BFGS starts from on a problem with two parameters

PHYSICS · Subjective · Fermi University Physics · Computational Physics · Scientific programming for physicists · Scientific programming for physicists

Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness

Problem

BFGS starts from B0−1=1B_0^{-1}=\mathbb{1} on a problem with two parameters. Its first step is s0=(1,0)\mathbf{s}_0=(1,0), and the gradient changes over that step by y0=(2,1)\mathbf{y}_0=(2,1). (a) Show from the chapter's inverse update that Bk+1−1yk=skB_{k+1}^{-1}\mathbf{y}_k=\mathbf{s}_k for any Bk−1B_k^{-1}, whenever ykTsk≠0\mathbf{y}_k^{\mathsf T}\mathbf{s}_k\ne0. (b) Compute B1−1B_1^{-1} and show that it is positive definite. (c) At the new point the gradient is ∇f=(0,1)\nabla f=(0,1). Find the BFGS direction d=−B1−1∇f\mathbf{d}=-B_1^{-1}\nabla f and the slope ∇f⋅d\nabla f\cdot\mathbf{d}.

Browse the problem library