A Two-Phase Method for Solving Continuous Rank-One Quadratic Knapsack Problems

In this paper, we propose a two-phase algorithm for solving continuous rank-one quadratic knapsack problems (R1QKP). In particular, we study the solution structure of the problem without the knapsack constraint. We propose an $O(n\log n)$ algorithm in this case. We then use the solution structure to propose an $O(n^2\log n)$ algorithm that finds an interval containing the optimal value of the Lagrangian dual of R1QKP. In the second phase, we solve the restricted Lagrangian dual problem using a traditional single-variable optimization method. We perform a computational test on random instances and compare our algorithm with the general solver CPLEX.