Lagrangean methods for the 0–1 Quadratic Knapsack Problem

In this paper, we propose an reactive method to solve the Quadratic Knapsack Problem (noted QKP). The quadratic knapsack problem is a well-studied combinatorial optimisation problem. In all variants of the quadratic knapsack problems, for a set of given items, profits are not only assigned to individual items but also to pairs of them. The pairwise profit is added to the quadratic objective value only when the two corresponding items are both included in the same knapsack. We let a knapsack with a capacity C and a set of candidate objects (or items), each item has a positive weight w <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , and profit Pi, if selected, generates an object profit p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> and a pairwise profit P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</inf> with any other selected object j. The objective of the QKP is to select a subset of objects to fill the knapsack so as to maximize the overall profit P, while the overall weight does not exceed the knapsack capacity C. This problem is known as quadratic knapsack problem and has been shown strongly NP-hard and it has been applied with a variety of important applications, such as, in the location of satellites, airports, railway stations or freight terminals. The proposed method is mainly based on two complementary phases. In the first phase we use a greedy algorithm for provide the starting solution. In the second phase we improved the quality of the starting solutions based on a reactive search with applying a destroy and repair process, the reactive phase is based both diversification and intensification strategy. The obtained results are compared to those reached by the Cplex solverl and literature. Consequently, the experimental results show the effectiveness of the proposed approach, and the computational results show that the algorithm is capable of solving instances of the QKP that cannot be solved by other methods.

In this paper we study solution methods for solving the dual problem corresponding to the Lagrangian Decomposition of two-stage stochastic mixed 0-1 models. We represent the two-stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangian Decomposition (LD) is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangian Decomposition schemes: the Subgradient Method, the Volume Algorithm, the Progressive Hedging Algorithm, and the Dynamic Constrained Cutting Plane scheme. We test the tightness of the LD bounds in a testbed of medium- and large-scale stochastic instances.

Decomposition approaches for the efficient solution of short-term scheduling problems

A simplified binary artificial fish swarm algorithm for 0–1 quadratic knapsack problems

In this paper, the Lagrangian decomposition and coordination technique is applied to flowshop scheduling problems in which the sum of the changeover cost and the tardiness penalty is minimized. The proposed method possesses a feature that the scheduling problem is decomposed not into single job-level subproblems but into single machine-level subproblems. By decomposing the problem into one-machine multi-operation subproblems, the changeover cost and/or changeover time can easily be embedded in the objective function. In the proposed method, each subproblem for a single machine is solved by combining the simulated annealing method and the neighborhood search algorithm. In order to avoid oscillations in multiplier values, a new Lagrangian function is used to solve each subproblem. The effectiveness of the proposed method is verified by comparing the results of the example problems solved by the proposed method with those solved by the simulated annealing method in which a schedule of the entire machine is successively improved.

In all the variants of the knapsack problems considered so far the profit of choosing a given item was independent of the other items chosen. In many real life applications as well as in problems with roots in graph theory it is natural to assume that the profit of a packing also should reflect how well the given items fit together. One possible formulation of such an interdependence is the quadratic knapsack problem (QKP) in which an item has a corresponding profit and an additional profit is redeemed if the item is selected together with another item. (QKP) was first introduced by Gallo, Hammer and Simeone [160] and has been studied intensively in the last decade due to its simple structure and challenging difficulty.

The Quadratic Knapsack Problem (QKP) deals with maximizing a quadratic objective function subject to given constraints on the capacity of the Knapsack. We assume all coefficients to be non-negative and all variables to be binary. Solution to QKP generalizes the problem of finding whether a graph contains a clique of given size. We propose in this paper a Novel Quantum Evolutionary Algorithm (NQEA) for QKPs. These algorithms are general enough and can be used for similar subsection of problems. We report in this paper solutions which lie in less than 1% of the optimal solutions. We also show that our algorithm is scalable to much larger problem sizes and is capable of exploiting the search space to its maximum.

Lagrangian relaxation is often an efficient tool to solve (large-scale) optimization problems, even nonconvex. However it introduces a duality gap, which should be small for the method to be really efficient. Here we make a geometric study of the duality gap. Given a nonconvex problem, we formulate in a first part a convex problem having the same dual. This formulation involves a convexification in the product of the three spaces containing respectively the variables, the objective and the constraints. We apply our results to several relaxation schemes, especially one called “Lagrangean decomposition” in the combinatorial-optimization community, or “operator splitting” elsewhere. We also study a specific application, highly nonlinear: the unit-commitment problem.

A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by benchmark problems in optimal topology design. The performance and novelty of the proposed method are compared with the popular commercial codes.

The convex hull heuristic is a heuristic for mixed-integer programming problems with a nonlinear objective function and linear constraints. It is a matheuristic in two ways: it is based on the mathematical programming algorithm called simplicial decomposition, or SD (von Hohenbalken in Math Program 13:49–68, 1977), and at each iteration, one solves a mixed-integer programming problem with a linear objective function and the original constraints, and a continuous problem with a nonlinear objective function and a single linear constraint. Its purpose is to produce quickly feasible and often near optimal or optimal solutions for convex and nonconvex problems. It is usually multi-start. We have tested it on a number of hard quadratic 0–1 optimization problems and present numerical results for generalized quadratic assignment problems, cross-dock door assignment problems, quadratic assignment problems and quadratic knapsack problems. We compare solution quality and solution times with results from the literature, when possible.

This paper presents a new Iterated Semi-Greedy Algorithm (ISGA) for the 0-1 Quadratic Knapsack Problem.The proposed ISGA is easier to implement, runs faster, and produces comparable results than state-of-the-art methods. Computational evaluation is performed over a benchmark with large instances of 1000 and 2000 objects.

The 0-1 quadratic knapsack problem (QKP) is a hard computational problem, which is a generalization of the knapsack problem (KP). In this paper, a mini-swarm system is presented. Each agent, realized with minor declarative knowledge and simple behavioral rules, searches on a structural landscape of the problem through the guided generate-and-test behavior under the law of socially biased individual learning, and cooperates with others by indirect interactions. The formal decomposition of behaviors allows understanding and reusing elemental operators, while utilizes the heuristic information on the landscape. The results on a collection of the QKP instances by mini-swarm versions are compared with that of both a branch-and-bound algorithm and a greedy genetic algorithm, which show its effectiveness

We have developed a method to extend the functionality of the Ising machine, which currently has an energy function limited to a binary-quadratic form. The proposed method utilizes auxiliary variables dependent on the decision variables of the problem to be solved so that it can be accelerated by hardware parallel processing. Auxiliary variables add third-order or higher terms or rectified linear unit-type nonlinear functions to the binary-quadratic energy function of the Ising machine, modifying the value of the energy difference resulting from the reversal of the decision variable. To enable parallel computation, the computation of the energy change uses information that can be accessed locally by the decision neurons. We confirmed that the proposed method works for the 0/1 linear knapsack problem, the quadratic knapsack problem, the 3-SAT problem, and the random 3-XORST problem.

In the conventional Lagrangian relaxation approach, the scheduling problems are decomposed into job-level sub-problems or operation-level sub-problems. However, these approaches are not applicable to the flowshop problems with the changeover cost which depends on the sequence of operations. In this paper, we propose a machine-oriented decomposition method for the flowshop problems using the Lagrangian decomposition and coordination technique. In the proposed method, each sub-problem for single machine is solved by the simulated annealing method. The solutions of the sub-problems are used to generate a feasible schedule by a heuristic procedure. The effectiveness of the proposed method is verified by comparing the results of the example problems solved by the proposed method with those solved by the conventional method. Furthermore, it has been shown that the proposed approach is easily applicable to the flow hop problem with resource constrains minimizing the changeover costs and due date penalties.

In the wireless sensor networks, deploying a small amount of heterogeneous nodes which directly communicate with the Sink node, can form shortcuts. The shortcuts can make the network to have small-world properties, which can save energy and improve network performance. In the large-scale networks, the optimal deployment of heterogeneous nodes is a NP-hard problem. Considering the total energy consumption of the network and the uniformity of each node energy consumption, this paper presents a clustering heterogeneous network routing algorithm which based on mixed integer programming (CHNMIP). Firstly, the algorithm supposes that the heterogeneous nodes can only be placed in the position of the common nodes and converts the optimal deployment of heterogeneous nodes into a mixed integer programming problem, and finds out the approximate optimal solution using Lagrangian relaxation and Benders decomposition algorithm. Then, it dynamically divides common nodes into clusters, and sets the appropriate transmission. Finally, it theoretically analyzes the performance of CHNMIP routing algorithm, and the simulation results also show that CHNMIP routing algorithm can better improve the network performance in different simulation environment.