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11qnewton

# 11qnewton - EE236C(Spring 2008-09 11 Quasi-Newton methods...

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Unformatted text preview: EE236C (Spring 2008-09) 11. Quasi-Newton methods • variable metric methods • quasi-Newton methods • BFGS update • limited-memory quasi-Newton methods 11–1 Newton method for unconstrained minimization minimize f ( x ) f convex, twice continously differentiable Newton method x + = x − t ∇ 2 f ( x ) − 1 ∇ f ( x ) • advantages: fast convergence, affine invariance • disadvantages: requires second derivatives, solution of linear equation can be too expensive for large scale applications Quasi-Newton methods 11–2 Variable metric methods x + = x − tH − 1 ∇ f ( x ) H ≻ is approximation of the Hessian at x , chosen to: • avoid calculation of second derivatives • simplify computation of search direction ‘variable metric’ interpretation (EE236B, lecture 10, page 11) Δ x = − H − 1 ∇ f ( x ) is steepest descent direction at x for quadratic norm bardbl z bardbl H = ( z T Hz ) 1 / 2 Quasi-Newton methods 11–3 Quasi-Newton methods given starting point x (0) ∈ dom f , H ≻ for k = 1 , 2 , . . . , until a stopping criterion is satisfied 1. compute quasi-Newton direction Δ x = − H − 1 k − 1 ∇ f ( x ( k − 1) ) 2. determine step size t ( e.g. , by backtracking line search) 3. compute x ( k ) = x ( k − 1) + t Δ x 4. compute H k • different methods use different rules for updating H in step 4 • can also propagate H − 1 k to simplify calculation of Δ x Quasi-Newton methods 11–4 Broyden-Fletcher-Goldfarb-Shanno (BFGS) update H k = H k − 1 + yy T...
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11qnewton - EE236C(Spring 2008-09 11 Quasi-Newton methods...

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