A reactive search for the quadratic knapsack problem

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

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.

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 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.

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.

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.

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.

Lagrangean methods for the 0–1 Quadratic Knapsack Problem

The differences in performance among binary-integer encodings in an Ising machine, which can solve combinatorial optimization problems, are investigated. Many combinatorial optimization problems can be mapped to find the lowest-energy (ground) state of an Ising model or its equivalent model, the Quadratic Unconstrained Binary Optimization (QUBO). Since the Ising model and QUBO consist of binary variables, they often express integers as binary when using Ising machines. A typical example is the combinatorial optimization problem under inequality constraints. Here, the quadratic knapsack problem is adopted as a prototypical problem with an inequality constraint. It is solved using typical binary-integer encodings: one-hot encoding, binary encoding, and unary encoding. Unary encoding shows the best performance for large-sized problems.

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 consider a chance-constrained quadratic knapsack problem (CQKP) where each item has a random size that is finitely distributed. We present a new convex 0-1 quadratic program reformulation for CQKP. This new reformulation improves the existing reformulation for general 0-1 quadratic program based on diagonal perturbation in the sense that the continuous relaxation of the new reformulation is tighter than or at least as tight as that of the existing reformulation. The improved reformulation is derived from a general matrix decomposition of the objective function and piecewise linearization of 0-1 variables. We show that the optimal parameters in the improved reformulation can be obtained by solving an SDP problem. Extension to k-item probabilistic quadratic knapsack problems is also discussed. Preliminary comparison results are reported to demonstrate the effectiveness of the improved reformulation.

Green House Gases are a necessary evil. They are necessary for the existence of life. There is an old age saying that anything excess is bad and dangerous. This adage is applicable in case of greenhouse gases .Green House Gases become an evil because of their abnormal increase in the atmosphere due to out of bound human activities. Railway station is a transportation hub, carrying passengers and essential commodities and goods from one location to another. Railway station is a point of mass contact. In Railway station, there exist various infrastructure for movement of trains, passage of Goods Trains, Freight terminals, various facilities of passenger amenities, transportation facilities for commutation of passengers and staff, Electronic Gadgets for controlling train movements, passenger boarding, deboarding facilities etc. Congregation of huge no. of people and connected infrastructure results in fluctuation of various parameters of environment and ecology. As such, there is a need to study the environmental condition of a Railway station. However, as of now, there is limited study on carbon balance assessment of individual Railway station. Analyzing various aspects of sources of emission within the defined boundary of station and then quantifying the measures to counter such emissions thus finding the balance and based on that precisely defining the carbon Index of each Railway station across world particularly India will go a long way in the protection of environment. The instant study assesses the carbon balance of Chandi mandir Railway Station , a small nonsuburban station under Ambala Division of Northern Railway, within its defined administrative and operational boundary. The evaluation of carbon Balance of this station is done as per the scope categories defined in Greenhouse Gas (GHG) Protocol and Life Cycle Assessment (LCA) principles. It categorizes Carbon emissions into Scope I, Scope II, and Scope III. Emissions from fuel use, electricity consumption (traction and non-traction), passenger and staff commutation, waste generation, and water use are quantified using suitable carbon conversion factors(CCF). Carbon sequestration from trees and grassland within station premises and carbon offsets from rooftop solar photovoltaic systems are also quantified. Then , carbon balance of the station is evaluated by subtracting total co2 sequestration from total carbon emission. Results obtained is that of total annual GHG emissions are approximately 103.7 tCO₂e, while sequestration potential is about 122.5 tCO₂e, thus, resulting in a net carbon-negative balance of −18.8 tCO₂e per year. The study brings out a scintillating fact that small railway stations can achieve carbon negativity through massive tree plantation, use of solar energy, EV, Biogas plant, Rainwater harvesting, adopting green building technology, optimum utilisation of natural light, reduction water wastage and ecofriendly waste management. A tenpoint Recourse plan is suggested to achieve a sustainable carbon neutral Railway station. Based on carbon balance, railway stations are proposed to be classified into three categories, GREEN STATION, YELLOW STATION and RED STATION. In this study, a general guideline for small railway station is also suggested to formulate polices to achieve carbon neutrality of stations which would greatly influence the net zero target of India for the moderation of nationally determined contribution (NDC) of our country. This study is first of its kind assessment in India having huge potential in long run.

Intensification and diversification in metaheuristics are two main strategies to enhance the search process and solution quality. In Meta-RaPS (Metaheuristic for Randomized Priority Search), a recent memoryless metaheuristic, intensification and diversification strategies are controlled only by the level of randomness specified by its parameters. We introduce in this paper a Path Relinking (PR) learning algorithm and integrate it into Meta-RaPS to intelligently enhance its intensification capability by learning “good” attributes of the best solutions. To evaluate its performance, the proposed Meta-RaPS PR is tested on the 0-1 Multidimensional Knapsack Problem (MKP). The results show that applying PR as an intensification strategy in Meta-RaPS is very effective as it outperformed other approaches used in the literature with this problem. The PR approach also transformed the memoryless nature of Meta-RaPS into an “intelligent” algorithm.

Availability of rail transport is limited, due to lack of infrastructure or unprofitable. Therefore the linkage between rail and bus network are the basis of an efficient public transport system. From a passenger’s perspective, the most important role plays spatial integration (in the context of transfer). The aim of the article is to evaluate the integration of the main railway and bus stations. The study is based on the voivodship cities, which are the most important transport nodes in this country. Author focus on the location of stations in the city space, their distance, but also the spatial barriers that may hinder a possible transfer.

Recently, Ghanbari and Mahdavi-Amiri (Appl Math Model 34:3363–3375, 2010) gave the general compromised solution of an LR fuzzy linear system using ABS algorithm. Here, using this general solution, we solve quadratic programming problems with fuzzy LR variables. We convert fuzzy quadratic programming problem to a crisp quadratic problem by using general solution of fuzzy linear system. By using this method, the crisp optimization problem has fewer variables in comparison with other methods, specially when rank of the coefficient matrix is full. Thus, solving the fuzzy quadratic programming problem by using our proposed method is computationally easier than the solving fuzzy quadratic programming problem by using ranking function. Also, we study the fuzzy quadratic programming problem with symmetric variables. We show that, in this case, the associate quadratic programming problem is a convex problem, and thus, we able to find the global optimal.