The Unconstrained Nonlinear Programming Solver: Broyden-Fletcher-Goldfarb-Shanno (BFGS) Algorithm :: SAS/OR(R) 9.2 User's Guide: Mathematical Programming

The Unconstrained Nonlinear Programming Solver

Broyden-Fletcher-Goldfarb-Shanno (BFGS) Algorithm

The NLPU solver implements large-scale limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithms (recursive and matrix forms). The matrix form is used for bound-constrained optimization, while the recursive loop is used for unconstrained optimization.

Recursive Hessian matrix update:

$h_k = (v^{\rm t}_{k - 1} ... v^{\rm t}_{k - m}) h^0_k (v_{k - m} ... v_{k - ... ...- 1}) + {} \[4mm] ... \[2mm] \rho_{k - 1} s_{k - 1} s^{\rm t}_{k - 1}$

Compact matrix update:

$h_k = & \gamma_k i + [s_k & \gamma_k y_k ] x {} \[4mm] & {} x [ r^{-t}_k (d_... ..._{k} & -r^{-t} \ -r^{-1} & 0 ] [ s^{\rm t}_k \ \gamma_k y^{\rm t}_k ]$

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