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Journal articles on the topic 'Unconstrained binary quadratic'

Author: Grafiati
Published: 11 January 2025

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1

Verma, Amit, and Mark Lewis. "Goal seeking Quadratic Unconstrained Binary Optimization." Results in Control and Optimization 7 (June 2022): 100125. http://dx.doi.org/10.1016/j.rico.2022.100125.

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2

Lewis, Mark, John Metcalfe, and Gary Kochenberger. "Robust optimisation of unconstrained binary quadratic problems." International Journal of Operational Research 36, no. 4 (2019): 441. http://dx.doi.org/10.1504/ijor.2019.10025701.

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3

Lewis, Mark, John Metcalfe, and Gary Kochenberger. "Robust optimisation of unconstrained binary quadratic problems." International Journal of Operational Research 36, no. 4 (2019): 441. http://dx.doi.org/10.1504/ijor.2019.104050.

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4

Boettcher, S. "Extremal Optimization for Quadratic Unconstrained Binary Problems." Physics Procedia 68 (2015): 16–19. http://dx.doi.org/10.1016/j.phpro.2015.07.102.

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5

Wang, Yang, Zhipeng Lü, Fred Glover, and Jin-Kao Hao. "Path relinking for unconstrained binary quadratic programming." European Journal of Operational Research 223, no. 3 (December 2012): 595–604. http://dx.doi.org/10.1016/j.ejor.2012.07.012.

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6

Kochenberger, Gary, Jin-Kao Hao, Fred Glover, Mark Lewis, Zhipeng Lü, Haibo Wang, and Yang Wang. "The unconstrained binary quadratic programming problem: a survey." Journal of Combinatorial Optimization 28, no. 1 (April 18, 2014): 58–81. http://dx.doi.org/10.1007/s10878-014-9734-0.

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7

Glover, Fred, and Jin-Kao Hao. "f-Flip strategies for unconstrained binary quadratic programming." Annals of Operations Research 238, no. 1-2 (December 11, 2015): 651–57. http://dx.doi.org/10.1007/s10479-015-2076-1.

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8

Rahmeh, Samer, and Adam Neumann. "HUBO & QUBO and Prime Factorization." International Journal of Bioinformatics and Intelligent Computing 3, no. 1 (February 20, 2024): 45–69. http://dx.doi.org/10.61797/ijbic.v3i1.301.

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This document details the methodology and steps taken to convert Higher Order Unconstrained Binary Optimization (HUBO) models into Quadratic Unconstrained Binary Optimization (QUBO) models. The focus is primarily on prime factorization problems; a critical and computationally intensive task relevant in various domains including cryptography, optimization, and number theory. The conversion from Higher-Order Binary Optimization (HUBO) to Quadratic Unconstrained Binary Optimization (QUBO) models is crucial for harnessing the capabilities of advanced computing methodologies, particularly quantum computing and DYNEX neuromorphic computing. Quantum computing offers potential exponential speedups for specific problems through its intrinsic parallelism capabilities. Conversely, DYNEX neuromorphic computing enhances efficiency and accelerates the resolution of intricate, pattern-oriented tasks by simulating memristors in GPUs, employing a highly decentralized approach, via Blockchain technology. This transformation enables the exploitation of these cutting-edge computing paradigms to address complex optimization challenges effectively. Through detailed explanations, mathematical formulations, and algorithmic strategies, this document aims to provide a comprehensive guide to understanding and implementing the conversion process from HUBO to QUBO. It underscores the importance of such transformations in making prime factorization computationally feasible on both existing classical computers and emerging computing technologies.
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Merz, Peter, and Kengo Katayama. "Memetic algorithms for the unconstrained binary quadratic programming problem." Biosystems 78, no. 1-3 (December 2004): 99–118. http://dx.doi.org/10.1016/j.biosystems.2004.08.002.

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10

Liefooghe, Arnaud, Sébastien Verel, and Jin-Kao Hao. "A hybrid metaheuristic for multiobjective unconstrained binary quadratic programming." Applied Soft Computing 16 (March 2014): 10–19. http://dx.doi.org/10.1016/j.asoc.2013.11.008.

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11

Wang, Di, and Robert Kleinberg. "Analyzing quadratic unconstrained binary optimization problems via multicommodity flows." Discrete Applied Mathematics 157, no. 18 (November 2009): 3746–53. http://dx.doi.org/10.1016/j.dam.2009.07.009.

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12

Mauri, Geraldo Regis, and Luiz Antonio Nogueira Lorena. "Lagrangean decompositions for the unconstrained binary quadratic programming problem." International Transactions in Operational Research 18, no. 2 (February 2, 2011): 257–70. http://dx.doi.org/10.1111/j.1475-3995.2009.00743.x.

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13

Boros, Endre, Peter L. Hammer, and Gabriel Tavares. "Local search heuristics for Quadratic Unconstrained Binary Optimization (QUBO)." Journal of Heuristics 13, no. 2 (February 21, 2007): 99–132. http://dx.doi.org/10.1007/s10732-007-9009-3.

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14

Glover, Fred, Zhipeng Lü, and Jin-Kao Hao. "Diversification-driven tabu search for unconstrained binary quadratic problems." 4OR 8, no. 3 (January 5, 2010): 239–53. http://dx.doi.org/10.1007/s10288-009-0115-y.

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15

Wang, Qiwei. "Support vector machine based on the quadratic unconstrained binary optimization model." Journal of Physics: Conference Series 2858, no. 1 (October 1, 2024): 012002. http://dx.doi.org/10.1088/1742-6596/2858/1/012002.

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Abstract Support vector machine (SVM) is a powerful supervised machine learning model that is often used in binary classification algorithms. As Moore’s Law approaches its theoretical limits and the demand for machine learning to handle large-scale, high-dimensional data analysis intensifies, the necessity of adopting non-traditional computational approaches becomes evident. Quantum computing, in particular, emerges as a vital solution for the effective training of SVM models, providing capabilities beyond those of classical computing systems. To solve the above problems, a QUBO (quadratic unconstrained binary optimization) model is proposed to transform the SVM machine learning model into a quadratic unconstrained binary optimization problem so that they can be effectively trained on the D-Wave platform using adiabatic quantum computer. The results show that the QUBO model can transform the SVM model into a simple quadratic programming problem, which makes it suitable for adiabatic quantum computer processing. When processing large-scale and high-dimensional data, this transformation shows a natural advantage and significantly improves computational efficiency. The application potential of this transformation technology is huge in the medical field.
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Palubeckis, Gintaras. "Iterated Tabu Search for the Unconstrained Binary Quadratic Optimization Problem." Informatica 17, no. 2 (January 1, 2006): 279–96. http://dx.doi.org/10.15388/informatica.2006.138.

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17

Glover, Fred, Bahram Alidaee, César Rego, and Gary Kochenberger. "One-pass heuristics for large-scale unconstrained binary quadratic problems." European Journal of Operational Research 137, no. 2 (March 2002): 272–87. http://dx.doi.org/10.1016/s0377-2217(01)00209-0.

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18

Lewis, Mark, and Fred Glover. "Quadratic unconstrained binary optimization problem preprocessing: Theory and empirical analysis." Networks 70, no. 2 (June 24, 2017): 79–97. http://dx.doi.org/10.1002/net.21751.

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19

Matsumoto, Yuki, and Shota Nakamura. "qualign: solving sequence alignment based on quadratic unconstrained binary optimisation." EMBnet.journal 28 (March 8, 2023): e1020. http://dx.doi.org/10.14806/ej.28.0.1020.

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Bioinformatics has, among others, the issue of solving complex computational problems with vast amounts of sequencing data. Recently, a new computing architecture, the annealing machine, has emerged that applies to actual problems and is available for practical use. This novel architecture can solve discrete optimisation problems by replacing algorithms designed under the von Neumann architecture. To perform computations on the annealing machine, quadratic unconstrained binary optimisation (QUBO) formulations should be constructed and optimised according to the application. In this study, we developed an algorithm under the annealing machine architecture to solve sequence alignment problems, a known fundamental process widely used in genetic analysis, such as mutation detection and genome assembly. We constructed a QUBO formulation based on dynamic programming to solve a pairwise sequence alignment and derived its general form. We compared with conventional methods to solve 40 bp of pairwise alignment problem. Our implementation, named qualign, solved sequence alignment problems with accuracy comparable to that of conventional methods. Although a small pairwise alignment was solved owing to the limited memory size of this method, this is the first step of the application of annealing machines. We showed that our QUBO formulation solved the sequencing alignment problem. In the future, increasing the memory size of annealing machine will allow annealing machines to impact a wide range of bioinformatics applications positively.Availability: the source code of qualign is available at https://github.com/ymatsumoto/qualign
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20

Ronagh, Pooya, Brad Woods, and Ehsan Iranmanesh. "Solving constrained quadratic binary problems via quantum adiabatic evolution." Quantum Information and Computation 16, no. 11&12 (September 2016): 1029–47. http://dx.doi.org/10.26421/qic16.11-12-6.

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Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems.
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21

Turkalj, Ivica, Mohammad Assadsolimani, Markus Braun, Pascal Halffmann, Niklas Hegemann, Sven Kerstan, Janik Maciejewski, Shivam Sharma, and Yuanheng Zhou. "Quadratic Unconstrained Binary Optimization Approach for Incorporating Solvency Capital into Portfolio Optimization." Risks 12, no. 2 (January 29, 2024): 23. http://dx.doi.org/10.3390/risks12020023.

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In this paper, we consider the inclusion of the solvency capital requirement (SCR) into portfolio optimization by the use of a quadratic proxy model. The Solvency II directive requires insurance companies to calculate their SCR based on the complete loss distribution for the upcoming year. Since this task is, in general, computationally challenging for insurance companies (and therefore, not taken into account during portfolio optimization), employing more feasible proxy models provides a potential solution to this computational difficulty. Here, we present an approach that is also suitable for future applications in quantum computing. We analyze the approximability of the solvency capital ratio in a quadratic form using machine learning techniques. This allows for an easier consideration of the SCR in the classical mean-variance analysis. In addition, it allows the problem to be formulated as a quadratic unconstrained binary optimization (QUBO), which benefits from the potential speedup of quantum computing. We provide a detailed description of our model and the translation into a QUBO. Furthermore, we investigate the performance of our approach through experimental studies.
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22

Papp, Dávid. "On the Complexity of Local Search in Unconstrained Quadratic Binary Optimization." SIAM Journal on Optimization 26, no. 2 (January 2016): 1257–61. http://dx.doi.org/10.1137/15m1047775.

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23

Douiri, Sidi Mohamed, and Souad Elbernoussi. "An unconstrained binary quadratic programming for the maximum independent set problem." Nonlinear Analysis: Modelling and Control 17, no. 4 (October 25, 2012): 410–17. http://dx.doi.org/10.15388/na.17.4.14047.

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For a given graph G = (V, E) the maximum independent set problem is to find the largest subset of pairwise nonadjacent vertices. We propose a new model which is a reformulation of the maximum independent set problem as an unconstrained quadratic binary programming, and we resolve it afterward by means of a genetic algorithm. The efficiency of the approach is confirmed by results of numerical experiments on DIMACS benchmarks.
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24

Glover, Fred, and Jin Kao Hao. "Fast two-flip move evaluations for binary unconstrained quadratic optimisation problems." International Journal of Metaheuristics 1, no. 2 (2010): 100. http://dx.doi.org/10.1504/ijmheur.2010.034201.

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25

Mauri, Geraldo Regis, and Luiz Antonio Nogueira Lorena. "A column generation approach for the unconstrained binary quadratic programming problem." European Journal of Operational Research 217, no. 1 (February 2012): 69–74. http://dx.doi.org/10.1016/j.ejor.2011.09.016.

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26

Palubeckis, Gintaras. "Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem." Annals of Operations Research 131, no. 1-4 (October 2004): 259–82. http://dx.doi.org/10.1023/b:anor.0000039522.58036.68.

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27

Pardalos, Panos M., Oleg A. Prokopyev, Oleg V. Shylo, and Vladimir P. Shylo. "Global equilibrium search applied to the unconstrained binary quadratic optimization problem." Optimization Methods and Software 23, no. 1 (February 2008): 129–40. http://dx.doi.org/10.1080/10556780701550083.

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28

Kochenberger, Gary A., Fred Glover, Bahram Alidaee, and Cesar Rego. "An Unconstrained Quadratic Binary Programming Approach to the Vertex Coloring Problem." Annals of Operations Research 139, no. 1 (October 2005): 229–41. http://dx.doi.org/10.1007/s10479-005-3449-7.

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29

Goudet, Olivier, Adrien Goëffon, and Jin-Kao Hao. "A large population island framework for the unconstrained binary quadratic problem." Computers & Operations Research 168 (August 2024): 106684. http://dx.doi.org/10.1016/j.cor.2024.106684.

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30

Mauri, Geraldo Regis, and Luiz Antonio Nogueira Lorena. "Improving a Lagrangian decomposition for the unconstrained binary quadratic programming problem." Computers & Operations Research 39, no. 7 (July 2012): 1577–81. http://dx.doi.org/10.1016/j.cor.2011.09.008.

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31

Lewis, Mark, and Gary Kochenberger. "Probabilistic multistart with path relinking for solving the unconstrained binary quadratic problem." International Journal of Operational Research 26, no. 1 (2016): 13. http://dx.doi.org/10.1504/ijor.2016.075647.

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32

Chen, Ming, Yuning Chen, Yonghao Du, Luona Wei, and Yingwu Chen. "Heuristic algorithms based on deep reinforcement learning for quadratic unconstrained binary optimization." Knowledge-Based Systems 207 (November 2020): 106366. http://dx.doi.org/10.1016/j.knosys.2020.106366.

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33

Shylo, V. P., and O. V. Shylo. "Systems Analysis; Solving unconstrained binary quadratic programming problem by global equilibrium search." Cybernetics and Systems Analysis 47, no. 6 (November 2011): 889–97. http://dx.doi.org/10.1007/s10559-011-9368-5.

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34

Vyskocil, Tomas, and Hristo Djidjev. "Embedding Equality Constraints of Optimization Problems into a Quantum Annealer." Algorithms 12, no. 4 (April 17, 2019): 77. http://dx.doi.org/10.3390/a12040077.

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Quantum annealers such as D-Wave machines are designed to propose solutions for quadratic unconstrained binary optimization (QUBO) problems by mapping them onto the quantum processing unit, which tries to find a solution by measuring the parameters of a minimum-energy state of the quantum system. While many NP-hard problems can be easily formulated as binary quadratic optimization problems, such formulations almost always contain one or more constraints, which are not allowed in a QUBO. Embedding such constraints as quadratic penalties is the standard approach for addressing this issue, but it has drawbacks such as the introduction of large coefficients and using too many additional qubits. In this paper, we propose an alternative approach for implementing constraints based on a combinatorial design and solving mixed-integer linear programming (MILP) problems in order to find better embeddings of constraints of the type ∑ x i = k for binary variables x i. Our approach is scalable to any number of variables and uses a linear number of ancillary variables for a fixed k.
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35

Gilliam, Austin, Stefan Woerner, and Constantin Gonciulea. "Grover Adaptive Search for Constrained Polynomial Binary Optimization." Quantum 5 (April 8, 2021): 428. http://dx.doi.org/10.22331/q-2021-04-08-428.

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In this paper we discuss Grover Adaptive Search (GAS) for Constrained Polynomial Binary Optimization (CPBO) problems, and in particular, Quadratic Unconstrained Binary Optimization (QUBO) problems, as a special case. GAS can provide a quadratic speed-up for combinatorial optimization problems compared to brute force search. However, this requires the development of efficient oracles to represent problems and flag states that satisfy certain search criteria. In general, this can be achieved using quantum arithmetic, however, this is expensive in terms of Toffoli gates as well as required ancilla qubits, which can be prohibitive in the near-term. Within this work, we develop a way to construct efficient oracles to solve CPBO problems using GAS algorithms. We demonstrate this approach and the potential speed-up for the portfolio optimization problem, i.e. a QUBO, using simulation and experimental results obtained on real quantum hardware. However, our approach applies to higher-degree polynomial objective functions as well as constrained optimization problems.
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36

Tosun, Umut. "A new tool for automated transformation of Quadratic Assignment Problem instances to Quadratic Unconstrained Binary Optimisation models." Expert Systems with Applications 201 (September 2022): 116953. http://dx.doi.org/10.1016/j.eswa.2022.116953.

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37

Mattesi, Mirko, Luca Asproni, Christian Mattia, Simone Tufano, Giacomo Ranieri, Davide Caputo, and Davide Corbelletto. "Diversifying Investments and Maximizing Sharpe Ratio: A Novel Quadratic Unconstrained Binary Optimization Formulation." Quantum Reports 6, no. 2 (May 27, 2024): 244–62. http://dx.doi.org/10.3390/quantum6020018.

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The optimization of investment portfolios represents a pivotal task within the field of financial economics. Its objective is to identify asset combinations that meet specified criteria for return and risk. Traditionally, the maximization of the Sharpe Ratio, often achieved through quadratic programming, has constituted a popular approach for this purpose. However, real-world scenarios frequently necessitate more complex considerations, particularly in relation to portfolio diversification with a view to mitigating sector-specific risks and enhancing stability. The incorporation of diversification alongside the Sharpe Ratio into the optimization model creates a joint optimization task, which can be formulated as Quadratic Unconstrained Binary Optimization (QUBO) and addressed using quantum annealing or hybrid computing techniques. These techniques offer promising solutions. We present a novel QUBO formulation for this optimization, detailing its mathematical formulation and demonstrating its advantages over classical methods, particularly in handling diversification objectives. By leveraging available QUBO solvers and hybrid approaches, we explore the feasibility of handling large-scale problems while highlighting the importance of diversification in achieving robust portfolio performance. We finally elaborate on the results showing the trade-off between the observed values of the portfolio’s Sharpe Ratio and diversification, as a natural consequence of solving a multi-objective optimization problem.
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38

Katayama, Kengo, and Hiroyuki Narihisa. "Performance of simulated annealing-based heuristic for the unconstrained binary quadratic programming problem." European Journal of Operational Research 134, no. 1 (October 2001): 103–19. http://dx.doi.org/10.1016/s0377-2217(00)00242-3.

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39

Gu, Shenshen, Tao Hao, and Hanmei Yao. "A pointer network based deep learning algorithm for unconstrained binary quadratic programming problem." Neurocomputing 390 (May 2020): 1–11. http://dx.doi.org/10.1016/j.neucom.2019.06.111.

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40

Cai, Yiqiao, Jiahai Wang, Jian Yin, and Yalan Zhou. "Memetic clonal selection algorithm with EDA vaccination for unconstrained binary quadratic programming problems." Expert Systems with Applications 38, no. 6 (June 2011): 7817–27. http://dx.doi.org/10.1016/j.eswa.2010.12.124.

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41

Zhou, Ying, Lingjing Kong, Yong Liu, Yiqiao Cai, and Shaopeng Liu. "A Single Deep Neural Network Model for Multiobjective Unconstrained Binary Quadratic Programming Problem." International Journal of Cognitive Informatics and Natural Intelligence 18, no. 1 (December 3, 2024): 1–17. https://doi.org/10.4018/ijcini.361012.

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The multiobjective unconstrained binary quadratic programming problem is an important combinatorial optimization problem with both theory and practical values. Until now, several efforts have been made to design metaheuristic methods to solve the problem. However, designing such effective methods is not trivial and heavily depends on experts' specific knowledge. Meanwhile, due to the iterative nature of metaheuristic methods, they require a long time to find high-quality solutions. From the perspective of machine learning, this paper proposes a deep reinforcement learning method to solve the problem. The method can automatically learn effective heuristics from a large amount of data, thus decreasing the need for experts' knowledge. Meanwhile, by leveraging the power of GPU, the method can quickly obtain high-quality solutions for a batch of instances. Experimental results show the proposed method outperforms two classical metaheuristic methods in terms of solution quality and running time for solving the problem.
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42

Alidaee, Bahram, Fred Glover, Gary A. Kochenberger, and Cesar Rego. "A new modeling and solution approach for the number partitioning problem." Journal of Applied Mathematics and Decision Sciences 2005, no. 2 (January 1, 2005): 113–21. http://dx.doi.org/10.1155/jamds.2005.113.

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The number partitioning problem has proven to be a challenging problem for both exact and heuristic solution methods. We present a new modeling and solution approach that consists of recasting the problem as an unconstrained quadratic binary program that can be solved by efficient metaheuristic methods. Our approach readily accommodates both the common two-subset partition case as well as the more general case of multiple subsets. Preliminary computational experience is presented illustrating the attractiveness of the method.
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43

Alidaee, Bahram, Haibo Wang, and Lutfu S. Sua. "An Efficient Closed-Form Formula for Evaluating r-Flip Moves in Quadratic Unconstrained Binary Optimization." Algorithms 16, no. 12 (December 5, 2023): 557. http://dx.doi.org/10.3390/a16120557.

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Quadratic unconstrained binary optimization (QUBO) is a classic NP-hard problem with an enormous number of applications. Local search strategy (LSS) is one of the most fundamental algorithmic concepts and has been successfully applied to a wide range of hard combinatorial optimization problems. One LSS that has gained the attention of researchers is the r-flip (also known as r-Opt) strategy. Given a binary solution with n variables, the r-flip strategy “flips” r binary variables to obtain a new solution if the changes improve the objective function. The main purpose of this paper is to develop several results for the implementation of r-flip moves in QUBO, including a necessary and sufficient condition that when a 1-flip search reaches local optimality, the number of candidates for implementation of the r-flip moves can be reduced significantly. The results of the substantial computational experiments are reported to compare an r-flip strategy-embedded algorithm and a multiple start tabu search algorithm on a set of benchmark instances and three very-large-scale QUBO instances. The r-flip strategy implemented within the algorithm makes the algorithm very efficient, leading to very high-quality solutions within a short CPU time.
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44

Kudo, Kazue. "Compressed sensing based on QUBO formulation." Journal of Physics: Conference Series 2207, no. 1 (March 1, 2022): 012033. http://dx.doi.org/10.1088/1742-6596/2207/1/012033.

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Abstract Ising machines efficiently solve the combinatorial optimization problems described by the Ising model or the quadratic unconstrained binary optimization (QUBO) formulation. A hybrid method based on the QUBO formulation for compressed sensing is proposed. The proposed method comprises alternative steps of discrete and continuous optimization. In the discrete optimization step, the objective function is described by the QUBO formulation. Successful examples obtained via the proposed method are demonstrated. The performance of the proposed method depends highly on the initial conditions.
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45

Teplukhin, Alexander, Brian K. Kendrick, Susan M. Mniszewski, Sergei Tretiak, and Pavel A. Dub. "Sampling electronic structure quadratic unconstrained binary optimization problems (QUBOs) with Ocean and Mukai solvers." PLOS ONE 17, no. 2 (February 11, 2022): e0263849. http://dx.doi.org/10.1371/journal.pone.0263849.

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The most advanced D-Wave Advantage quantum annealer has 5000+ qubits, however, every qubit is connected to a small number of neighbors. As such, implementation of a fully-connected graph results in an order of magnitude reduction in qubit count. To compensate for the reduced number of qubits, one has to rely on special heuristic software such as qbsolv, the purpose of which is to decompose a large quadratic unconstrained binary optimization (QUBO) problem into smaller pieces that fit onto a quantum annealer. In this work, we compare the performance of the open-source qbsolv which is a part of the D-Wave Ocean tools and a new Mukai QUBO solver from Quantum Computing Inc. (QCI). The comparison is done for solving the electronic structure problem and is implemented in a classical mode (Tabu search techniques). The Quantum Annealer Eigensolver is used to map the electronic structure eigenvalue-eigenvector equation to a QUBO problem, solvable on a D-Wave annealer. We find that the Mukai QUBO solver outperforms the Ocean qbsolv with one to two orders of magnitude more accurate energies for all calculations done in the present work, both the ground and excited state calculations. This work stimulates the further development of software to assist in the utilization of modern quantum annealers.
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46

Zhou, Ying, Lingjing Kong, Ziyan Wu, Shaopeng Liu, Yiqiao Cai, and Ye Liu. "Ensemble of multi-objective metaheuristic algorithms for multi-objective unconstrained binary quadratic programming problem." Applied Soft Computing 81 (August 2019): 105485. http://dx.doi.org/10.1016/j.asoc.2019.105485.

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47

Wang, Jiahai. "Discrete Hopfield network combined with estimation of distribution for unconstrained binary quadratic programming problem." Expert Systems with Applications 37, no. 8 (August 2010): 5758–74. http://dx.doi.org/10.1016/j.eswa.2010.02.032.

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48

Boros, Endre, Peter L. Hammer, Richard Sun, and Gabriel Tavares. "A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO)." Discrete Optimization 5, no. 2 (May 2008): 501–29. http://dx.doi.org/10.1016/j.disopt.2007.02.001.

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49

Hanafi, Saïd, Ahmed-Riadh Rebai, and Michel Vasquez. "Several versions of the devour digest tidy-up heuristic for unconstrained binary quadratic problems." Journal of Heuristics 19, no. 4 (June 8, 2011): 645–77. http://dx.doi.org/10.1007/s10732-011-9169-z.

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50

Wang, Jiahai, Ying Zhou, and Jian Yin. "Combining tabu Hopfield network and estimation of distribution for unconstrained binary quadratic programming problem." Expert Systems with Applications 38, no. 12 (November 2011): 14870–81. http://dx.doi.org/10.1016/j.eswa.2011.05.060.

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