Lagrangian Relaxation and Pegging Test for Linear Ordering Problems
نویسندگان
چکیده
We develop an algorithm for the linear ordering problem, which has a large number of applications such as triangulation of input-output matrices, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences. The algorithm is based on the Lagrangian relaxation of a binary integer linear programming formulation of the problem. Since the number of the constraints is proportional to the third power of the number of items and grows rapidly, we propose a modified subgradient method that temporarily ignores a large part of the constraints and gradually adds constraints whose Lagrangian multipliers are likely to be positive at an optimal multiplier vector. We also propose an improvement on the ordinary pegging test by using the problem structure.
منابع مشابه
Lagrangian relaxation and pegging test for the clique partitioning problem
The clique partitioning problem is an NP-hard combinatorial optimization problem with applications to data analysis such as clustering. Though a binary integer linear programming formulation has been known for years, one needs to deal with a huge number of variables and constraints when solving a large instance. In this paper, we propose a size reduction algorithm which is based on the Lagrangi...
متن کاملApproximation Algorithms for the Multiply Constrained Assignment Problem
We consider the multiply constrained assignment problem (MCAP), which is a variation of the linear assignment problem with m additional constraints. First, we derive upper and lower bounds respectively by heuristic and relaxation methods. The latter also implies the Lagrangian relaxation, as well. Next, we introduce the pegging test to MCAP, and further a virtual pegging test to reduce MCAP int...
متن کاملA reduction approach to the repeated assignment problem
We consider the repeated assignment problem (RAP), which is a K-fold repetition of the n n linear assignment problem (LAP), with the additional requirement that no assignment can be repeated more than once. In actual applications K is typically much smaller than n. First, we derive upper and lower bounds respectively by a heuristic together with local search, and an efficient method solving the...
متن کاملAn exact algorithm for the fixed-charge multiple knapsack problem
We formulate the fixed-charge multiple knapsack problem (FCMKP) as an extension of the multiple knapsack problem (MKP). The Lagrangian relaxation problem is easily solved, and together with a greedy heuristic we obtain a pair of upper and lower bounds quickly. We make use of these bounds in the pegging test to reduce the problem size. We also present a branch-and-bound (B&B) algorithm to solve ...
متن کاملA pegging approach to the precedence-constrained knapsack problem
The knapsack problem (KP) is generalized to the case where items are partially ordered through a set of precedence relations. As in ordinary KPs, each item is associated with profit and weight, the knapsack has a fixed capacity, and the problem is to determine the set of items to be packed in the knapsack. However, each item can be accepted only when all the preceding items have been included i...
متن کامل