perturbed problem min f (x) Bx=c+u as in Fig. 9. Its solution x(u) and the corresponding vector of Lagrange multipliers λ(u) are fully determined by the KKT conditions A BT B O x(u) b = , λ(u) c+u so that x(u) A BT = B O λ(u) −1 b A BT = B O c+u −1 b A BT + B O c −1 o . 4 Equality Constrained Problems 47 ∇f (x)T d(u) = −(BT λ)T d(u) = −λT Bd(u) = −λT u. It follows that −[λ]i can be used to approximate the change of the optimal cost due to the violation of the ith constraint by [u]i .

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