The general solution of a mixed integer linear programme over a cone

We give a general method of finding the optimal objective, and solution, values of a Mixed Integer Linear Programme over a Cone (MILPC) as a function of the coefficients (objective, matrix and right- hand side). In order to do this we first convert the matrix of constraint coefficients to a Normal Form (Modified Hermite Normal Form (MHNF)). Then we project out all the variables leaving an (attainable) bound on the optimal objective value. For (M)IPs, including MILPC, projection is more complex, than in the Linear programming (LP) case, yielding the optimal objective value as a finite disjunction of inequalities The method can also be interpreted as finding the 'minimal' strengthening of the constraints of the LP relaxation which yields an integer solution to the associated LP.