Solving 0–1 Programming Problems by a Penalty Approach

In this paper, we discuss the application of a penalty method to the numerical solution of a general class of large scale 0-1 combinatorial optimization problems. In this approach, a transformation based on a quadratic function is first introduced. Imposing this quadratic function as an equality constraint, the 0-1 optimization problem is converted into an equivalent optimization problem involving continuous decision variables. To avoid the involvement of the additional equality quadratic constraint in the optimization problem, we append the quadratic function to the cost to form an augmented cost function. For a linear 0-1 optimization problem, this penalty approach is particularly useful, as the constraints of the transformed problem remain linear. Numerical experiments are carried out for an NP-hard problem with up to 1000 decision variables. The numerical results have confirmed the effectiveness of the method.