The performance of the modified subgradient algorithm on solving the 0-1 quadratic knapsack problem

In this study, the performance of the modified subgradient algorithm (MSG) to solve the 0-1 quadratic knapsack problem (QKP) is examined. The MSG is proposed by Gasimov for solving dual problems constructed with respect to sharp Augmented Lagrangian function. The MSG has some important proven properties. For example, it is convergent, and it guarantees the zero duality gap for the problems such that its objective and constraint functions are all Lipschtz. Besides it doesn't need to be convexity or differentiability conditions on the primal problem. The MSG has successfully used for solving nonconvex continuous and some combinatorial problems with equality constraints since it was suggested. In this study, the MSG is used to solve the QKP which is one of the well known combinatorial optimization problems with inequality constraint. Firstly, zero-one nonlinear problem is converted into continuous nonlinear problem by adding only one constraint and not adding new variables, then to solve the continuous QKP, dual problem with "zero duality gap" is constructed by using the sharp Augmented Lagrangian function. Finally, to solve the dual problem, the MSG is used by considering the equality constraint in the computation of the norm. The proposed approach is applied for some test problems. The results are also presented and discussed.