Unconstrained Optimization Model Research Articles

Support vector machine based on the quadratic unconstrained binary optimization model

Abstract Support vector machine (SVM) is a powerful supervised machine learning model that is often used in binary classification algorithms. As Moore’s Law approaches its theoretical limits and the demand for machine learning to handle large-scale, high-dimensional data analysis intensifies, the necessity of adopting non-traditional computational approaches becomes evident. Quantum computing, in particular, emerges as a vital solution for the effective training of SVM models, providing capabilities beyond those of classical computing systems. To solve the above problems, a QUBO (quadratic unconstrained binary optimization) model is proposed to transform the SVM machine learning model into a quadratic unconstrained binary optimization problem so that they can be effectively trained on the D-Wave platform using adiabatic quantum computer. The results show that the QUBO model can transform the SVM model into a simple quadratic programming problem, which makes it suitable for adiabatic quantum computer processing. When processing large-scale and high-dimensional data, this transformation shows a natural advantage and significantly improves computational efficiency. The application potential of this transformation technology is huge in the medical field.

Solving the resource constrained project scheduling problem with quantum annealing

Quantum annealing emerges as a promising approach for tackling complex scheduling problems such as the resource-constrained project scheduling problem (RCPSP). This study represents the first application of quantum annealing to solve the RCPSP, analyzing 12 well-known mixed integer linear programming (MILP) formulations and converting the most qubit-efficient one into a quadratic unconstrained binary optimization (QUBO) model. We then solve this model using the D-wave advantage 6.3 quantum annealer, comparing its performance against classical computer solvers. Our results indicate significant potential, particularly for small to medium-sized instances. Further, we introduce time-to-target and Atos Q-score metrics to evaluate the effectiveness of quantum annealing and reverse quantum annealing. The paper also explores advanced quantum optimization techniques, such as customized anneal schedules, enhancing our understanding and application of quantum computing in operations research.

Recovery of bandlimited graph signals based on the reproducing kernel Hilbert space

Signal recovery on graphs is attracting more and more attentions. Based on the smoothness assumption, the signal recovery problem can be formulated as an unconstrained optimization model. Although the model can be solved analytically, the computational cost is too expensive. In this paper, to reduce the computational complexity, we skillfully construct a kernel function, such that the smoothness item is equivalent to the norm of the signal modulo a constant in the reproducing kernel Hilbert space. Based on this, by using the Representer Theorem of reproducing kernel Hilbert space, the optimization problem for recovering the signal from a given bandlimited space, can be solved in a lower-dimensional subspace corresponding to the known labeled set. Experiments for the recovery of artificial graph signals in GSPBOX and real data in the recommendation system show very encouraging results.

Quantum‐Based Combinatorial Optimization for Optimal Sensor Placement in Civil Structures

Over the last decade, concepts such as industry 4.0 and the Internet of Things (IoT) have contributed to the increase in the availability and affordability of sensing technology. In this context, structural health monitoring (SHM) arises as an especially interesting field to integrate and develop these new sensing capabilities, given the criticality of structural application for human life and the elevated costs of manual monitoring. Due to the scale of structural systems, one of the main challenges when designing a modern monitoring system is the optimal sensor placement (OSP) problem. The OSP problem is combinatorial in nature, making its exact solution infeasible in most practical cases, usually requiring the use of metaheuristic optimization techniques. While approaches such as genetic algorithms (GAs) have been able to produce significant results in many practical case studies, their ability to scale up to more complex structures is still an area of open research. This study proposes a novel quantum‐based combinatorial optimization approach to solve the OSP problem approximately, within the context of SHM. For this purpose, a quadratic unconstrained binary optimization (QUBO) model formulation is developed, taking as a starting point of the modal strain energy (MSE) of the structure. The framework is tested using numerical simulations of Warren truss bridges of varying scales. The results obtained using the proposed framework are compared against exhaustive search approaches to verify their performance. More importantly, a detailed discussion of the current limitations of the technology and the future paths of research in the area is presented to the reader.

An alternative three-dimensional subspace method based on conic model for unconstrained optimization

In this paper, a three-dimensional subspace conjugate gradient method is proposed, in which the search direction is generated by minimizing the approximation model of the objective function in a three-dimensional subspace. The approximation model is not unique and is alternative between quadratic model and conic model by the specific criterions. The strategy of initial stepsize and nonmonotone line search are adopted, and the global convergence of the presented algorithm is established under mild assumptions. In numerical experiments, we use a collection of 80 unconstrained optimization test problems to show the competitive performance of the presented method.

New advances for quantum‐inspired optimization

AbstractAdvances in quantum computing with applications in combinatorial optimization have evolved at an increasing rate in recent years. The quadratic unconstrained binary optimization (QUBO) model is at the center of these developments and has become recognized as an effective alternative method for representing a wide variety of combinatorial optimization problems. Additional momentum has resulted from the arrival of quantum computers and their ability to solve the Ising spin glass problem, another form of the QUBO model. This paper highlights advances in solving QUBO models and extensions to more general polynomial unconstrained binary optimization (PUBO) models as important alternatives to traditional approaches. Computational experience is provided that compares the performance of unique quantum‐inspired metaheuristic solvers—the Next Generation Quantum (NGQ) solver for QUBO models and the NGQ‐PUBO solver for PUBO models—with the performance of CPLEX and the Dwave quantum advantage solver. Extensive results, including experiments with a set of large set partitioning problems representing the largest QUBO models reported in the literature to date, along with maximum diversity and max cut problem sets, disclose that our solvers outperform both CPLEX and Dwave by a wide margin in terms of both computational time and solution quality.

Quafu-Qcover: Explore combinatorial optimization problems on cloud-based quantum computers

We introduce Quafu-Qcover, an open-source cloud-based software package developed for solving combinatorial optimization problems using quantum simulators and hardware backends. Quafu-Qcover provides a standardized and comprehensive workflow that utilizes the quantum approximate optimization algorithm (QAOA). It facilitates the automatic conversion of the original problem into a quadratic unconstrained binary optimization (QUBO) model and its corresponding Ising model, which can be subsequently transformed into a weight graph. The core of Qcover relies on a graph decomposition-based classical algorithm, which efficiently derives the optimal parameters for the shallow QAOA circuit. Quafu-Qcover incorporates a dedicated compiler capable of translating QAOA circuits into physical quantum circuits that can be executed on Quafu cloud quantum computers. Compared to a general-purpose compiler, our compiler demonstrates the ability to generate shorter circuit depths, while also exhibiting superior speed performance. Additionally, the Qcover compiler has the capability to dynamically create a library of qubits coupling substructures in real-time, utilizing the most recent calibration data from the superconducting quantum devices. This ensures that computational tasks can be assigned to connected physical qubits with the highest fidelity. The Quafu-Qcover allows us to retrieve quantum computing sampling results using a task ID at any time, enabling asynchronous processing. Moreover, it incorporates modules for results preprocessing and visualization, facilitating an intuitive display of solutions for combinatorial optimization problems. We hope that Quafu-Qcover can serve as an instructive illustration for how to explore application problems on the Quafu cloud quantum computers.

Decentralized Coordination of DERs for Dynamic Economic Emission Dispatch

This paper focuses on the dynamic economic emission dispatch (DEED) problem, to coordinate the distributed energy resources (DERs) in a power system and achieve economical and environmental operation. Distributed energy storages (ESs) are introduced into problem formulation in which charging/discharging efficiency is taken into account. By relaxing the nonconvexity induced by the charging/discharging model of ESs and network losses, we convert the non-convex DEED problem into its convex equivalency. Then, through a Lagrangian duality reformulation, an equivalent unconstrained consensus optimization model is established—a novel consensus-based decentralized algorithm, where the incremental cost is chosen as the consensus variable. At each iteration, only one primal variable requires sub-optimization, and it is completely locally updated. This is different from the well-known alternating direction method of multiplier (ADMM)-based algorithms where more than one subproblem needs to be solved at each iteration. The results of the comparative experiments also reflect the algorithm’s advantage in terms of computational efficiency. The simulation results validate the effectiveness of the proposed algorithm, achieving a balance between emissions and economic considerations.

Optimal configuration of concentrating solar power generation in power system with high share of renewable energy resources

Under the worldwide carbon neutralization targets, concentrating solar power (CSP) is arousing great attention. With the thermal energy storage (TES), CSP is friendly to the power system operation by supplying controllable renewable energy. The capacities of its solar field and TES are essential parameters for maximizing the profit of a CSP plant. This paper formulates an explicit expression of the CSP plant's profit instead of using a simulation-based method. Then, an unconstrained optimization model is proposed to calculate its optimal configuration directly. This model provides insights into the optimal configuration of CSP with different penetrations of wind power in the case study. The results show that to obtain a better profit for the CSP plant, large solar multiple (more than 3.0) and TES (more than 13 h) are preferred to collaborate with high penetration of wind and photovoltaic plants. The effectiveness of the proposed method is verified compared to the enumeration searching method. The economy and feasibility of installing an electric heater (EH) in CSP are also demonstrated. Generally, the optimal investment in EH is linearly correlated to the penetration of variable energy resources.

A monocular visual single-axis rotation measurement method of multi-rudders based on cooperative targets

To tackle the problem that the rotation angle of multiple rudders needs to be measured in real time and high precision, a measurement method based on cooperative targets is proposed in this paper. Firstly, to detect and encode multiple similar rudders with sparse feature points in complex backgrounds, we design a novel visual cooperative target with encoding information, and study a cooperative target localization algorithm based on contour extraction as well as a corner point extraction algorithm based on a checkerboard grid. Then, through the PnP algorithms and numerous corner points extracted from the cooperative target, an unconstrained optimization model is developed based on the re-projection error, and the Levenberg-Marquard (LM) algorithm is applied to estimate the rotation angle. Finally, an experiment platform is set up to evaluate the accuracy and speed of the algorithm. Compared with the existing methods, the experimental results show that the proposed cooperative target exhibits favorable localization under various typical experimental environments with a detection success rate better than 98%, meanwhile, the actual measurement accuracy of the angle estimation model is better than 0.05° and the average execution time of each measurement is less than 20 ms. Moreover, the proposed method is easy to install and operate in actual measurement, and can achieve simultaneous measurement of the spatial rotation angle of multiple rudders, which is well suited to actual industrial production needs.

HUBO and QUBO models for prime factorization

The security of the RSA cryptosystem is based on the difficulty of factoring a large number N into prime numbers p and q satisfying N=ptimes q. This paper presents a prime factorization method using a D-Wave quantum computer that could threaten the RSA cryptosystem in the future. The starting point for this method is very simple, representing two prime numbers as qubits. Then, we set the difference between the product of the two prime numbers expressed in qubits and N as a cost function, and we find the solution when the cost function is minimized. D-Wave's quantum annealer can find the minimum value of any quadratic problem. However, the cost function must be a higher-order unconstrained optimization (HUBO) model because it contains second- or higher-order terms. We used a hybrid solver accessible via Leap, D-Wave’s real-time quantum cloud service, and the dimod package provided by the D-Wave Ocean software development kit (SDK) to solve the HUBO problem. We also successfully factorized 102,454,763 with 26 logical qubits. In addition, we factorized 1,000,070,001,221 using the range-dependent Hamiltonian algorithm.

Nonmonotone Alternative Direction Method Based on Simple Conic Model for Unconstrained Optimization

In this paper, we propose a nonmonotone Alternative Direction Method (ADM) based on simple conic model for unconstrained optimization. Unlike traditional trust region method, the subproblem in our method is a simple conic model, where the Hessian of the objective function is replaced by a scalar approximation, the trust region subproblem is solved by ADM which was first proposed by Zhu and Ni. When the trial point isn’t accepted by trust region, line search technique is used to find an acceptable point instead of resolving the trust region subproblem. The new method needs less memory capacitance and computational complexity. The global convergence of the algorithm is established under some mild conditions. Numerical results on a series of standard test problems are reported to show the new method is effective and robust.

HUBO formulations for solving the eigenvalue problem

Solving the eigenvalue problem is particularly important in almost all fields of science and engineering. With the development of quantum computers, multiple algorithms have been proposed for this purpose. However, such methods are usually only applicable to matrices of specific types, such as unitary or Hermitian matrices. The quantum annealer of the D-Wave, a quantum computer, returns the minimum value of the quadratic unconstrained binary optimization (QUBO) model. Thus, quantum annealers can be leveraged to solve arbitrary eigenvalue problems by formulating corresponding QUBO models. In this paper, we propose two higher-order unconstrained optimization (HUBO) formulations to solve eigenvalue problems involving n×n general matrices. In addition, we use a formula to reduce the order and convert the HUBO model into a QUBO model. Further, by using a quantum approximate optimization algorithm, this method can be extended to a gate-model quantum computer.

Three-dimensional point cloud registration algorithm based on the correlation coefficient

Point cloud registration is an important research problem in machine vision. One of the most widely used methods for point set registration is the iterative closest point (ICP) algorithm. However, ICP is known to be susceptible to local minima because it is based on the local iterative optimization technique and its performance critically relies on the quality of the initialization. To reduce the possibility of the algorithm falling into local minima, we propose an iterative algorithm based on geometric distance and local feature weighting named weighted correlation coefficient iterative closest point algorithm (WCC_ICP). The algorithm first establishes the corresponding relationship of points using linear representation and constructs an optimization model with constraints. Subsequently, the constrained optimization model is converted into an unconstrained optimization model by Lagrangian multiplication. Finally, an iterative technique is used to solve the optimal rigid body transformation of the optimized model. In addition, in each iteration, the identification ability of the local feature operator of points is used to repair the corresponding relation of points, which reduces the risk of the iterative algorithm falling into the local minima. Numerical experiments show that the proposed WCC_ICP algorithm reduces the risk of the iterative algorithm falling into local minima.

Quantum Annealing in the NISQ Era: Railway Conflict Management

We are in the noisy intermediate-scale quantum (NISQ) devices' era, in which quantum hardware has become available for application in real-world problems. However, demonstrations of the usefulness of such NISQ devices are still rare. In this work, we consider a practical railway dispatching problem: delay and conflict management on single-track railway lines. We examine the train dispatching consequences of the arrival of an already delayed train to a given network segment. This problem is computationally hard and needs to be solved almost in real time. We introduce a quadratic unconstrained binary optimization (QUBO) model of this problem, which is compatible with the emerging quantum annealing technology. The model's instances can be executed on present-day quantum annealers. As a proof-of-concept, we solve selected real-life problems from the Polish railway network using D-Wave quantum annealers. As a reference, we also provide solutions calculated with classical methods, including the conventional solution of a linear integer version of the model as well as the solution of the QUBO model using a tensor network-based algorithm. Our preliminary results illustrate the degree of difficulty of real-life railway instances for the current quantum annealing technology. Moreover, our analysis shows that the new generation of quantum annealers (the advantage system) does not perform well on those instances, either.

AI-Based Intelligent Model to Predict Epidemics Using Machine Learning Technique

The immediate international spread of severe acute respiratory syndrome revealed the potential threat of infectious diseases in a closely integrated and interdependent world. When an outbreak occurs, each country must have a well-coordinated and preventative plan to address the situation. Information and Communication Technologies have provided innovative approaches to dealing with numerous facets of daily living. Although intelligent devices and applications have become a vital part of our everyday lives, smart gadgets have also led to several physical and psychological health problems in modern society. Here, we used an artificial intelligence AI-based system for disease prediction using an Artificial Neural Network (ANN). The ANN improved the regularization of the classification model, hence increasing its accuracy. The unconstrained optimization model reduced the classifier’s cost function to obtain the lowest possible cost. To verify the performance of the intelligent system, we compared the outcomes of the suggested scheme with the results of previously proposed models. The proposed intelligent system achieved an accuracy of 0.89, and the miss rate 0.11 was higher than in previously proposed models.

Optimization with photonic wave-based annealers.

Many NP-hard combinatorial optimization (CO) problems can be cast as a quadratic unconstrained binary optimization model, which maps naturally to an Ising model. The final spin configuration in the Ising model can adiabatically arrive at a solution to a Hamiltonian, given a known set of interactions between spins. We enhance two photonic Ising machines (PIMs) and compare their performance against classical (Gurobi) and quantum (D-Wave) solvers. The temporal multiplexed coherent Ising machine (TMCIM) uses the bistable response of an electro-optic modulator to mimic the spin up and down states. We compare TMCIM performance on Max-cut problems. A spatial photonic Ising machine (SPIM) convolves the wavefront of a coherent laser beam with the pixel distribution of a spatial light modulator to adiabatically achieve a minimum energy configuration, and solve a number partitioning problem (NPP). Our computational results on Max-cut indicate that classical solvers are still a better choice, while our NPP results show that SPIM is better as the problem size increases. In both cases, connectivity in Ising hardware is crucial for performance. Our results also highlight the importance of better understanding which CO problems are most likely to benefit from which type of PIM. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Unconstrained model and merge sorted serpentine arrangement method for dynamic reconfiguration of large PV arrays

• An unconstrained optimization model, and a heuristic method namely, Merge Sorted Serpentine Arrangement, are proposed for modelling and solving the problem of finding the optimal reconfiguration of a photovoltaic array to minimize partial shading losses. • The proposed unconstrained optimization model formulation enables the application of different meta-heuristic and heuristic optimization algorithms for solving the problem of finding the optimal reconfiguration of a photovoltaic array to minimize partial shading losses. • Extensive case studies show that the proposed method reduces average computation time by up to 80 % when compared with other methods. • The proposed method consistently reduces partial shading losses by up to 11 % under dynamic and static partial shading conditions due to its lower computation time as demonstrated by extensive case studies. • The main drawback of the proposed method is that it increases memory requirements by up 100 % when compared with other methods. The problem of finding the optimal interconnection between photovoltaic modules in a dynamic photovoltaic array to minimize partial shading losses is a Non-deterministic Polynomial time-hard (i.e., NP-hard) problem. Therefore, no known deterministic method can solve the problem in polynomial time. However, meta-heuristic and heuristic methods can provide solutions to this problem. In this paper, an unconstrained optimization model and a heuristic method, namely, Merge Sorted Serpentine Arrangement (MSA), are proposed for the first time for modelling and solving the problem. In the proposed method, the irradiance levels of the PV modules are accurately estimated using the double diode model and then sorted according to the merge sorting algorithm. After that, the PV modules are arranged in the PV array according to their sorted irradiance levels in a serpentine way using the proposed switching matrix. The proposed method has consistently resulted in lower computational time and partial shading losses when compared with the other methods in the literature, due to the application of the Merge Sorting Algorithm, which has an average time complexity of O (n log(n)). However, the proposed method has resulted in increased memory, switches, and sensor requirements when compared with some of the other methods in the literature.

Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing

In this work, we introduce a new workflow to solve portfolio optimization problems on annealing platforms. We combine a classical preprocessing step with a modified unconstrained binary optimization (QUBO) model and evaluate it using simulated annealing (classical computer), digital annealing (Fujitsu’s Digital Annealing Unit), and quantum annealing (D-Wave Advantage). Starting from Markowitz’s theory on portfolio optimization, our classical preprocessing step finds the most promising assets within a set of possible assets to choose from. We then modify existing QUBO models for portfolio optimization, such that there are no limitations on the number of assets that can be invested in. Furthermore, our QUBO model enables an investor to also place an arbitrary amount of money into each asset. We apply this modified QUBO to the set of promising asset candidates we generated previously via classical preprocessing. A solution to our QUBO model contains information about what percentage of the whole available capital should be invested into which asset. For the evaluation, we have used publicly available real-world data sets of stocks of the New York Stock Exchange as well as common ETFs. Finally, we have compared the respective annealing results with randomly generated portfolios by using the return, variance, and diversification of the created portfolios as measures. The results show that our QUBO formulation is capable of creating well-diversified portfolios that respect certain criteria given by an investor, such as maximizing return, minimizing risk, or sticking to a certain budget.

Solving the Adaptive Cubic Regularization Sub-Problem Using the Lanczos Method

The adaptive cubic regularization method solves an unconstrained optimization model by using a three-order regularization term to approximate the objective function at each iteration. Similar to the trust-region method, the calculation of the sub-problem highly affects the computing efficiency. The Lanczos method is an useful tool for simplifying the objective function in the sub-problem. In this paper, we implement the adaptive cubic regularization method with the aid of the Lanczos method, and analyze the error of Lanczos approximation. We show that both the error between the Lanczos objective function and the original cubic term, and the error between the solution of the Lanczos approximation and the solution of the original cubic sub-problem are bounded up by the condition number of the optimal Hessian matrix. Furthermore, we compare the numerical performances of the adaptive cubic regularization algorithm when using the Lanczos approximation method and the adaptive cubic regularization algorithm without using the Lanczos approximation for unconstrained optimization problems. Numerical experiments show that the Lanczos method improves the computation efficiency of the adaptive cubic method remarkably.