Quadratic Formulation Research Articles

Reallocation of time after an exogenous reduction in mandatory travel: transport, work, and leisure in Chilean two-worker two-gender households

A two-step model framework is proposed and applied to analyze empirically the impact of a travel time reduction on labor supply in two-worker two-gender households, using a nationwide data set for time use collected in 2015 in Chile. First, a system of structural equations (SSE) is estimated to reveal which activities can be considered as committed, and to unveil the hierarchy of activities by gender undertaken by the household workers to help defining the labor supply model. Then a quadratic formulation including committed time for each individual and committed expenses for the family is applied to the time-use data to estimate a labor supply model, considering the findings with the SSE (hierarchy and committed activities). Results were obtained controlling for household size, region, and age, showing systematically that labor supply diminishes with the wage rates of either working member of the household (a forward falling shape), that committed expenses induces more work, and that diminishing mandatory travel time induces an increase in working hours that varies between 27 and 64 percent of the liberated time, in line with 2023 reports on the reallocation of liberated travel time due to remote working and confirming the theoretical findings by Jara-Diaz and Contreras (2024).

Quantum annealing for nearest neighbour compliance problem

Quantum Computing has emerged as a promising alternative, utilising quantum mechanics for faster computations. This paper explores the nearest neighbour compliance (NNC) Problem in Gate-based Quantum Computers, where quantum gates are constrained to operate on physically adjacent qubits. The NNC problem aims to optimise the insertion of SWAP-gates to ensure compliance with these constraints while minimising their count. This work introduces Quantum Annealing to tackle the NNC problem, proposing two Quadratic Unconstrained Optimisation Problem formulations. The formulations are tested on a contemporary Quantum Annealer, and their performance is compared with previous methods. It shows that the prospect of using Quantum Annealing is promising, however, the current state of the hardware makes that finding the embedding is the limiting factor.

Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation

In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

Smarter facility layout design: leveraging worker localisation data to minimise travel time and alleviate congestion

This paper introduces a novel methodology leveraging worker localisation data from ultrawide-band sensors to formulate alternative facility layouts aimed at minimising travel time and congestion in labour-intensive manufacturing systems. The system preprocesses sensor data to discern flow patterns between existing stations within the production facility, such as machine tools, workbenches, and stores. This information about the movement of people and materials informs the generation of optimised layouts through scenario-based optimisation. We explored two methods to devise these new layouts: a mixed-integer linear programming method and a simulated annealing metaheuristic, the latter being specifically developed to find high-quality solutions to the quadratic layout design formulation. Both methods employ biobjective formulations, focussing on the minimisation of travel time and the reduction of congestion risk on the manufacturing floor, an aspect often neglected in prior studies. Our methodology, applied to a real-world manual assembly line case study, demonstrated the potential to reduce travel time by a minimum of 32% and alleviate congestion while maintaining significant safety distances between facilities. This was achieved by automatically identifying design features that position high-traffic facilities closely and align them to eliminate movement overlaps.

Analytical game strategies for active UAV defense considering response delays

In the realm of aerial warfare, the protection of Unmanned Aerial Vehicles (UAVs) against adversarial threats is crucial. In order to balance the impact of response delays and the demand for onboard applications, this paper derives three analytical game strategies for the active defense of UAVs from differential game theory, accommodating the first-order dynamic delays. The targeted UAV executes evasive maneuvers and launches a defending missile to intercept the attacking missile, which constitutes a UAV-Missile-Defender (UMD) three-body game problem. We explore two distinct operational paradigms: the first involves the UAV and the defender working collaboratively to intercept the incoming threat, while the second prioritizes UAV self-preservation, with independent maneuvering away from potentially sacrificial engagements. Starting with model linearization and order reduction, the Collaborative Interception Strategy (CIS) is first derived via a linear quadratic differential game formulation. Building upon CIS, we further explore two distinct strategies: the Informed Defender Interception Strategy (IDIS), which utilizes UAV maneuvering information, and the Unassisted Defender Interception Strategy (UDIS), which does not rely on UAV maneuvering information. Additionally, we investigate the conditions for the existence of saddle point solutions and their relationship with vehicle maneuverability and response agility. The simulations demonstrate the effectiveness and advantages of the proposed strategies.

Green synthesized silver nanoparticles in two stages: Box Behnken Design to machine learning

In order to solve the modeling issues due to data scarcity problems in the disciplines utilizing statistical approximations, a novel two-stage idea is proposed. As a use case, nanoparticle biosynthesis was selected, for which an environmentally friendly process is of vital importance. First, Box Behnken Design was used for experimental setup, quadratic model formulation and data generation. The second stage consists of Machine Learning, in which the data generated in the previous stage were fed into a Neural Network to determine the relationship between the parameters. Obtained results showed that the proposed combined strategy provided better nanoparticle size estimations than the statistical approach alone. In the absence of publicly available databases, data generation using experimental design and machine learning, as proposed here, could be a faster, lower-cost, and greener solution. Our proposed method can be applied to a wide range of biotechnology and bioengineering applications with significant advanced knowledge.

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.

Carbonation resistance of recycled fine aggregate concrete reinforced by calcium sulfate whiskers

This study investigates mechanical and carbonation resistance tests conducted on Recycled Fine Aggregate Concrete (RFAC) with three different Recycled Fine Aggregate (RFA) replacement rates and four calcium sulfate whisker (CSW) contents, and investigates the effects of RFA replacement rate and CSW content on the mechanical properties of RFAC and the carbonation resistance of RFAC with calcium sulfate whiskers at different ages of carbonation. The microstructure of CSW-modified RFAC was observed and analyzed by SEM electron microscopy to explore further enhancement and toughening mechanisms of CSW in RFAC, as well as the optimal content of CSW in RFAC. In addition, the carbonation depth of recycled concrete with calcium sulfate beard is modeled based on a quadratic fitting formulation. The test results show that the mechanical properties and carbonation resistance of RFAC would decrease with the increase of the RFA replacement rate. Incorporating CSW can effectively improve the mechanical properties and carbonation resistance of RFAC. At 60 % RFA substitution, CSW at 2 % content has the most significant effect on the mechanical properties and carbonation resistance of RFAC. Microscopic analysis shows that CSW has a bridging effect at the micro-cracks in the concrete matrix, preventing the creation and expansion of micro-cracks in the concrete matrix, thus improving RFAC mechanical properties of RFAC and resistance to carbonation. The predicted results of the carbonation depth model which are constructed from the test data coincide with the results. We can use them to predict and analyze the carbonation depth of the carbonation resistance of recycled concrete.

Carbon Footprint of the Food Industry and ESG-Related Business Promotion

The carbon emissions of the food industry account for a significant proportion of the overall carbon emissions. Figuring out the carbon footprint of the food industry is a great way to find out the reason for the large amount of carbon emissions. In this part, the paper selected several variations that may have some influence on carbon emissions and did some intensive studies to figure out which one is the major factor. By analyzing through life cycle assessment and regression by Random Forest, it was figured that fertilizer is the most influential factor in carbon emissions, and focusing more on the usage of compound fertilizer and decreasing the usage of other kinds of fertilizer can effectively reduce the carbon emissions. Then the paper tried to figure out whether the reduction is realistic, and the paper did research in the financial field. The paper did an ESG score ranking of the fertilizer company, cultivated the investment weight in the portfolio, and did a risk rating of them. The research adopted the Grey Forecasting Model and quadratic Optimization Formulation to process data. Through the simple study of the carbon footprint of fertilizers and an analysis of the investment ratio of investors, it is finally found that to economically promote and support the development of carbon-reducing compound fertilizers by business is essential. Besides, it coincides with investors’ interests.

State estimators for discrete‐time descriptor linear systems with mixed uncertainties and state constraints

AbstractThis article presents two novel mixed‐uncertainty state estimators for discrete‐time descriptor linear systems, namely linear time‐varying mixed‐uncertainty filter (LTVMF) and linear time‐invariant mixed‐uncertainty filter (LTIMF). The former is based on the minimum‐variance approach, from which quadratic and explicit (special case of quadratic) formulations are derived and addressed to LTV systems. Both formulations incorporate the knowledge of state linear constraints, such as equality (in the descriptor form) and inequality, to mitigate precision and accuracy issues related to initialization and evolution of the state estimates. The LTIMF algorithm is based on the mixed criterion and addressed to LTI systems, with the low‐cost computation being its motivation. Both LTVMF and LTIMF algorithms solve state‐estimation problems in which the uncertainties are combined to yield the so‐called mixed‐uncertainty vector, which is composed by set‐bounded uncertainties, characterized by constrained zonotopes, and stochastic uncertainties, characterized by Gaussian random vectors. As mixed‐uncertainty vectors imply biobjective optimization problems, we innovatively present multiobjective arguments to justify the choice of the solution on the Pareto‐optimal front. In order to discuss the advantages and drawbacks, the state estimators are tested in two numerical examples.

QuadWave1D: An optimized quadratic formulation for spectral prediction of coastal waves

Spectral information of coastal waves and the associated statistical parameters (e.g., the significant wave height and mean wave period) over large spatial scales is essential for many applications (e.g., coastal safety assessments, coastal management and developments, etc.). This demand explains the necessity for accurate yet effective models. A well-known efficient modelling approach is the quadratic approach (often referred to as frequency-domain models, weakly nonlinear mild-slope models, amplitude models, etc.). The efficiency of this approach is achieved through modelling reduction of the original governing equations (e.g., Euler equations). Most significantly, wave nonlinearity is described solely by a single quadratic mode-coupling term. Therefore, doubts arise with regard to the predictive capabilities of the quadratic approach to reliably describe the nonlinear development of waves in the coastal environment where nonlinearity is typically significant. This study attempts to push the limit of the prediction capabilities of nonlinear coastal waves based on the quadratic approach. To this end, an optimization process is proposed, striving to extract the quadratic formulation which describes most adequately nonlinear wave developments over water depths and bathymetrical structures which characterize the coastal environment. The outcome is the model QuadWave1D: a fully dispersive quadratic model for coastal wave prediction in one-dimension. Based on a wide set of examples (including monochromatic, bichromatic and irregular wave conditions) and comparing to other representative quadratic formulations, it is found that QuadWave1D presents superior predictive capabilities of both the sea-swell components and the infragravity field.

A quadratic formulation for the optimal allocation of fault indicators

Faults are an inherent part of power distribution systems, and fault indicators play a crucial role in self-healing smart distribution systems by assisting in fault detection and location. However, the effectiveness of fault indicators depends on their number and location in the distribution system. This paper addresses the optimal decisions on the allocation of fault indicators in distribution networks to minimize the expected fault location time.To build upon the existing knowledge, a comprehensive review of the literature on the allocation of fault indicators is provided. Additionally, a novel approach is given by formulating the fault location time as a quadratic function of the network’s section sizes. From a relationship established between the fault indicator allocation problem and connected edge-partitions, a mathematical model and a metaheuristic are developed.To evaluate the proposed methodologies, a benchmark comprising 19 networks is employed, including the largest network solved for this problem to date. The results demonstrate the superiority of the new methodologies over previous approaches, as evidenced by the quality of the solutions and their optimality gaps.

The impact of modulational instability on coastal wave forecasting using quadratic models

Coastal wave forecasting over large spatial scales is essential for many applications (e.g., coastal safety assessments, coastal management and developments, etc.). This demand explains the necessity for accurate yet effective models. A well-known efficient modelling approach is the quadratic approach (often referred to as frequency-domain models, nonlinear mild-slope models, amplitude models, etc.). The efficiency of this approach stems from a significant modelling reduction of the original governing equations (e.g., Euler equations). Most significantly, the description of wave nonlinearity essentially collapses into a single mode coupling term determined by the quadratic interaction coefficients. As a result, it is expected that the efficiency achieved by the quadratic approach is accompanied by a decrease in prediction accuracy. In order to gain further insight into the predictive capabilities of this modelling approach, this study examines six different quadratic formulations, three of which are of the Boussinesq type and the other three are referred to as fully dispersive. It is found that while the Boussinesq formulations reliably predict the evolution of coastal waves, the predictions by the fully dispersive formulations tend to be affected by false developments of modulational instability. Consequently, the predicted wave fields by the fully dispersive formulations are characterized by unexpectedly strong modulations of the sea-swell part and associated unexpected infragravity response. The impact of the modulational instability on wave prediction based on the quadratic approach is further demonstrated using existing laboratory results of bichromatic and irregular wave conditions.

Sailing league problems

AbstractWe describe a class of combinatorial design problems which typically occur in professional sailing league competitions. We discuss connections to resolvable block designs and equitable coverings and to scheduling problems in operations research. We in particular give suitable boolean quadratic and integer linear optimization problem formulations, as well as further heuristics and restrictions, that can be used to solve sailing league problems in practice. We apply those techniques to three case studies obtained from real sailing leagues and compare the results with previously used tournament plans.

Exact mixed-integer quadratic formulation and solution for large-scale thermal unit commitment

Abstract Thermal unit commitment (UC) is a nonlinear combinatorial optimization problem that minimizes total operating costs while considering system load balance, on/off restrictions and other constraints. Successfully solving the thermal UC problem contributes to a more reliable power system and reduces thermal costs. This paper presents an exact mixed-integer quadratic programming (EMIQP) method for large-scale thermal UC problems. EMIQP revolutionizes the landscape by seamlessly translating the intricate nonlinear combinatorial optimization problem of UC into an exact mixed-integer quadratic formulation. This approach also elegantly reimagines on/off constraints as mixed-integer linear equations, employing both the sum and respective approaches. Our case studies unequivocally demonstrate the exceptional prowess of the EMIQP method, consistently securing the global optimum. Moreover, the mathematical-based EMIQP method produces identical results at each run, which is extremely important for UC in the real world.

Stationary and Non-Stationary Periodic Flows Mathematical Modelling using Various Vortex Viscosity Models

Introduction. Mathematical modelling of currents is an urgent research topic in the field of hydrodynamics and oceanography. Despite ongoing research in the field of developing accurate and efficient numerical methods for solving Navier-Stokes equations that take into account vortex viscosity, the problems of accurate prediction and control of turbulence remain unresolved. The influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions also remains relevant. The aim of the study is to study the influence of linearized and quadratic bottom friction and two turbulence models on the numerical solution of stationary and non-stationary periodic flows. Special emphasis is placed on comparing numerical results with analytical solutions within the framework of using various models of bottom friction.Materials and Methods. The computational models used in this study are based on a simplified two-dimensional wave model and full three-dimensional Navier-Stokes equations. The classical model of shallow water motion and the 2D model without taking into account dynamic changes in the geometry of the reservoir surface are derived from a system of equations for a spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics of a shallow reservoir. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear effects – nonlinearities caused by higher-order terms in the equations of motion, i. e. terms of advective acceleration and friction, and nonlinear effects caused by geometric nonlinearities, this is due, for example, to different water depths and reservoir widths, which will be important when modelling a real sea.Results. The results of modeling stationary and non-stationary periodic flows in a schematized rectangular basin using linearized bottom friction are presented. The influence of linearization on the numerical solution is investigated in comparison with analytical profiles using models calculating bottom friction in a quadratic formulation. In combination with quadratic bottom friction, two turbulence models are studied: the constant vortex viscosity and the Prandtl mixing length model. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modeling and analytical solutions averaged in depth.Discussion and Conclusion. New approaches to modelling and studying flows with variable vortex viscosity are proposed, including analysis of the influence of linearization and the use of various turbulence models. For the linearized and quadratic formulations of bottom friction, it is proved that the numerical results for the case of stationary flow show great similarity with analytical solutions, since the surface height is much less than the water depth and advection can be neglected. The numerical results for the unsteady flow also show a good agreement with the theory. Unlike analytical solutions, numerical modelling has minor deviations in the long run. The study of flows, within the framework of using various turbulence models, will make it possible to take into account the influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions. The results obtained make it possible to better understand and describe the physical processes occurring in shallow waters. This opens up new possibilities for applying mathematical modelling to predict and analyze the impact of human activities on the marine environment and to solve other problems in the field of oceanology and geophysics.

A Quadratic Formulation of ESS Degradation and Optimal DC Microgrid Operation Strategy Using Quadratic Programming

A Quadratic Formulation of ESS Degradation and Optimal DC Microgrid Operation Strategy Using Quadratic Programming

Formulating a quadratic frequency nadir constraint in unit commitment based on energy conservation law

This paper proposes a quadratic formulation for the frequency nadir constraint (FNC) based on the energy conservation law. Incorporating the formulation into the unit commitment (UC) model, a mixed-integer quadratically constrained programming (MIQCP) model for frequency nadir constrained UC can be obtained, which can be solved efficiently by the MIQCP solver. The distinct feature of the formulation is that it considers the inertia difference among generators, which has a significant effect on the on/off schedule of generators. Besides, the formulation has a clear physical meaning and does not have to be linearized. The test on the WSCC 3-machine 9-bus benchmark system validates the accuracy of the formulation, and the case studies on a New England 10-machine 39-bus system confirm the effectiveness of the proposed formulation.

The price-elastic knapsack problem

This paper introduces the price-elastic knapsack problem (PEKP), an extension of the classic knapsack problem where instead of fixed item characteristics, the weight of each item and the profit from including an item in the knapsack are a function of a parameter, namely the price. PEKP is first formulated as a generic nonlinear optimization problem and three special cases are investigated. First, we show a polynomial-time solvable case. Next, we formulate the case where the item weights are affine-linear functions as a quadratic program. The computational results show that solving the quadratic program to optimality is computationally challenging, and thus, an approach that decomposes the problem into three mixed integer programs is proposed. Similarly, the case where the weights of the items are piecewise-linear functions is investigated and a quadratic formulation is proposed. A solution approach based on decomposing the problem into three mixed integer programs that are solved independently is also proposed. Using randomly generated instances of varying sizes, the computational results show that the proposed decomposition leads to significant computational advantages compared to solving the quadratic program in the cases of affine-linear and piecewise-linear functions.

3D Electromagnetic Forward Modeling in Anisotropic Medium Using the Secondary Field Formula

The electromagnetic (EM) method is an essential geophysical method to detect underground geological structures. With the increasing requirements for the reliability of data interpretation, we need an accurate anisotropic forward algorithm, which is applied several times in the inversion. At the same time, modeling the anisotropic response is an advantageous method for designing actual survey lines. In this paper, we utilize the quadratic field formulation in anisotropic media to derive the governing equations for electromagnetic fields. The governing equation is discretized using the finite difference method. Experimental results indicate that our algorithm is able to accurately model both the magnetotelluric (MT) and the controlled-source electromagnetic (CSEM) response.