A Subgradient Algorithm for Nonlinear Integer Programming

This paper describes a subgradient approach to nonlinear integer programming and, in particular, nonlinear 0-1 integer programming. In this approach, the objective function for a nonlinear integer program is considered as a nonsmooth function over the integer points. The subgradient and the supporting plane for the function are deened, and a necessary and suucient condition for the optimal solution is established, based on the theory of nonsmooth analysis. A new algorithm, called the subgradient algorithm , is developed. The algorithm is in some sense an extension of Newton's method to discrete problems: The algorithm searches for a solution iteratively among the integer points. In each iteration, it generates the next point by solving the problem for a local piecewise linear model. Each local model is constructed using the supporting planes for the objective function at a set of previously generated integer points. A solution is found when either the optimality condition is satissed or an iterate is repeated. In either case, the algorithm terminates in nite steps. The theory and the algorithm are presented. The methods for computing the supporting planes and solving the linear subproblems are described. Test results for a small set of problems are given.