Quadratic Integer Programming: Complexity and Equivalent Forms

min f (x) = 1 2 xQx + cx s.t. x 2 D (1) 36 where D is a polyhedron in R, c 2 R. Without any 37 loss of generality, we can assume that Q is a real sym38 metric (n n)-matrix. If this is not the case, then the 39 matrix Q can be converted to symmetric form by re40 placing Q by (Q + QT)/2, which does not change the 41 value of the objective function f (x). Note that if Q is 42 positive semidefinite, then Problem (1) is considered to 43 be a convex minimization problem. When Q is negative 44 semidefinite, Problem (1) is considered to be a concave 45 minimization problem. When Q has at least one positive 46 and one negative eigenvalue (i. e., Q is indefinite), Prob47 lem (1) is considered to be an indefinite quadratic pro48 gramming problem. We know that in the case of convex 49 minimization problem, every Kuhn-Tucker point is a lo50 cal minimum, which is also a global minimum. In this 51 case, there are a number of classical optimization meth52 ods that can obtain the globally optimal solutions of 53 quadratic convex programming problems. These meth54 ods can be found in many places in the literature. In 55 the case of concave minimization over polytopes, it is 56 well known that if the problem has an optimal solution, 57 then an optimal solution is attained at a vertex of D. On 58 the other hand, the global minimum is not necessarily 59 attained at a vertex of D for infinite quadratic program60 ming problems. In this case, from second order opti61 mality conditions, the global minimum is attained at the 62 boundary of the feasible domain. In this research, with63 out loss of generality, we are interested in developing 64 solution techniques to solve general (convex, concave 65 and indefinite) quadratic programming problems. 66