Binary Quadratic Programming Problem Research Articles

New techniques to improve neighborhood exploration in pareto local search

Pareto Local Search (PLS) is an extension of local search to multiobjective combinatorial optimization problems (MCOPs). It is widely used as a building block in many multiobjective metaheuristics. However, it suffers from poor anytime behavior and long convergence time. In this paper, we focus on improving the neighborhood exploration mechanism to accelerate PLS. In total, we propose three new search strategies, namely, Promising List, Don’t Look Bits and Continuing Search and implement them on the multiobjective Traveling Salesman Problem (mTSP) and the multiobjective Unconstrained Binary Quadratic Programming (mUBQP) problem. In Promising List, features of problem structure are utilized to guide the local search. In Don’t Look Bits, neighborhood moves that are less promising to produce improving solutions are skipped. In Continuing Search, the agent will start from where previous neighborhood exploration stops to avoid repetitive neighborhood moves. The experimental results verify the effectiveness of the three techniques both individually and collaboratively on most of the test instances. Besides, one proposed PLS variant outperforms two state-of-the-art PLS-based algorithms.

A Dual Bounding Framework Through Cost Splitting for Binary Quadratic Optimization

Binary quadratic programming (BQP) is a class of combinatorial optimization problems comprising binary variables, quadratic objective functions, and linear/nonlinear constraints. This paper examines a unified framework to reformulate a BQP problem with linear constraints to a new BQP with an exponential number of variables defined on a graph. This framework relies on the concept of stars in the graph to split the quadratic costs into adjacent and nonadjacent components indicating in-star and out-of-star interactions. We exploit the star-based structure of the new reformulation to develop a decomposition-based column generation algorithm. In our computational experiments, we evaluate the performance of our methodology on different applications with different quadratic structures. The quadratic component of the problem is dealt with in the column generation master problem and its subproblem. Results indicate the superiority of the framework over one of the state-of-the-art solvers, GUROBI, when applied to various benchmark reformulations with adjacent-only or sparse quadratic cost matrices. The framework outperforms GUROBI in terms of both dual bound and computational time in almost all instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: The authors thank the Mitacs Accelerate Program for providing funding for this project. In addition, B. Rostami gratefully acknowledges the funding provided by the Canadian Natural Sciences and Engineering Research Council (NSERC) under a Discovery Grant [Grant RGPIN-2020-05395]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2021.0186 .

Walsh-based surrogate-assisted multi-objective combinatorial optimization: A fine-grained analysis for pseudo-boolean functions

The aim of this paper is to study surrogate-assisted algorithms for expensive multiobjective combinatorial optimization problems. Targeting pseudo-boolean domains, we provide a fine-grained analysis of an optimization framework using the Walsh basis as a core surrogate model. The considered framework uses decomposition in the objective space, and integrates three different components, namely, (i) an inner optimizer for searching promising solutions with respect to the so-constructed surrogate, (ii) a selection strategy to decide which solution is to be evaluated by the expensive objectives, and (iii) the strategy used to setup the Walsh order hyper-parameter. Based on extensive experiments using two benchmark problems, namely bi-objective NK-landscapes and unconstrained binary quadratic programming problems (UBQP), we conduct a comprehensive in-depth analysis of the combined effects of the considered components on search performance, and provide evidence on the effectiveness of the proposed search strategies. As a by-product, our work shed more light on the key challenges for designing a successful surrogate-assisted multi-objective combinatorial search process.

Benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization

Recently, inspired by quantum annealing, many solvers specialized for unconstrained binary quadratic programming problems have been developed. For further improvement and application of these solvers, it is important to clarify the differences in their performance for various types of problems. In this study, the performance of four quadratic unconstrained binary optimization problem solvers, namely D-Wave Hybrid Solver Service (HSS), Toshiba Simulated Bifurcation Machine (SBM), Fujitsu Digital Annealer (DA), and simulated annealing on a personal computer, was benchmarked. The problems used for benchmarking were instances of real problems in MQLib, instances of the SAT-UNSAT phase transition point of random not-all-equal 3-SAT (NAE 3-SAT), and the Ising spin glass Sherrington-Kirkpatrick (SK) model. Concerning MQLib instances, the HSS performance ranked first; for NAE 3-SAT, DA performance ranked first; and regarding the SK model, SBM performance ranked first. These results may help understand the strengths and weaknesses of these solvers.

On-Off Scheduling for Electric Vehicle Charging in Two-Links Charging Stations Using Binary Optimization Approaches.

In this study, we deal with the problem of scheduling charging periods of electrical vehicles (EVs) to satisfy the users’ demands for energy consumption as well as to optimally utilize the available power. We assume three-phase EV charging stations, each equipped with two charging ports (links) that can serve up to two EVs in the scheduling period but not simultaneously. Considering such a specification, we propose an on–off scheduling scheme wherein control over an energy flow is achieved by flexibly switching the ports in each station on and off in a manner such as to satisfy the energy demand of each EV, flatten the high energy-consuming load on the whole farm, and to minimize the number of switching operations. To satisfy these needs, the on–off scheduling scheme is formulated in terms of a binary linear programming problem, which is then extended to a quadratic version to incorporate the smoothness constraints. Various algorithmic approaches are used for solving a binary quadratic programming problem, including the Frank–Wolfe algorithm and successive linear approximations. The numerical simulations demonstrate that the latter is scalable, efficient, and flexible in a charging procedure, and it shaves the load peak while maintaining smooth charging profiles.

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.

Multi-objectivization inspired metaheuristics for the sum-of-the-parts combinatorial optimization problems

Multi-objectivization is a term used to describe strategies developed for optimizing single-objective problems by multi-objective algorithms. This paper focuses on multi-objectivizing the sum-of-the-parts combinatorial optimization problems, which include the traveling salesman problem, the unconstrained binary quadratic programming and other well-known combinatorial optimization problem. For a sum-of-the-parts combinatorial optimization problem, we propose to decompose its original objective into two sub-objectives with controllable correlation. Based on the decomposition method, two new multi-objectivization inspired single-objective optimization techniques called non-dominance search and non-dominance exploitation are developed, respectively. Non-dominance search is combined with two metaheuristics, namely iterated local search and iterated tabu search, while non-dominance exploitation is embedded within the iterated Lin–Kernighan metaheuristic. The resultant metaheuristics are called ILS+NDS, ITS+NDS and ILK+NDE, respectively. Empirical studies on some TSP and UBQP instances show that with appropriate correlation between the sub-objectives, there are more chances to escape from local optima when new starting solution is selected from the non-dominated solutions defined by the decomposed sub-objectives. Experimental results also show that ILS+NDS, ITS+NDS and ILK+NDE all significantly outperform their counterparts on most of the test instances.

A Doubly Graduated Method for Inference in Markov Random Field

Maximum a posteriori (MAP) inference in Markov random field (MRF) lays the foundation for many computer vision tasks, which can be formulated by a binary quadratic programming (BQP) problem. Compar...

Heuristic algorithms based on deep reinforcement learning for quadratic unconstrained binary optimization

The unconstrained binary quadratic programming (UBQP) problem is a difficult combinatorial optimization problem that has been intensively studied in the past decades. Due to its NP-hardness, many heuristic algorithms have been developed for the solution of the UBQP. These algorithms are usually problem-tailored, which lack generality and scalability. To address these issues, a heuristic algorithm based on deep reinforcement learning (DRLH) is proposed in this paper. It features in inputting specific features and using a neural network model called NN to guild the selection of variable at each solution construction step. Also, to improve the algorithm speed and efficiency, two algorithm variants named simplified DRLH (DRLS) and DRLS with hill climbing (DRLS-HC) are developed as well. These three algorithms are examined through extensive experiments in comparison with famous heuristic algorithms from the literature. Experimental results show that the DRLH, DRLS, and DRLS-HC outperform their competitors in terms of both solution quality and computational efficiency. Precisely, the DRLH achieves the best-quality results, while DRLS offers a high-quality solution in a very short time. By adding a hill-climbing procedure to DRLS, the resulting DRLS-HC algorithm is able to obtain almost the same quality result as DRLH with however 5 times less computing time on average. We conducted additional experiments on large-scale instances and various data distributions to verify the generality and scalability of the proposed algorithms, and the results on benchmark instances indicate the ability of the algorithms to be applied to practical problems.

A Fast Binary Quadratic Programming Solver Based on Stochastic Neighborhood Search.

Many image processing and pattern recognition problems can be formulated as binary quadratic programming (BQP) problems. However, solving a large BQP problem with a good quality solution and low computational time is still a challenging unsolved problem. Current methodologies either adopt an independent random search in a semi-definite space or perform search in a relaxed biconvex space. However, the independent search has great computation cost as many different trials are needed to get a good solution. The biconvex search only searches the solution in a local convex ball, which can be a local optimal solution. In this paper, we propose a BQP solver that alternatingly applies a deterministic search and a stochastic neighborhood search. The deterministic search iteratively improves the solution quality until it satisfies the KKT optimality conditions. The stochastic search performs bootstrapping sampling to the objective function constructed from the potential solution to find a stochastic neighborhood vector. These two steps are repeated until the obtained solution is better than many of its stochastic neighborhood vectors. We compare the proposed solver with several state-of-the-art methods for a range of image processing and pattern recognition problems. Experimental results showed that the proposed solver not only outperformed them in solution quality but also with the lowest computational complexity.

IntMA: Dynamic Interaction-aware resource allocation for containerized microservices in cloud environments

The Information Technology sector has undergone tremendous changes arising due to the emergence and prevalence of Cloud Computing. Microservice Architectures have also been attracting attention from several industries and researchers. Due to the suitability of microservices for the Cloud environments, an increasing number of Cloud applications are now provided as microservices. However, this transition to microservices brings a wide range of infrastructural orchestration challenges. Though several research works have discussed the engineering of microservice-based applications, there is an inevitable need for research on handling the operational phases of the microservice components. Microservice application deployment in containerized datacenters must be optimized to enhance the overall system performance. In this research work, the deployment of microservice application modules on the Cloud infrastructure is first modelled as a Binary Quadratic Programming Problem. In order to reduce the adverse impact of communication latencies on the response time, the interaction pattern between the microservice components is modelled as an undirected doubly weighted complete Interaction Graph. A novel, robust heuristic approach IntMA is also proposed for deploying the microservices in an interaction-aware manner with the aid of the interaction information obtained from the Interaction Graph. The proposed allocation policies are implemented in Kubernetes. The effectiveness of the proposed approach is evaluated on the Google Cloud Platform, using different microservice reference applications. Experimental results indicate that the proposed approach improves the response time and throughput of the microservice-based systems.

A pointer network based deep learning algorithm for unconstrained binary quadratic programming problem

Combinatorial optimization problems have been widely used in various fields. And many types of combinatorial optimization problems can be generalized into the model of unconstrained binary quadratic programming (UBQP). Therefore, designing an effective and efficient algorithm for UBQP problems will also contribute to solving other combinatorial optimization problems. Pointer network is an end-to-end sequential decision structure and combines with deep learning technology. With the utilization of the structural characteristics of combinatorial optimization problems and the ability to extract the rule behind the data by deep learning, pointer network has been successfully applied to solve several classical combinatorial optimization problems. In this paper, a pointer network based algorithm is designed to solve UBQP problems. The network model is trained by supervised learning (SL) and deep reinforcement learning (DRL) respectively. Trained pointer network models are evaluated by self-generated benchmark dataset and ORLIB dataset respectively. Experimental results show that pointer network model trained by SL has strong learning ability to specific distributed dataset. Pointer network model trained by DRL can learn more general distribution data characteristics. In other words, it can quickly solve problems with great generalization ability. As a result, the framework proposed in this paper for UBQP has great potential to solve large scale combinatorial optimization problems.

Closed-form formulas for evaluating r-flip moves to the unconstrained binary quadratic programming problem

The Unconstrained Binary Quadratic Programming problem (UBQP) belongs to the NP-hard class and has become a framework for modeling a variety of combinatorial optimization problems. The methods most commonly used to solve instances of the UBQP explore the concept of neighborhood of a solution. Given a binary vector x ∈ {0, 1}n, solution to a UBQP instance, a neighborhood of x can be defined by flip moves. Flip moves consist on selecting one or more elements (positions) of x and “flip” their values to their complementary values (i.e., from 1 to 0 or from 0 to 1). Normally, those methods compute a large number of flip moves, and so the whole process to solve an instance can be quite time consuming. In order to reduce this time, some works have proposed ways to efficiently evaluate one or two flip moves, and also extensions to higher order moves. In this paper we propose two closed-form formulas for evaluating quickly any order of flip moves. To test our theoretical findings, we executed an extensive set of computational experiments over well-known instances for the problem. Against common belief, our results show that it is possible to compute high order flip moves very fast.

Ensemble of multi-objective metaheuristic algorithms for multi-objective unconstrained binary quadratic programming problem

Metaheuristics have been widely utilized for solving NP-hard optimization problems. However, these algorithms usually perform differently from one problem to another, i.e., one may be effective on a problem but performs badly on another problem. Therefore, it is difficult to choose the best algorithm in advance for a given problem. In contrast to selecting the best algorithm for a problem, selection hyper-heuristics aim at performing well on a set of problems (instances). This paper proposes a selection hyper-heuristic based algorithm for multi-objective optimization problems. In the proposed algorithm, multiple metaheuristics exhibiting different search behaviors are managed and controlled as low-level metaheuristics in an algorithm pool, and the most appropriate metaheuristic is selected by means of a performance indicator at each search stage. To assess the performance of the proposed algorithm, an implementation of the algorithm containing four metaheuristics is proposed and tested for solving multi-objective unconstrained binary quadratic programming problem. Experimental results on 50 benchmark instances show that the proposed algorithm can provide better overall performance than single metaheuristics, which demonstrates the effectiveness of the proposed algorithm.

Practical integer-to-binary mapping for quantum annealers

Recent advancements in quantum annealing hardware and numerous studies in this area suggests that quantum annealers have the potential to be effective in solving unconstrained binary quadratic programming problems. Naturally, one may desire to expand the application domain of these machines to problems with general discrete variables. In this paper, we explore the possibility of employing quantum annealers to solve unconstrained quadratic programming problems over a bounded integer domain. We present an approach for encoding integer variables into binary ones, thereby representing unconstrained integer quadratic programming problems as unconstrained binary quadratic programming problems. To respect some of the limitations of the currently developed quantum annealers, we propose an integer encoding, named bounded- coefficient encoding, in which we limit the size of the coefficients that appear in the encoding. Furthermore, we propose an algorithm for finding the upper bound on the coefficients of the encoding using the precision of the machine and the coefficients of the original integer problem. Finally, we experimentally show that this approach is far more resilient to the noise of the quantum annealers compared to traditional approaches for the encoding of integers in base two.

Computing optimal discrete readout weights in reservoir computing is NP-hard

We show NP-hardness of a generalized quadratic programming problem, which we called unconstrained n-ary quadratic programming (UNQP). This problem has recently become practically relevant in the context of novel memristor-based neuromorphic microchip designs, where solving the UNQP is a key operation for on-chip training of the neural network implemented on the chip. UNQP is the problem of finding a vector v ∈ SN which minimizes vTQv+vTc, where S={s1,…,sn}⊂Z is a given set of eligible parameters for v, Q∈ZN×N is positive semi-definite, and c∈ZN. In memristor-based neuromorphic hardware, S is physically given by a finite (and small) number of possible memristor states. The proof of NP-hardness is by reduction from the unconstrained binary quadratic programming problem, which is a special case of UNQP where S={0,1} and which is known to be NP-hard.

PPLS/D: Parallel Pareto Local Search Based on Decomposition.

Pareto local search (PLS) is a basic building block in many metaheuristics for a multiobjective combinatorial optimization problem. In this paper, an enhanced PLS variant called parallel PLS based on decomposition (PPLS/D) is proposed. PPLS/D improves the efficiency of PLS using the techniques of parallel computation and problem decomposition. It decomposes the original search space into L subregions and executes L parallel processes searching in these subregions simultaneously. Inside each subregion, the PPLS/D process is guided by a unique scalar objective function. PPLS/D differs from the well-known two phase PLS in that it uses the scalar objective function to guide every move of the PLS procedure in a fine-grained manner. In the experimental studies, PPLS/D is compared against the basic PLS and a recently proposed PLS variant on the multiobjective unconstrained binary quadratic programming problems and the multiobjective traveling salesman problems with, at most, four objectives. The experimental results show that regardless of whether the initial solutions are randomly generated or generated by heuristic methods, PPLS/D always performs significantly better than the other two PLS variants.

A note on the sufficient initial condition ensuring the convergence of directly extended 3-block ADMM for special semidefinite programming

ABSTRACTIn this note, we consider three types of problems, H-weighted nearest correlation matrix problem and two types of important doubly non-negative semidefinite programming, derived from the binary integer quadratic programming and maximum cut problem. The dual of these three types of problems is a 3-block separable convex optimization problem with a coupling linear equation constraint. It is known that, the directly extended 3-block alternating direction method of multipliers (ADMM3d) is more efficient than many of its variants for solving these convex optimization, but its convergence is not guaranteed. By choosing initial points properly, we obtain the convergence of ADMM3d for solving the dual of these three types of problems. Furthermore, we simplify the iterative scheme of ADMM3d and show the equivalence of ADMM3d to the 2-block semi-proximal ADMM for solving the dual's reformulation, under these initial conditions.

Finite-Control-Set Model Predictive Control for DFIG Wind Turbines

This paper presents a time efficient finite-control-set model predictive control (FCS-MPC) scheme for the doubly fed induction generator system. In this scheme, the switching states of the rotor side converter are directly taken as control inputs. This way, the optimized control action can be directly applied to the converter. Compared with the existing FCS-MPC approaches, the salient feature of the proposed scheme is the reduction of the computation time. By introducing a set of augmented decision variables, the original intractable binary quadratic programming problem in FCS-MPC can be analytically transformed to a binary linear programming problem, which can be solved efficiently. By this means, the computation time of the proposed scheme is much less than that of the existing schemes. This reduction in computation time enables FCS-MPC with longer prediction horizons, thus yielding better control performance. Note to Practitioners —This paper was motivated by the problem of improving the control performance for the doubly fed induction generator (DFIG). Because of its advantages, such as high energy efficiency, low acoustic noise, variable speed operation, and reduced converter rating, DFIG has emerged as a promising solution for the wind power generation. However, existing DFIG controllers have limitations, such as difficult to tune the parameters, inefficient to handle constraints, and do not consider the discrete operation of the power converters. To overcome these limitations, this paper proposes a new DFIG control scheme based on finite-control-set model predictive control (FCS-MPC), which has the abilities of online optimal control, explicitly handling constraints, and directly generating the switching signals for the power converters. A computationally efficient algorithm is proposed to reformulate the FCS-MPC problem as a linear program, thereby significantly reducing the computational efforts and rendering the proposed scheme more practical for implementation. Simulation results validate the effectiveness of the proposed control scheme. This scheme can be implemented in the field-programmable gate array, which will be our future work.

The generalized vertex cover problem and some variations

In this paper we study the generalized vertex cover problem (GVC), which is a generalization of various well studied combinatorial optimization problems. GVC is shown to be equivalent to the unconstrained binary quadratic programming problem and also equivalent to some other variations of the general GVC. Some solvable cases are identified and approximation algorithms are suggested for special cases. We also study GVC on bipartite graphs and identify some polynomially solvable cases. We show that GVC on bipartite graphs is equivalent to the bipartite unconstrained 0-1 quadratic programming problem. Integer programming formulations of GVC and related problems are presented and establish half-integrality property on some variables for the corresponding linear programming relaxations. We also discuss special cases of GVC where all feasible solutions are independent sets or vertex covers. These problems are observed to be equivalent to the maximum weight independent set problem or minimum weight vertex cover problem along with some algorithmic results.