L-BFGS-B

L-BFGS-B extends L-BFGS to bound constraints $\ell \le \sigma \le u$. In EIT these are ${\sigma }_{\min }\le \sigma \le {\sigma }_{\max }$ (see Box Constraints on Conductivity). Each iteration:

1. Generalised Cauchy point. Follow the projected steepest-descent path $P(\sigma -t\nabla \Phi )$, where $P$ clips to the box, and find the first local minimiser of the quadratic L-BFGS model along it. This identifies the active set: variables that sit at a bound.
2. Subspace minimisation. Minimise the quadratic model over the free variables with the active ones fixed. Then project back into the box, or truncate the step.
3. Line search along the resulting direction, satisfying the Wolfe conditions.

The compact limited-memory representation keeps the cost per iteration at $\mathcal{O}(mn)$.

Why it matters in EIT. Bounds keep the conductivity physical and the forward problem coercive. Simply clipping after an unconstrained step breaks the quasi-Newton curvature information. L-BFGS-B handles the bounds consistently.

In ModularEIT.jl: LBFGS, minimize.

References

1. R. H. Byrd, P. Lu, J. Nocedal, C. Zhu (1995). A Limited Memory Algorithm for Bound Constrained Optimization. SIAM J. Sci. Comput. 16(5), 1190–1208. doi:10.1137/0916069
2. C. Zhu, R. H. Byrd, P. Lu, J. Nocedal (1997). Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization. ACM Trans. Math. Softw. 23(4), 550–560. doi:10.1145/279232.279236