  This numerical simulation study investigates solving the L1-norm fitted and
regularized (L1-L1) linear programming (LP) problem to find a well-localized
volumetric current density in transcranial electrical stimulation (tES), where
a current pattern is attached through contact electrodes attached to the skin
to create a stimulus in a targeted brain region. We consider a metaheuristic
optimization process where the problem parameters are selected so that the
final solution found is optimal with respect to given metacriteria, e.g., the
intensity and focality of the volumetric current density in the brain. We focus
on the effect of the LP algorithm on the solution. We examine interior-point
and simplex algorithms, which constitute two major alternative ways to solve an
LP task; the interior-point algorithms are based on determining a feasible
solution set to allow finding an optimizer via Newton's method, while the
simplex methods take steps along the edges of a polytope, subdividing the set
of candidate solutions into simplicial polygons. Interior-point methods stand
out among them for their complexity and predictability in terms of convergence
properties at the same time. To find the current pattern, we apply five
alternative optimization toolboxes: Matlab, MOSEK, Gurobi, SDPT3, and SeDuMi.
We suggest that the mutual differences between these vary based on the
placement of the target as well as the resolution of the two-parameter lattice.
In the numerical experiments, we investigate maximizing the focality and
intensity of the L1-L1 optimized stimulation current and, in the latter regard,
examine its relationship to the reciprocal current pattern, maximizing the
focused current density.
