  The properties of the ground state of the simplest frustrated system, the
dilute Ising chain in a magnetic field, are rigorously investigated over the
entire range of concentrations of charged non-magnetic impurities. Analytical
methods are proposed for calculating the residual entropy of frustrated states,
including states at phase boundaries, which are based on the Markov property of
the system and involve solving a linear optimization problem for energy and a
nonlinear optimization problem for entropy. These methods allow obvious
generalizations for one-dimensional pseudospin models with anisotropic
interactions. We calculate the composition, entropy and magnetization for the
ground state phases. We prove the absence of pseudo-transitions in the dilute
Ising chain, since the residual entropy of states at phase boundaries is always
higher than the entropy of adjacent phases. The concentration dependencies of
magnetization at the phase boundaries are obtained, and unlike linear
dependencies for adjacent phases, they have nonlinear behavior. Field-induced
transitions between ground states and entropy jumps associated with them are
also considered, and in particular, it is shown that the field-induced
transition from an antiferromagnetic state to a frustrated one is accompanied
by charge ordering.
