Global optimal solution to quadratic discrete programming problem with inequality constraints

This paper presents a canonical dual method for solving a quadratic discrete value selection problem subjected to inequality constraints. By using a linear transformation, the problem is first reformulated as a standard quadratic 0–1 integer programming problem. Then, by the canonical duality theory, this challenging problem is converted to a concave maximization over a convex feasible set in continuous space. It is proved that if this canonical dual problem has a solution in its feasible space, the corresponding global solution to the primal problem can be obtained directly by a general analytical form. Otherwise, the problem could be NP-hard. In this case, a quadratic perturbation method and an associated canonical primal-dual algorithm are proposed. Numerical examples are illustrated to demonstrate the efficiency of the proposed method and algorithm.