  We propose an efficient approach to accurately finding the ground-state
structures in alloys based on the cluster expansion method. In this approach, a
small number of candidate ground-state structures are obtained without any
information of the energy. To generate the candidates, we employ the convex
hull constructed from the correlation functions of all possible structures by
using an efficient algorithm. This approach is applicable to not only simple
lattices but also complex lattices. Firstly, we evaluate the convex hulls for
binary alloys with four types of simple lattice. Then we discuss the structures
on the vertices. To examine the accuracy of this approach, we perform a set of
density functional theory calculations and the cluster expansion for Ag-Au
alloy and compare the formation energies of the vertex structures with those of
all possible structures. As applications, the ground-state structures of the
intermetallic compounds CuAu, CuAg, CuPd, AuAg, AuPd, AgPd, MoTa, MoW and TaW
are similarly evaluated. Finally, the energy distribution is obtained for
different cation arrangements in MgAl$_2$O$_4$ spinel, for which long-range
interactions are essential for the accurate description of its energetics.
