The authors study the nonlinear integer programming problem with arbitrary (not necessarily linear) objective function and constraints. The first two sections present the relevant literature on this problem as well as the necessary definitions and background theorems. The third section presents an auxiliary function that can be used in an algorithm, which is described in the following section, to converge to a solution, while at the same time escape areas of local minimizers by varying the value of a parameter. The algorithm is proven to converge with probability one. In the fifth and last section the authors report the results of the computational experimentation performed using this algorithm. The article concludes with a complete description of the test problems used and a list of useful references.