  We propose a new exact approach for solving integer linear programming (ILP)
problems which we will call projective splitting algorithms (PSAs). Unlike
classical methods for solving ILP problems, PSAs conduct the search for the
optimal solution by generating candidate solutions tailored to specific values
of the objective function. As a consequence of this strategy, the number of
variables in the original ILP problem is systematically reduced without adding
any additional constraint to the initial formulation. This is the first of a
two-part series on PSAs. In this paper we focus on the resolution of pure
integer linear programming (PILP) problems, leaving the treatment of mixed
integer linear programming (MILP) formulations to the second part of this
series. The proposed algorithm was tested against the IBM ILOG CPLEX [2]
optimizer on instances of the 0-1 Multidimensional Knapsack Problem (0-1MKP),
showing satisfactory results on instances with a large number of variables.
