Binary Optimization Problem Research Articles

Quantum Annealing versus Digital Computing

Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.

A two-phase intensification tabu search algorithm for the maximum min-sum dispersion problem

In this paper, we study the maximum min-sum dispersion problem (Max-Minsum DP for short) which is a classical binary optimization problem proven to be NP-hard and with numerous real-world applications. For solving this computationally challenging problem, we propose a two-phase intensification tabu search algorithm (TPITS), which integrates several distinguishing features, such as an intensification tabu search to avoid visiting the previous encountered solutions and an attribute-based tabu search to refine the search in the second phase. Tested on seven sets of totally 160 public instances in the literature, the study demonstrates the efficacy of the proposed TPITS algorithm in terms of both solution quality and computational efficiency. Specifically, our proposed TPITS algorithm is able to improve the previous best known results for 69 instances, while matching the previous best known results for 74 ones. We also provide experimental evidences to highlight the beneficial effect of the important components in the TPITS algorithm.

Benchmark Test of Black-box Optimization Using D-Wave Quantum Annealer

In solving optimization problems, objective functions generally need to be minimized or maximized. However, objective functions cannot always be formulated explicitly in a mathematical form for complicated problem settings. Although several regression techniques infer the approximate forms of objective functions, they are at times expensive to evaluate. Optimal points of "black-box" objective functions are computed in such scenarios, while effectively using a small number of clues. Recently, an efficient method by use of inference by sparse prior for a black-box objective function with binary variables has been proposed. In this method, a surrogate model was proposed in the form of a quadratic unconstrained binary optimization (QUBO) problem, and was iteratively solved to obtain the optimal solution of the black-box objective function. In the present study, we employ the D-Wave 2000Q quantum annealer, which can solve QUBO by driving the binary variables by quantum fluctuations. The D-Wave 2000Q quantum annealer does not necessarily output the ground state at the end of the protocol due to freezing effect during the process. We investigate effects from the output of the D-Wave quantum annealer in performing black-box optimization. We demonstrate a benchmark test by employing the sparse Sherrington-Kirkpatrick (SK) model as the black-box objective function, by introducing a parameter controlling the sparseness of the interaction coefficients. Comparing the results of the D-Wave quantum annealer to those of the simulated annealing (SA) and semidefinite programming (SDP), our results by the D-Wave quantum annealer and SA exhibit superiority in black-box optimization with SDP. On the other hand, we did not find any advantage of the D-Wave quantum annealer over the simulated annealing. As far as in our case, any effects by quantum fluctuation are not found.

Multivehicle Task Assignment Based on Collaborative Neurodynamic Optimization With Discrete Hopfield Networks.

This article presents a collaborative neurodynamic optimization (CNO) approach to multivehicle task assignments (TAs). The original combinatorial quadratic optimization problem for TA is reformulated as a quadratic unconstrained binary optimization (QUBO) problem with a quadratic utility function and a penalty function for handling load capacity and cooperation constraints. In the framework of CNO with a population of discrete Hopfield networks (DHNs), a TA algorithm is proposed for solving the formulated QUBO problem. Superior experimental results in four typical multivehicle operation scenarios are reported to substantiate the efficacy of the proposed neurodynamics-based TA approach.

Signals classification based on IA-optimal CNN.

The versatility of the existing A-optimal-based CNN for solving multiple types of signals classification problems has not been verified by different signals datasets. Moreover, the existing A-optimal-based CNN uses a simplified approximate function as the optimization objective function instead of precise analytical function, which affects the signals classification accuracy to a certain extent. In this paper, a classification method called IA-optimal CNN is proposed. To improve the stability of the classifier, the trace of the covariance matrix of the weights of the fully connected layer is used as the optimization objective function, and the parameter optimization model is established without any simplification of the optimization objective function. In addition, to avoid the difficulty of not being able to obtain the analytical expression formula of the partial derivative of the inverse matrix with regard to the networks parameters, a novel dual function is introduced to transform the optimization problem into an equivalent binary function optimization problem. Furthermore, based on the above analytical solution results, the parameters are updated using the alternate iterative optimization method and the accurate weight update formula is deduced in detail. Five signals datasets are used to test the universality of the IA-optimal CNN in signals classification fields. The performance of IA-optimal CNN is showed, and the experimental results are compared with the existing A-optimal-based classification algorithm. Lastly, the following conclusion is proved theoretically: For the A-optimal-based CNN, the trace of the covariance matrix will continue to decrease and approach a convergence value in the iterative process, but it is impossible for the networks to strictly reach the A-optimal state.

Binary crow search algorithm for the uncapacitated facility location problem

The crow search algorithm (CSA) is a recently proposed population-based optimization algorithm for continuous optimization. Since the original CSA searches for a feasible solution in a continuous search space, it cannot handle binary optimization problems directly. A few binary variants of CSA are presented in the literature. However, these variants search for a new solution in the continuous domain and need transfer functions to adapt the solution to the binary domain. This may cause poor exploration, making some regions in the search space impossible to discover. This paper proposes an effective binary CSA (BinCSA) using bitwise operations that directly searches for a feasible solution in the binary search space. For this purpose, the original update mechanism of the CSA is improved using exclusive-OR and AND logical operators in order to provide a good balance between exploration and exploitation in the binary search space. The effectiveness of the proposed BinCSA is evaluated on the uncapacitated facility location problem (UFLP), one of the most widely investigated pure binary optimization problems. The performance of BinCSA is examined using two different UFLP datasets, ORLIB and M*. The experimental results show that BinCSA obtained the optimal solution for 13 out of 15 instances of ORLIB and 12 out of 20 instances of M*. Moreover, BinCSA exhibits superior performance on ORLIB instances when compared to other methods and is very competitive on M* instances in terms of solution quality and robustness. The source code for BinCSA, as used for the UFLP, is available at https://github.com/3mrullah/BinCSA.

Binary Jaya algorithm based on binary similarity measure for feature selection

Feature selection (FS) has become an indispensable data preprocessing task because of the huge amount of high dimensional data being generated by current technologies. These high dimensional data contains irrelevant, redundant, and noisy features that deteriorate classification accuracy. FS reduces dimensionality by removing the unwanted features thus improves classification accuracy. FS can be considered as a binary optimization problem. In order to solve this problem, this work proposes a new wrapper feature selection technique based on the Jaya algorithm. Three binary variants of the Jaya algorithm are proposed, the first and second ones are based on transfer functions namely BJaya-S and BJaya-V. The third variant (BJaya-JS) explores the search space on the basis of the Jaccard Similarity index. In addition, a probability-based local search technique, namely Neighbourhood Search is proposed to balance the exploration and exploitation. The variants of Jaya algorithm are evaluated and the best variant is selected. The best variant is further compared with six state-of-the-art feature selection techniques. All the performances are tested on 18 high dimensional standard UCI datasets. Experimental result comparison shows that the proposed feature selection technique performs better than other competitors.

Qubit-efficient encoding schemes for binary optimisation problems

We propose and analyze a set of variational quantum algorithms for solving quadratic unconstrained binary optimization problems where a problem consisting of nc classical variables can be implemented on O(log⁡nc) number of qubits. The underlying encoding scheme allows for a systematic increase in correlations among the classical variables captured by a variational quantum state by progressively increasing the number of qubits involved. We first examine the simplest limit where all correlations are neglected, i.e. when the quantum state can only describe statistically independent classical variables. We apply this minimal encoding to find approximate solutions of a general problem instance comprised of 64 classical variables using 7 qubits. Next, we show how two-body correlations between the classical variables can be incorporated in the variational quantum state and how it can improve the quality of the approximate solutions. We give an example by solving a 42-variable Max-Cut problem using only 8 qubits where we exploit the specific topology of the problem. We analyze whether these cases can be optimized efficiently given the limited resources available in state-of-the-art quantum platforms. Lastly, we present the general framework for extending the expressibility of the probability distribution to any multi-body correlations.

Hash Bit Selection via Collaborative Neurodynamic Optimization With Discrete Hopfield Networks.

Hash bit selection (HBS) aims to find the most discriminative and informative hash bits from a hash pool generated by using different hashing algorithms. It is usually formulated as a binary quadratic programming problem with an information-theoretic objective function and a string-length constraint. In this article, it is equivalently reformulated in the form of a quadratic unconstrained binary optimization problem by augmenting the objective function with a penalty function. The reformulated problem is solved via collaborative neurodynamic optimization (CNO) with a population of classic discrete Hopfield networks. The two most important hyperparameters of the CNO approach are determined based on Monte Carlo test results. Experimental results on three benchmark data sets are elaborated to substantiate the superiority of the collaborative neurodynamic approach to several existing methods for HBS.

Grover Adaptive Search for Constrained Polynomial Binary Optimization

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.

Semi-supervised time series classification method for quantum computing

In this paper we develop methods to solve two problems related to time series (TS) analysis using quantum computing: reconstruction and classification. We formulate the task of reconstructing a given TS from a training set of data as an unconstrained binary optimization (QUBO) problem, which can be solved by both quantum annealers and gate-model quantum processors. We accomplish this by discretizing the TS and converting the reconstruction to a set cover problem, allowing us to perform a one-versus-all method of reconstruction. Using the solution to the reconstruction problem, we show how to extend this method to perform semi-supervised classification of TS data. We present results indicating our method is competitive with current semi- and unsupervised classification techniques, but using less data than classical techniques.

Solving knapsack problems using a binary gaining sharing knowledge-based optimization algorithm

This article proposes a novel binary version of recently developed Gaining Sharing knowledge-based optimization algorithm (GSK) to solve binary optimization problems. GSK algorithm is based on the concept of how humans acquire and share knowledge during their life span. A binary version of GSK named novel binary Gaining Sharing knowledge-based optimization algorithm (NBGSK) depends on mainly two binary stages: binary junior gaining sharing stage and binary senior gaining sharing stage with knowledge factor 1. These two stages enable NBGSK for exploring and exploitation of the search space efficiently and effectively to solve problems in binary space. Moreover, to enhance the performance of NBGSK and prevent the solutions from trapping into local optima, NBGSK with population size reduction (PR-NBGSK) is introduced. It decreases the population size gradually with a linear function. The proposed NBGSK and PR-NBGSK applied to set of knapsack instances with small and large dimensions, which shows that NBGSK and PR-NBGSK are more efficient and effective in terms of convergence, robustness, and accuracy.

Heuristic based binary grasshopper optimization algorithm to solve unit commitment problem

The unit commitment problem in power system is a highly nonlinear, nonconvex, multiconstrained, complex,highly dimensional, mixed integer and combinatorial generation selection problem. The phenomenon of committing anddecommitting represents a discrete problem that requires binary/discrete optimization techniques to tackle with unitcommitment optimization problem. The key functions of the unit commitment optimization problem involve decidingwhich units to commit and then to decide their optimum power (economic dispatch). This paper confers a binarygrasshopper optimization algorithm to solve the unit commitment optimization problem under multiple constraints.The grasshopper optimization algorithm is a metaheuristic, multiple solutions-based algorithm inspired by the naturalswarming behavior of grasshopper towards food. For solving the binary unit commitment optimization problem, thereal/continues value grasshopper optimization algorithm is mapped into binary/discrete search-space by using an S-shaped sigmoid function. The proposed algorithm is tested on IEEE benchmark systems of 4, 5, 6, 10, 20, 26, 40, 60,80, and 100 generating units including the IEEE 118-bus system and the results are compared with different classical,heuristics, metaheuristics, quantum, and hybrid approaches. The results confer better performance of binary grasshopperoptimization algorithm to solve the unit commitment optimization problem.

A survey on the Artificial Bee Colony algorithm variants for binary, integer and mixed integer programming problems

Most of the optimization problems encountered in the real world are discrete type which involves decision variables defined in the discrete search space. Binary optimization problems, integer and mixed integer programming problems are of this category, and they require suitable solution representation and search operators to be solved by nature-inspired algorithms. One of the widely-used and well-known nature-inspired algorithms is Artificial Bee Colony (ABC) that has been originally proposed to solve the problems in the continuous domain, and hence, its standard version employs the search operators to exploit the information of the solution vectors encoded in the continuous domain. To be able to cope with the discrete problems, particularly binary, integer and mixed integer programming problems, which are also a group of numeric optimization problems, various encoding types, search operators and selection operators have been integrated into ABC. In this paper, we review the studies proposing new ABC variants to solve discrete numeric optimization problems. To the best of our knowledge, this will be the first comprehensive survey study on this topic. Therefore, we hope that this study would be beneficial to the readers interested in the use of ABC for the binary, integer and mixed integer discrete optimization problems.

UTF: Upgrade transfer function for binary meta-heuristic algorithms

In the real world, many optimization problems are discrete and very complex to solve. Some of them are in the class of NP-hard problems and their search spaces grow exponentially with the problem size. As a result, an exhaustive search will be impractical using exact algorithms. In the last decades, meta-heuristic algorithms as approximate algorithms have shown superior performance in solving these problems. The majority of these algorithms have been designed for continuous search spaces and are not able to solve binary optimization problems. Therefore, a transfer function is applied to convert the continuous search space to the binary one. The performance of such binary algorithms depends on their ability of exploration, exploitation and transfer function. Several transfer functions have been introduced so far but they have shown poor exploration and exploitation in solving some problems. In this study, a novel adaptive transfer function, based on two linear functions, is proposed to overcome the shortcomings of existing transfer functions. The proposed method called upgrade transfer function (UTF) adapts itself during running the algorithm to switch from exploration to exploitation. This capability also covers disadvantages of metaheuristic algorithms in terms of poor exploration and exploitation. The performance of UTF has been evaluated by three discrete optimization problems: function optimization, feature selection and the 0–1 multi-knapsack problem (MKP). The results of binary particle swarm optimization (BPSO), binary artificial bee colony (BABC) and several improved BPSO and BABC have been compared with those of UTF–BPSO and UTF–BABC using function optimization problems. Also, the efficiency of UTF–BPSO and UTF–BABC and some binary meta-heuristic algorithms such as binary salp swarm algorithm (BSSA) and binary gray wolf optimization (bGWO), binary dragon algorithm (BDA), binary multi-neighborhood artificial bee colony (BMNABC), binary hybrid topology particle swarm optimization quadratic interpolation (BHTPSO-QI), binary ant lion optimizer (bALO) and binary gravitational search algorithm (BGSA) have been evaluated by feature selection problems. Moreover, UTF and seventeen transfer functions have been applied in original PSO, ABC, SSA and GWO algorithms to solve low and high dimensions 0–1 MKP benchmark instances. The results showed that the new transfer function significantly enhances the performance of algorithms to achieve the best solution in the binary search space.

Binary fish migration optimization for solving unit commitment

Inspired by migratory graying, Pan et al. proposed the fish migration optimization (FMO) algorithm. It integrates the models of migration and swim into the optimization process. This paper firstly proposes a binary version of FMO, called BFMO. In order to improve the search ability of BFMO, ABFMO is introduced to solve the problems of stagnation and falling into local traps. The transfer function is responsible for mapping the continuous search space to the binary space. It plays a critical factor in the binary meta-heuristics. This paper brings a new transfer function and compares it with the transfer functions used by BPSO, BGSA and BGWO. Experiments prove that the new transfer function has realized good results in the solving quality. Unit commitment (UC) is a NP-hard binary optimization problem. BFMO and ABFMO are tested with the IEEE benchmark systems consisting of various generating units with 24-h demand horizon. The effectivenesses of BFMO and ABFMO are compared with seven binary evolutionary algorithms. The simulation results and non-parametric tests verify that they achieve great performance.

A binary artificial bee colony algorithm and its performance assessment

Artificial bee colony algorithm, ABC for short, is a swarm-based optimization algorithm proposed for solving continuous optimization problems. Due to its simple but effective structure, some binary versions of the algorithm have been developed. In this study, we focus on modification of its xor-based binary version, called as binABC. The solution update rule of basic ABC is replaced with a xor logic gate in binABC algorithm, and binABC works on discretely-structured solution space. The rest of components in binABC are the same as with the basic ABC algorithm. In order to improve local search capability and convergence characteristics of binABC, a stigmergic behavior-based update rule for onlooker bees of binABC and extended version of xor-based update rule are proposed in the present study. The developed version of binABC is applied to solve a modern benchmark problem set (CEC2015). To validate the performance of proposed algorithm, a series of comparisons are conducted on this problem set. The proposed algorithm is first compared with the basic ABC and binABC on CEC2015 set. After its performance validation, six binary versions of ABC algorithm are considered for comparison of the algorithms, and a comprehensive comparison among the state-of-art variants of swarm intelligence or evolutionary computation algorithms is conducted on this set of functions. Finally, an uncapacitated facility location problem set, a pure binary optimization problem, is considered for the comparison of the proposed algorithm and binary variants of ABC algorithm. The experimental results and comparisons show that the proposed algorithm is successful and effective in solving binary optimization problems as its basic version in solving continuous optimization problems.

A Novel Binary QUasi-Affine TRansformation Evolutionary (QUATRE) Algorithm

QUasi-Affine TRansformation Evolutionary (QUATRE) algorithm generalized differential evolution (DE) algorithm to matrix form. QUATRE was originally designed for a continuous search space, but many practical applications are binary optimization problems. Therefore, we designed a novel binary version of QUATRE. The proposed binary algorithm is implemented using two different approaches. In the first approach, the new individuals produced by mutation and crossover operation are binarized. In the second approach, binarization is done after mutation, then cross operation with other individuals is performed. Transfer functions are critical to binarization, so four families of transfer functions are introduced for the proposed algorithm. Then, the analysis is performed and an improved transfer function is proposed. Furthermore, in order to balance exploration and exploitation, a new liner increment scale factor is proposed. Experiments on 23 benchmark functions show that the proposed two approaches are superior to state-of-the-art algorithms. Moreover, we applied it for dimensionality reduction of hyperspectral image (HSI) in order to test the ability of the proposed algorithm to solve practical problems. The experimental results on HSI imply that the proposed methods are better than Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA).

Binary team game algorithm based on modulo operation for knapsack problem with a single continuous variable

For solving the knapsack problem with a single continuous variable (KPC), a binary team game algorithm (TGA) with one-way mutation strategy is proposed. Firstly, without changing the evolution mode of TGA, three basic operations are reconstructed based on modulo 2 operation. Then, a binary TGA (BTGA) suitable for solving binary optimization problem is proposed. In order to use BTGA to solve KPC problem effectively, a one-way mutation strategy to improve individual quality is subsequently developed. Finally, based on BTGA and the existing repair and optimization algorithm of eliminating infeasible solutions, a novel algorithm MOBTGA for solving KPC problem is proposed. For validating the performance of MOBTGA, Kruskal–Wallis test is used to determine the reasonable values of its parameters. The comparison of experimental results obtained by different algorithms for two sets of KPC instances is subsequently executed. The comparison results show that MOBTGA has better performance than the existing heuristic algorithms ETDE, S-HBDE and B-HBDE in terms of solution accuracy and stability for solving KPC. The same time, the solving speed of MOBTGA also has a certain degree of competitiveness. Thus, MOBTGA is a quick and efficient heuristic algorithm for solving large-scale KPC instances.

Exact Facetial Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization

The exact solution of the NP-hard (nondeterministic polynomial-time hard) maximum cut problem is important in many applications across, for example, physics, chemistry, neuroscience, and circuit layout—which is also due to its equivalence to the unconstrained binary quadratic optimization problem. Leading solution methods are based on linear or semidefinite programming and require the separation of the so-called odd-cycle inequalities. In their groundbreaking research, F. Barahona and A. R. Mahjoub have given an informal description of a polynomial-time algorithm for this problem. As pointed out recently, however, additional effort is necessary to guarantee that the inequalities obtained correspond to facets of the cut polytope. In this paper, we shed more light on a so enhanced separation procedure and investigate experimentally how it performs in comparison with an ideal setting where one could even employ the sparsest, most violated, or geometrically most promising facet-defining odd-cycle inequalities. Summary of Contribution: This paper aims at a better capability to solve binary quadratic optimization or maximum cut problems and their various applications using integer programming techniques. To this end, the paper describes enhancements to a well-known algorithm for the central separation problem arising in this context; it is demonstrated experimentally that these enhancements are worthwhile from a computational point of view. The linear relaxations of the aforementioned problems are typically solved using fewer iterations and cutting planes than with a nonenhanced approach. It is also shown that the enhanced procedure is only slightly inferior to an ideal, enumerative, and, in practice, intractable global cutting-plane selection.