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By Armand P., Gilbert J.C.

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Then xˆ minimizes (·, λ) ˆ Proof. 2), we obtain 0 ≤ −λˆ c(x j ) − (λ j ) (c(ˆx ) + s j ) + m 2 1 j c + mµ j + ≤ −λˆ w j − (λ j ) c(ˆx ) + (λˆ − λ j ) s j + m 2 1 j c j l j x j − xˆ + mµ + j l x j − xˆ , A BFGS-IP algorithm for convex optimization 423 where w j := c(x j ) + s j . 2, {x j } and {λ j } are bounded, and by definition of B and N: c(i) (ˆx ) = 0 for i ∈ B, and λˆ (i) = 0 for i ∈ N. Hence j j λˆ N w N + λ B c B (ˆx ) ≤ mµ j + O( Now, using s j = O( tions r j = o(µ j ) and j j j ) + O( s j ).

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