Binary Quadratic Programming Research Articles

Adaptive Beamforming With Continuous/Discrete Phase Shifters via Convex Relaxation

Adaptive antenna array employing phase shifters only can reduce the hardware cost and power consumption, but the related optimization is well understood to be difficult to solve. In this paper, we examine the use of both continuous phase shifters (CPS) and discrete phase shifters (DPS) in antenna array for adaptive interference suppression. First, we formulate the phase-only tuning problem with CPS via quadratically constrained quadratic program (QCQP), which can be approximated through semidefinite relaxation (SDR) and solved by convex-optimization solvers efficiently. We also demonstrate that quantizing such results directly may not necessarily give acceptable performance. Then, we propose to use binary quadratic program (BQP) for the optimization of the adaptive beamformer utilizing DPS with an arbitrary number of control bits to specify the phase-shift levels. The resultant BQP can be relaxed and solved efficiently too. Moreover, due to the similarity between the CPS and DPS formulation, we can mix them in a single beamforming scheme to provide additional compromises between the output signal-to-interference and noise ratio (SINR) performance and implementation cost. Simulations show that in the considered interference-limited environment, our proposed beamformers with CPS, DPS, and hybrid phase shifters (HPS) give desirable results as expected.

Solving binary semidefinite programming problems and binary linear programming problems via multi-objective programming

‎In recent years the binary quadratic program has grown in‎ ‎combinatorial optimization‎. ‎Quadratic programming can‎ ‎be formulated as a semidefinite programming problem‎. ‎In this paper‎, ‎we consider the general form of‎ ‎binary semidefinite programming problems (BSDP)‎.‎ ‎We show the optimal solutions of the BSDP belong to the efficient set of a semidefinite multiobjective programming problem (SDMOP)‎. ‎Although‎ ‎finding all efficient points for multiobjective is not an easy problem‎, ‎but‎ ‎solving a continuous problem would be easier than a discrete variable problem‎. ‎In this paper‎, ‎we solve an SDMOP‎, ‎as an auxiliary‎, ‎instead of BSDP‎. We show the performance of our method by generating and solving random problems.

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.

Correction to: Inductive linearization for binary quadratic programs with linear constraints

Correction to: Inductive linearization for binary quadratic programs with linear constraints

A swarm intelligence graph-based pathfinding algorithm (SIGPA) for multi-objective route planning

Personalized tourist route planning (TRP) and navigation are online or real-time applications whose mathematical modeling leads to complex optimization problems. These problems are usually formulated with mathematical programming and can be described as NP hard problems. Moreover, the state-of-the-art (SOA) path search algorithms do not perform efficiently in solving multi-objective optimization (MO) problems making them inappropriate for real-time processing. To address the above limitations and the need for online processing, a swarm intelligence graph-based pathfinding algorithm (SIGPA) for MO route planning was developed. SIGPA generates a population whose individuals move in a greedy approach based on A∗ algorithm to search the solution space from different directions. It can be used to find an optimal path for every graph-based problem under various objectives. To test SIGPA, a generic MOTRP formulation is proposed. A generic TRP formulation remains a challenge since it has not been studied thoroughly in the literature. To this end, a novel mixed binary quadratic programming model is proposed for generating personalized TRP based on multi-objective criteria and user preferences, supporting, also, electric vehicles or sensitive social groups in outdoor cultural environments. The model targets to optimize the route under various factors that the user can choose, such as travelled distance, smoothness of route without multiple deviations, safety and cultural interest. The proposed model was compared to five SOA models for addressing TRP problems in 120 various scenarios solved with CPLEX solver and SIGPA. SIGPA was also tested in real scenarios with A* algorithm. The results proved the effectiveness of our model in terms of optimality but also the efficiency of SIGPA in terms of computing time. The convergence and the fitness landscape analysis showed that SIGPA achieved quality solutions with stable convergence.

Partial neighborhood local searches

AbstractIn this work, we study partial neighborhood local search (PNLS) techniques, which consist of adaptive walks where moves are chosen in a random subset of the current solution neighborhood. PNLSs balance between intensification and diversification is mainly determined by its single parameter λ designing the subset size. We analyze and discuss three PNLSs variants, using the abstraction of several combinatorial optimization problems into fitness landscapes: NK landscapes, Unconstrained Binary Quadratic Programming, Flow‐shop scheduling, and Quadratic Assignment. Our empirical study first analyses the structure of these landscapes through indicators. Then, we perform a parameter study of PNLSs for two computational budgets to study the impact of the sample size on the balance between intensification and diversification on different landscapes. Moreover, these experiments allow us to set an appropriate parameter value to compare the ability of PNLSs to reach good‐quality solutions accurately. Finally, we compare PNLS variants with two classical metaheuristics, identifying links between landscape characteristics and algorithms behavior.

Finding optimal solutions to several gray pattern instances

In this paper we compare two new binary linear formulations to a standard quadratic binary program for the gray pattern problem and solved all three by the Gurobi solver. One formulation performed significantly better and obtained seven optimal solutions that were not proven optimal before. It is interesting that the formulation that performed best is based on significantly more variables and constraints.

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.

Allocating and scheduling resources for a mobile photo enforcement program

We present a scheduling model for an urban mobile photo radar speed enforcement (MPE) program. An MPE program utilizes automated photo radar technology to capture speed limit violators, towards an aim to reduce speeding and thus, improve traffic safety. We propose a binary quadratic program that determines where (by location visit tasks) and when (in shifts) to send enforcement resources (operators and equipment) over a month-long schedule. The aim of this program is to minimize violations of enforcement time halos (and thus, efficiency losses) in attaining traffic-safety focused program goals. We solved this problem using a combined Dantzig-Wolfe and column generation optimization approach, after exploring several different types of commonly-used solution methods. The model is applied to a currently operational MPE program in the City of Edmonton, Canada. Using a five-day enforcement time halo, we find that our model results improve resource utilization by 24% over historical program deployment. The scheduling model is the final step of a larger unified MPE program framework that systematically and efficiently connects high-level urban traffic safety goals down shift-level allocations of highly limited enforcement resources. It provides a data-driven and transparent framework that both contributes to cities’ pursuits of Vision Zero and combatting negative public perceptions of MPE programs. Additionally, the framework can be easily adapted for, and transferred to, other automated traffic enforcement technologies across jurisdictions.

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...

Multiproduct Newsvendor Problem with Customer-Driven Demand Substitution: A Stochastic Integer Program Perspective

This paper studies a multiproduct newsvendor problem with customer-driven demand substitution, where each product, once run out of stock, can be proportionally substituted by the others. This problem has been widely studied in the literature; however, because of nonconvexity and intractability, only limited analytical properties have been reported and no efficient approaches have been proposed. This paper first completely characterizes the optimal order policy when the demand is known and reformulates this nonconvex problem as a binary quadratic program. When the demand is random, we formulate the problem as a two-stage stochastic integer program, derive several necessary optimality conditions, prove the submodularity of the profit function, and also develop polynomial-time approximation algorithms and show their performance guarantees. We further propose a tight upper bound via nonanticipativity dual, which is proven to be very close to the optimal value and can yield a good-quality feasible solution under a mild condition. Our numerical investigation demonstrates effectiveness of the proposed algorithms. Moreover, several useful findings and managerial insights are revealed from a series of sensitivity analyses.

Inductive linearization for binary quadratic programs with linear constraints

A linearization technique for binary quadratic programs (BQPs) that comprise linear constraints is presented. The technique, called “inductive linearization”, extends concepts for BQPs with particular equation constraints, that have been referred to as “compact linearization” before, to the general case. Quadratic terms may occur in the objective function, in the set of constraints, or in both. For several relevant applications, the linear programming relaxations obtained from applying the technique are proven to be at least as strong as the one obtained with a well-known classical linearization. It is also shown how to obtain an inductive linearization automatically. This might be used, e.g., by general-purpose mixed-integer programming solvers.

Integrated mathematical model based on a heuristic method for loading and routing of vehicles: application in a tobacco company

The vehicle-routing problem (VRP) combined with freight-loading problem is a complex and relatively recent issue studied by the scientific literature. This paper presents the formulation of a mathematical model and a procedure to solve this problem in a Cuban tobacco company aiming to determine the quantity of merchandise to be loaded on vehicles and the best route to be taken. For this purpose, a decomposition’s heuristic method was used and it was integrated with multiobjective programming by-goals and mixed binary quadratic programming. This approach allowed simplifying the problem and offering a satisfactory solution based on the demand fulfillment, the vehicles’ rational use and for searching the local optimums of the traffic load indicator. The model was tested in a case study and its feasibility evaluated based on a real operational situation in a tobacco company. Although the results of the application of the developed model does not imply reaching the optimal solution to the problem studied, it represents an opportunity for company’s performance improvement and it could be adapted and applied to other institutions dedicated to the same activities.

BDD-based optimization for the quadratic stable set problem

The quadratic stable set problem (QSSP) is a natural extension of the well-known maximum stable set problem. The QSSP is NP-hard and can be formulated as a binary quadratic program, which makes it an interesting case study to be tackled from different optimization paradigms. In this paper, we propose a novel representation for the QSSP through binary decision diagrams (BDDs) and adapt a hybrid optimization approach which integrates BDDs and mixed-integer programming (MIP) for solving the QSSP. The exact framework highlights the modeling flexibility offered through decision diagrams to handle nonlinear problems. In addition, the hybrid approach leverages two different representations by exploring, in a complementary way, the solution space with BDD and MIP technologies. Machine learning then becomes a valuable component within the method to guide the search mechanisms. In the numerical experiments, the hybrid approach shows to be superior, by at least one order of magnitude, than two leading commercial MIP solvers with quadratic programming capabilities and a semidefinite-based branch-and-bound solver.

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.

A note on scheduling on uniform parallel machines with unavailability constraints

We show by counterexample that the binary integer quadratic programming model proposed by Kaabi and Harrath [2019. “Scheduling on Uniform Parallel Machines with Periodic Unavailability Constraints.” International Journal of Production Research 57: 216–227] is incorrect. We fix this model and propose two linear ones.

Modeling an Uncertain Productivity Learning Process Using an Interval Fuzzy Methodology

Existing methods for forecasting the productivity of a factory are subject to a major drawback—the lower and upper bounds of productivity are usually determined by a few extreme cases, which unacceptably widens the productivity range. To address this drawback, an interval fuzzy number (IFN)-based mixed binary quadratic programming (MBQP)–ordered weighted average (OWA) approach is proposed in this study for modeling an uncertain productivity learning process. In the proposed methodology, the productivity range is divided into the inner and outer sections, which correspond to the lower and upper membership functions of an IFN-based fuzzy productivity forecast, respectively. In this manner, all actual values are included in the outer section, whereas most of the values are included within the inner section to fulfill different managerial purposes. According to the percentages of outlier cases, a suitable forecasting strategy can be selected. To derive the values of parameters in the IFN-based fuzzy productivity learning model, an MBQP model is proposed and optimized. Subsequently, according to the selected forecasting strategy, the OWA method is applied to defuzzify a fuzzy productivity forecast. The proposed methodology has been applied to the real case of a dynamic random access memory factory to evaluate its effectiveness. The experimental results indicate that the proposed methodology was superior to several existing methods, especially in terms of mean absolute error, mean absolute percentage error, and root mean square error in evaluating the forecasting accuracy. The forecasting precision achieved using the proposed methodology was also satisfactory.

Query batching optimization in database systems

Techniques based on sharing data and computation among queries have been an active research topic in database systems. While work in this area developed algorithms and systems that are shown to be effective, there is a lack of rigorous modeling and theoretical study for query processing and optimization. Query batching in database systems has strong resemblance to order batching in the warehousing context. While the latter is a well-studied problem, the literature on optimization techniques for query batching problem is quite scarce/nonexistent. In this study, we develop a Mixed Binary Quadratic Program (MBQP) to optimize query-batching in a database system. This model uses the coefficients of a linear regression on the query retrieval time, trained by a large set of randomly generated query sets. We also propose two heuristics, the so-called restricted-cardinality search methods I and II, for solving the proposed MBQP. To demonstrate the effectiveness of our proposed techniques, we conduct a comprehensive computational study over randomly generated instances of three well-known database benchmarks. The computational results show that when the proposed MBQP is solved using the designed heuristics, an improvement of up to 61.8% in retrieving time is achieved.