Hessian Updates

The hessian keyword selects how the Hessian (or inverse Hessian) matrix is stored and updated at each optimization step.

Available Methods

direct_psb — Powell-Symmetric-Broyden (Default)

hessian = direct_psb

Stores the direct Hessian $\mathbf{H}$ (units: Ha/Ų) and solves the linear system $\mathbf{H}\cdot {\mathbf{d}}_k=-\mathbf{g}$ at each step via LU decomposition.

The PSB update formula preserves the secant condition while maintaining symmetry:

$${\mathbf{H}}_{k+1}={\mathbf{H}}_k+\frac{(\mathbf{y}-{\mathbf{H}}_k\mathbf{s}){\mathbf{s}}^{\top }+\mathbf{s}(\mathbf{y}-{\mathbf{H}}_k\mathbf{s}{)}^{\top }}{{\mathbf{s}}^{\top }\mathbf{s}}-\frac{(\mathbf{y}-{\mathbf{H}}_k\mathbf{s}{)}^{\top }\mathbf{s}}{{\left({\mathbf{s}}^{\top }\mathbf{s}\right)}^2}\mathbf{s}{\mathbf{s}}^{\top }$$

where $\mathbf{s}={\mathbf{R}}_{k+1}-{\mathbf{R}}_k$ and $\mathbf{y}={\mathbf{g}}_{k+1}-{\mathbf{g}}_k$.

Required when use_gediis = blend. Recommended for all production runs.

inverse_bfgs — Inverse Hessian BFGS

hessian = inverse_bfgs

Stores the inverse Hessian ${\mathbf{H}}^{-1}$ (units: Ų/Ha) and computes steps as ${\mathbf{d}}_k=-{\mathbf{H}}^{-1}\mathbf{g}$ via matrix-vector multiply (no linear solve required).

Uses the standard BFGS rank-2 update:

$${\mathbf{H}}_{k+1}^{-1}=\left(\mathbf{I}-{\rho }_k\mathbf{s}{\mathbf{y}}^{\top }\right){\mathbf{H}}_k^{-1}\left(\mathbf{I}-{\rho }_k\mathbf{y}{\mathbf{s}}^{\top }\right)+{\rho }_k\mathbf{s}{\mathbf{s}}^{\top },{\rho }_k=\frac{1}{{\mathbf{y}}^{\top }\mathbf{s}}$$

Note: Incompatible with use_gediis = blend. Use direct_psb instead for the blend optimizer.

bofill — Bofill Update (Experimental)

hessian = bofill

Blends the Powell (SR1) and Murtagh–Sargent updates using a Bofill-style weighting parameter $\varphi \in [0,1]$:

$${\mathbf{H}}_{k+1}=\varphi \cdot {\mathbf{H}}_{k+1}^{\text{Powell}}+(1-\varphi )\cdot {\mathbf{H}}_{k+1}^{\text{MS}}$$

The weight $\varphi$ is computed from the curvature: $\varphi$ is close to 1 when the gradient change is well-described by a symmetric rank-1 correction (Powell), and close to 0 otherwise (Murtagh–Sargent).

Recommended for: TS-like crossing points where the Hessian may have negative eigenvalues.

powell — Powell Symmetric Rank-1 (Experimental)

hessian = powell

Uses the symmetric SR1 update. Unlike BFGS, SR1 can build up negative curvature, which is useful for crossing points with Hessians that have negative eigenvalues.

bfgs_powell_mix — Adaptive Blend (Experimental)

hessian = bfgs_powell_mix

Adaptively switches between BFGS and Powell updates using Bofill-style weighting at each step. Inherits the robustness of Powell near negative curvature and the superlinear convergence of BFGS near positive-definite regions.

Choosing a Method