Binary Quadratic Problem Research Articles

A large population island framework for the unconstrained binary quadratic problem

The unconstrained binary quadratic problem is an NP-hard problem and has applications in many fields. Recently, the problem has attracted much interest in the field of quantum optimization, as it is directly related to the Ising problem in physics and the development of quantum computers. However, effectively solving large instances of this problem remains a major challenge for existing solution methods. To advance the state of the art in solving the problem on a large scale, we propose an evolutionary algorithm with a very large population organized in different islands and integrating a new pairing and recombination method to produce promising offspring in each generation. Numerous experiments are conducted to evaluate the effects of different pairing strategies, crossovers, and migration topologies. This research has led to the discovery of new bounds for difficult instances of the maximum cut problem, which has been transformed using the binary quadratic formulation.

New approach to solve unconstrained binary quadratic problem

New approach to solve unconstrained binary quadratic problem

Efficient Algorithm for Binary Quadratic Problem by Column Generation and Quantum Annealing

Efficient Algorithm for Binary Quadratic Problem by Column Generation and Quantum Annealing

An Irregular Tiled Array Technique for Microwave Wireless Power Transmission

An irregular tiled array technique with the maximization of beam collection efficiency (BCE) for microwave wireless power transmission (WPT) is addressed in this paper, where the radiating elements are irregularly partitioned into multiple subarrays for reducing the number of radio frequency (RF) chain. The system architectures of WPT based on irregular tiled arrays are analyzed for the exhibition of its low-cost advantages. To achieve a good trade-off between the manufacturing cost and beampattern, a mixed-integer optimization problem (MIOP) accounting for the maximization of BCE satisfying the subarray tiling configuration, scanning direction and peak sidelobe level (SLL) constraints is established. By equivalently reformulating the beampattern, the MIOP in terms of subarray tiling configuration and subarray excitations is decoupled, and a two-stage solution method can be implemented. Firstly, resort to ‘generalized BCE’, the binary quadratic optimization problem (BQOP) for maximizing the BCE at considered scanning angles while guaranteeing the full aperture coverage is reduced to a binary second-order cone optimization problem. After the subarray tiling configuration is determined, the non-convex Rayleigh entropy maximization problem at a given scanning angle with respect to subarray complex excitations while satisfying peak SLL constraint is globally solved by using iterative convex approximation algorithm. Owing to the high efficiency of convex optimization, the BCE is able to monotonically increase to a stable value. Several numerical results with different array size and subarray shapes are provided to assess the effectiveness of the proposed method by comparing to the competitive counterparts in the open literatures.

Mixed-Integer Nonlinear Programming for State-Based Non-Intrusive Load Monitoring

Energy disaggregation, known in the literature as Non-Intrusive Load Monitoring (NILM), is the task of inferring the energy consumption of each appliance given the aggregate signal recorded by a single smart meter. In this paper, we propose a novel two-stage optimization-based approach for energy disaggregation. In the first phase, a small training set consisting of disaggregated power profiles is used to estimate the parameters and the power states by solving a mixed integer programming problem. Once the model parameters are estimated, the energy disaggregation problem is formulated as a constrained binary quadratic optimization problem. We incorporate penalty terms that exploit prior knowledge on how the disaggregated traces are generated, and appliance-specific constraints characterizing the signature of different types of appliances operating simultaneously. Our approach is compared with existing optimization-based algorithms both on a synthetic dataset and on three real-world datasets. The proposed formulation is computationally efficient, able to disambiguate loads with similar consumption patterns, and successfully reconstruct the signatures of known appliances despite the presence of unmetered devices, thus overcoming the main drawbacks of the optimization-based methods available in the literature.

BiqBin: A Parallel Branch-and-bound Solver for Binary Quadratic Problems with Linear Constraints

We present BiqBin, an exact solver for linearly constrained binary quadratic problems. Our approach is based on an exact penalty method to first efficiently transform the original problem into an instance of Max-Cut, and then to solve the Max-Cut problem by a branch-and-bound algorithm. All the main ingredients are carefully developed using new semidefinite programming relaxations obtained by strengthening the existing relaxations with a set of hypermetric inequalities, applying the bundle method as the bounding routine and using new strategies for exploring the branch-and-bound tree.Furthermore, an efficient C implementation of a sequential and a parallel branch-and-bound algorithm is presented. The latter is based on a load coordinator-worker scheme using MPI for multi-node parallelization and is evaluated on a high-performance computer.The new solver is benchmarked against BiqCrunch, GUROBI, and SCIP on four families of (linearly constrained) binary quadratic problems. Numerical results demonstrate that BiqBin is a highly competitive solver. The serial version outperforms the other three solvers on the majority of the benchmark instances. We also evaluate the parallel solver and show that it has good scaling properties. The general audience can use it as an on-line service available athttp://www.biqbin.eu.

A Novel Method for Maximum Directivity Synthesis of Irregular Phased Arrays

A novel and effective method for the design of maximum directivity irregular phased arrays is proposed in this article. By decomposing the radiation pattern of irregular arrays (IAs) into the product of subarray pattern and subarray array factor, the principal factor for the reduction of directivity and the appearance of high sidelobes is analyzed. Aimed at maximizing directivity of IAs, a binary quadratic optimization problem (BQOP) is established for the design of subarray tiling configuration, where the directivities at considered scanning angles satisfying given peak sidelobes level (SLL) is maximized. To consider the directivities at all the scanning angles within given scan ranges, a concept of “generalized directivity” is introduced and defined. Meanwhile, it is known that high directivity usually enables to suppress high SLL to a great extent. Therefore, the BQOP is reduced to a simplified binary quadratic problem (SBQP) for the maximization of the “generalized directivity” without the high-dimensional constraint on SLL. The associated complex excitations at subarray level are solved using convex optimization after the subarray tiling configuration is determined. By comparing to the reported state-of-the-art methods, several representative numerical examples are implemented to assess the effectiveness of the proposed method.

Airport Gate Assignment as a Nash Equilibrium Problem

The airport gate assignment problem addresses the optimal assignment of a set of aircraft to a set of stands. The underlying combinatorial optimization problem is usually modeled as a binary quadratic assignment problem, whereby the stand assignment of a certain aircraft depends on the stand assignment of all other aircraft. The solving time of the optimization problem may increase exponentially with the number of aircraft and stands considered. For this reason, real-case scenarios can be solved only by heuristics in due time. In this paper, we propose a novel approach on modeling and solving the airport gate assignment problem by making use of the game theory. The aim is to identify a Nash equilibrium as a solution of the airport gate assignment problem in the following sense: no aircraft can improve its stand assignment by a sole deviation from its assigned stand. The algorithm is capable of delivering an assignment for a real-case scenario in minutes instead of hours. The performance of the algorithm is demonstrated by modeling and solving a real-case scenario for terminal 2 of the Munich Airport.

Dantzig–Wolfe reformulations for binary quadratic problems

The purpose of this paper is to provide strong reformulations for binary quadratic problems. We propose a first methodological analysis on a family of reformulations combining Dantzig–Wolfe and Quadratic Convex optimization principles. We show that a few reformulations of our family yield continuous relaxations that are strong in terms of dual bounds and computationally efficient to optimize. As a representative case study, we apply them to a cardinality constrained quadratic knapsack problem, providing extensive experimental insights. We report and analyze in depth a particular reformulation providing continuous relaxations whose solutions turn out to be integer optima in all our tests.

Multi-spin-flip engineering in an Ising machine

A merge process is proposed to engineer a multi-spin flip in an Ising machine. The merge process deforms the Hamiltonian (energy function) of the Ising model. We prove a theorem for the merge process and show that a single-spin flip in the deformed Hamiltonian is equivalent to a multi-spin flip in the original Hamiltonian. A merge process induces a transition within the subspace of feasible solutions. We propose a hybrid simulated annealing (SA) algorithm with the merge process. The hybrid algorithm outperforms the conventional SA algorithm, genetic algorithm, and tabu search in the binary quadratic knapsack problems (QKP) and the quadratic assignment problems (QAP). Finally, the hybrid merge process is used in a real Ising machine. The performance is improved in QKP and QAP instances. The merge process is generally applicable to existing Ising machine hardware because the deformed Hamiltonian keeps the format of the Ising model.

The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore–Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. We also present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem.

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.

Market Graph Clustering via QUBO and Digital Annealing

We present a novel technique for cardinality-constrained index-tracking, a common task in the financial industry. Our approach is based on market graph models. We model our reference indices as market graphs and express the index-tracking problem as a quadratic K-medoids clustering problem. We take advantage of a purpose-built hardware architecture to circumvent the NP-hard nature of the problem and solve our formulation efficiently. The main contributions of this article are bridging three separate areas of the literature, market graph models, K-medoid clustering and quadratic binary optimization modeling, to formulate the index-tracking problem as a binary quadratic K-medoid graph-clustering problem. Our initial results show we accurately replicate the returns of various market indices, using only a small subset of their constituent assets. Moreover, our binary quadratic formulation allows us to take advantage of recent hardware advances to overcome the NP-hard nature of the problem and obtain solutions faster than with traditional architectures and solvers.

Conic optimization: A survey with special focus on copositive optimization and binary quadratic problems

• We survey recent progress in conic optimization. • We discuss second order cone programs (SOCP), semidefinite problems (SDP), and copositive problems. • Special emphasis is given to modelling issues. • The survey contains more than 110 references. A conic optimization problem is a problem involving a constraint that the optimization variable be in some closed convex cone. Prominent examples are linear programs (LP), second order cone programs (SOCP), semidefinite problems (SDP), and copositive problems. We survey recent progress made in this area. In particular, we highlight the connections between nonconvex quadratic problems, binary quadratic problems, and copositive optimization. We review how tight bounds can be obtained by relaxing the copositivity constraint to semidefiniteness, and we discuss the effect that different modelling techniques have on the quality of the bounds. We also provide some new techniques for lifting linear constraints and show how these can be used for stable set and coloring relaxations.

A Study on Reconfigurable Nanomagnetic Array and Effect of Gilbert Damping on Reconfigurability

A 2-D grid of circular nanomagnets (NMs) has been effectively used to solve binary quadratic optimization problems via magnetic interaction in the X-Y plane. Reconfiguring this architecture is challenging due to the geometry and spacing constraints imposed by the computational algorithm, but can be achieved by destabilizing the uniform and unidirectional magnetic influence from the non-computing cell. In this paper, we studied how the straightforward spin transfer torque (STT) can induce a rotational coupling field between the non-computing and computing cells to impair static interaction. This mechanism is also better than spin orbital torque (SOT)-based reconfiguration in terms of power, speed and system integration. Our study shows that STT-based solution requires 14× less current desnsity and is 25% faster than the SOT-based counterpart. We extended the study to achieve low-power reconfiguration by altering the Gilbert damping. A lower damping can further reduce 60% of the required current than the high damping case.

Interference Minimization in 5G Physical-Layer Network Slicing

The interference resulting from densification of access points and the coexistence of different numerologies within the same spectrum severely hinders inter-slice isolation. We propose a slice allocation policy that enforces inter-slice isolation by minimizing the inter-slice interference suffered by each virtual operator. In detail, we design a binary quadratic non-convex optimization problem that minimizes i) the inter-slice interference generated by interfering base stations and ii) the inter-slice interference generated by the multiplexing of spectrum slices having different numerologies. We also provide a heuristic algorithm to render the solution scalable in practical scenarios. We assess the performance of both approaches by evaluating the signal-to-interference-plus-noise ratio (SINR) associated to each slice through simulations. Results reveal that the heuristic algorithm provides a solution comparable with the optimal one on different minimization scenarios. Moreover, a considerable SINR improvement is observed with respect to a base-line scheme that does not account for inter-slice interference.

A binary quadratic Titchmarsh divisor problem

We consider a binary quadratic variant of the Titchmarsh divisor problem and give an asymptotic formula for $\sum _{p^2+q^2\leq N} \tau (p^2+q^2+1)$, where $p,q$ are primes.

A modified descent method-based heuristic for binary quadratic knapsack problems with conflict graphs

The knapsack problem arises in a variety of real world applications, including flexible manufacturing systems, railway stations, hydrological studies and others. In this paper, we propose a descent method-based heuristic for tackling a special knapsack problem: the binary quadratic knapsack with conflict graphs. The proposed method combines (i) an intensification search with a descent method for enhancing the accuracy of the solutions and (ii) a diversification strategy which is used for enlarging the search space. The method uses degrading and re-optimization strategies in order to reach a series of diversified solutions. The performance of the proposed method is evaluated on benchmark instances taken from the literature, where its achieved results are compared to those reached by both GLPK solver and the best method available in the literature. The method seems very competitive, where it is able to achieve 37 new lower bounds.

Embedding Equality Constraints of Optimization Problems into a Quantum Annealer

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.

Robust optimisation of unconstrained binary quadratic problems

In this paper we focus on the unconstrained binary quadratic optimisation model, maximise xtQx, x binary, and consider the problem of identifying optimal solutions that are robust with respect to perturbations in the Q matrix. We are motivated to find robust, or stable, solutions because of the uncertainty inherent in the big data origins of Q and limitations in computer numerical precision, particularly in a new class of quantum annealing computers. Experimental design techniques are used to generate a diverse subset of possible scenarios, from which robust solutions are identified. An illustrative example with practical application to business decision making is examined. The approach presented also generates a surface response equation which is used to estimate upper bounds in constant time for Q instantiations within the scenario extremes. In addition, a theoretical framework for the robustness of individual xi variables is considered by examining the range of Q values over which the xi are predetermined.