Multiple Integer Research Articles

Optimizing a linear function over the efficient set of a multiple objective integer quadratic program

In this paper, we investigate the optimization of a function over the efficient set of a Multiple Objective Integer Quadratic Program (MOIQP). The method described employs a branch-and-cut approach to address the problem. At every node within the search tree, a profound cut is derived by considering the ascending trends of the criteria. This cut enables the removal of branches where the associated sub-problems don’t contain efficient solutions. The proposed methodology is illustrated in an example. Extensive numerical experiments are conducted to demonstrate the method’s effectiveness against state-of-the-art methods.

An Analysis of Anesthesia for C - Section Patients Using Neutrosophic Approach

This paper vividly describes the multiple object linear integer programming problem, using imprecision method which is applied to solve the mundane life problems, diagnosis and survey-based studies-with the help of neutrosophic fuzzy environment. The empirical intention of the article is to single out the most apt anaesthesia for c- section out of three categorized anaesthesia namely general, regional and local that fulfils the nominal four constraints such as preference of patients, safety, duration of pain tolerance and cost effective. The objectives and constraints to define a problem are created using the data surveyed from authentic anaesthesiologist’s and derived in the Neutrosophic number form.

Strongly coupled edge states in a graphene quantum Hall interferometer

Electronic interferometers using the chiral, one-dimensional (1D) edge channels of the quantum Hall effect (QHE) can demonstrate a wealth of fundamental phenomena. The recent observation of phase jumps in a Fabry-Pérot (FP) interferometer revealed anyonic quasiparticle exchange statistics in the fractional QHE. When multiple integer edge channels are involved, FP interferometers have exhibited anomalous Aharonov-Bohm (AB) interference frequency doubling, suggesting putative pairing of electrons into 2e\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\boldsymbol{2}}}{{\\boldsymbol{e}}}$$\\end{document} quasiparticles. Here, we use a highly tunable graphene-based QHE FP interferometer to observe the connection between interference phase jumps and AB frequency doubling, unveiling how strong repulsive interaction between edge channels leads to the apparent pairing phenomena. By tuning electron density in-situ from filling factor ν<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\boldsymbol{\ u }}} \\, < \\, {{\\boldsymbol{2}}}$$\\end{document} to ν>7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\boldsymbol{\ u }}} \\, > \\, {{\\boldsymbol{7}}}$$\\end{document}, we tune the interaction strength and observe periodic interference phase jumps leading to AB frequency doubling. Our observations demonstrate that the combination of repulsive interaction between the spin-split ν=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\boldsymbol{\ u }}}={{\\boldsymbol{2}}}$$\\end{document} edge channels and charge quantization is sufficient to explain the frequency doubling, through a near-perfect charge screening between the localized and extended edge channels. Our results show that interferometers are sensitive probes of microscopic interactions and enable future experiments studying correlated electrons in 1D channels using density-tunable graphene.

Multiple integer candidates ambiguity resolution: a unification ambiguity resolution algorithm

Among all the ambiguity resolution techniques, the Full Ambiguity Resolution (FAR), Partial Ambiguity Resolution (PAR) and Best Integer Equivariant (BIE) estimator are widely used. Although the researches have been done on the different classes of ambiguity resolution, we still hope to find the relationships among these specific algorithms. In this work, we unify the PAR and FAR algorithms under a whole framework of BIE by applying multiple integer candidates. A concise estimation formula of the variance of Gaussian BIE estimator based on the variance of float solution and the probability distribution of the candidates is first derived. Then, we propose an algorithm named Multiple Integer Candidates Ambiguity Resolution (MICAR) to discover as many ambiguities in the BIE as possible that can be estimated more precisely by PAR (FAR) algorithm instead of BIE. In the experiments, we utilize the simulated data of GPS (Global Positioning System) + BDS (BeiDou Navigation Satellite System) + Galileo (Galileo navigation satellite system) to contrast the effects of MICAR and single candidate estimator, i.e., FAR. By taking the threshold of 5 cm at 95% confidence level as an example, MICAR accelerates the convergence process by about 3.0 min. When the positioning sequence converges, MICAR reduces the root mean square of the positioning error by 9.8% in horizontal directions and 3.5% in vertical direction, which is attributed to more fixed NL.

Dynamic feedback algorithm based on spatial corner fitness for solving the three-dimensional multiple bin-size bin packing problem

To improve cargo loading efficiency and achieve diverse needs of companies for the loading process, this paper innovatively establishes a multiple objective mixed integer programming model for the three-dimensional multiple bin-size bin packing problem (3D-MBSBPP). The model is designed to maximize container space utilization rate and cargo load balance, minimize container usage costs, and incorporates some practical constraints. On this basis, we propose a novel dynamic feedback algorithm based on spatial corner fitness (SCF_DFA) to solve this model, which consists of three stages. Specifically, Stage 1 employs a heuristic algorithm based on spatial corner fitness to optimize the search of the remaining spaces. Stage 2 employs a container type selection algorithm to dynamically adjust and optimize container types. Stage 3 uses an improved genetic algorithm to improve the quality of the solutions of the first two stages. We demonstrate the effectiveness of the proposed algorithm through comparative experiments on benchmark instances, and apply it to solve the real-life instances for the 3D-MBSBPP. The results show that the proposed algorithm can make the average container space utilization rate reach 85.38%, which is 1.48% higher than that of baseline method, while the loading results obtained are more balanced, indicating the advantages of the SCF_DFA in solving 3D-MBSBPP. Furthermore, we conduct ablation experiments to confirm the effectiveness of each component within the algorithm.

Mordell–Tornheim Zeta Values, Their Alternating Version, and Their Finite Analogs

The purpose of this paper is two-fold. First, we consider the classical Mordell–Tornheim zeta values and their alternating version. It is well-known that these values can be expressed as rational linear combinations of multiple zeta values (MZVs) and the alternating MZVs, respectively. We show that, however, the spaces generated by the Mordell–Tornheim zeta values over the rational numbers are in general much smaller than the MZV space and the alternating MZV space, respectively, which disproves a conjecture of Bachmann, Takeyama and Tasaka. Second, we study supercongruences of some finite sums of multiple integer variables. This kind of congruences is a variation of the so called finite multiple zeta values when the moduli are primes instead of prime powers. In general, these objects can be transformed to finite analogs of the Mordell–Tornheim sums which can be reduced to multiple harmonic sums. This approach not only simplifies the proof of a few previous results but also generalizes some of them. At the end of the paper, we provide a general conjecture involving this type of sums, which is supported by strong numerical evidence.

Multiple electron pumping

The need to pump single electrons with a high degree of accuracy and fidelity has led to the development of a range of different pump and turnstile designs. Previous pumping mechanisms have all demonstrated that pumping more than one electron per cycle degrades the quantisation of the measured current. This unreliable delivery of multiple electrons per cycle has limited the use of on-demand single electron sources in electron quantum optic experiments. We present highly quantised current with multiple electrons pumped per cycle. We experimentally demonstrate that in our pumps an increase in electron throughput per cycle does not lead to an appreciable degradation in the accuracy of the produced current.Our pump is realised in an aluminium gallium arsenide two-dimensional electron gas, where electrons are pumped through a one-dimensional split-gate confinement potential under the influence of an applied source-drain voltage V_{text{SD}}, and where the pump is driven by a trapezoidal arbitrary waveform. This combination of a split-gate potential, V_{text{SD}} bias and trapezoidal wave form has led to the observation of robust quantised plateaus where not just a single electron, but a multiple integer number of electrons are pumped per cycle with a high degree of robustness and without the need of a magnetic field. For seven electrons per cycle, we report an increase of over two orders of magnitude in pumping accuracy from 2.72 times 10^{-2} in devices operating in the conventional pumping regime, to 1.64 times 10^{-4} in pumps operating in what we call the long plateau regime, a regime accessed under a change in a split-gate pumps applied V_{text{SD}} voltage. This pump will find direct use in quantum transport measurements where the metrological accuracy of single electrons pumped per cycle is not required and the low throughput per cycle of electrons is limiting.

A branch‐and‐price algorithm for identical parallel machine scheduling with multiple milestones

AbstractThis article considers an identical parallel‐machine task scheduling problem motivated by operations management of online services. A task with an integer processing time can be split into sub‐tasks with integer processing times. Each task has multiple integer milestones and at each milestone a nonnegative penalty will occur. The penalty value of a task at a milestone is a convex nonincreasing function of the completed amount by this milestone. Our objective is to determine a feasible schedule for all the tasks on given identical parallel machines, such that the sum of all tasks' total penalty at all milestones is minimized. We prove the NP‐hardness of this problem in the ordinary sense and develop a branch‐and‐price algorithm. Computational experiments utilizing data from an online service operations survey show that this algorithm is singularly efficient and promising.

DRL-Based Secure Aggregation and Resource Orchestration in MEC-Enabled Hierarchical Federated Learning

Federated learning (FL) provides a new paradigm for protecting data privacy by enabling model training at devices and model aggregation at servers. However, data information may be leaked to honest-but-curious aggregation servers by updated model parameters. The existing secure methods do not fully exploit the potentiality of data characteristics in enhancing security, which makes it impossible to optimize limited system resources overall to achieve secure, fair and efficient FL systems. In this paper, a DRL-based joint secure aggregation and resource orchestration scheme is proposed to guarantee security and fairness, and improve efficiency for hierarchical FL assisted by untrusted mobile-edge computing (MEC) servers. We formulate a joint optimization problem of data size, payment and resource orchestration, to maximize the long-term social welfare subject to secure aggregation and limited resources. Since the formulated problem is a complex mixed integer dynamic optimization problem with NP-hardness, where multiple mixed integer optimization variables are highly coupled in time-varying constraints and objective function, it is difficult to obtain its optimal solution via traditional optimization methods. Thus, we propose a hierarchical reward function-based DRL algorithm (MATD3) to guide the agents to approach the optimal policy of secure aggregation and resource orchestration. Simulation results show that the proposed algorithm MATD3 can achieve superior performance over comparison algorithms and the MEC-enabled HFL framework outperforms two-layer FL frameworks.

On relaxations of the max k-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.

Magnetic Field-Stabilized Wigner Crystal States in a Graphene Moiré Superlattice.

ABC-stacked trilayer graphene on boron nitride (ABC-TLG/hBN) moiré superlattices provides a tunable platform for exploring Wigner crystal states in which the electron correlation can be controlled by electric and magnetic fields. Here we report the observation of magnetic field-stabilized Wigner crystal states in a ABC-TLG/hBN. We show that correlated insulating states emerge at multiple fractional and integer fillings corresponding to ν = 1/3, 2/3, 1, 4/3, 5/3, and 2 electrons per moiré lattice site under a magnetic field. These correlated insulating states can be attributed to generalized Mott states for the integer fillings and generalized Wigner crystal states for the fractional fillings. The generalized Wigner crystal states are stabilized by a vertical magnetic field and are strongest at one magnetic flux quantum per three moiré superlattices. The ν = 2 insulating state persists up to 30 T, which can be described by a Mott-Hofstadter transition at a high magnetic field.

Tunable spin and valley excitations of correlated insulators in Γ-valley moiré bands.

Moiré superlattices formed from transition metal dichalcogenides support a variety of quantum electronic phases that are highly tunable using applied electromagnetic fields. While the valley degree of freedom affects optoelectronic properties in the constituent transition metal dichalcogenides, it has yet to be fully explored in moiré systems. Here we establish twisted double-bilayer WSe2 as an experimental platform to study electronic correlations within Γ-valley moiré bands. Through local and global electronic compressibility measurements, we identify charge-ordered phases at multiple integer and fractional moiré fillings. By measuring the magnetic field dependence of their energy gaps and the chemical potential upon doping, we reveal spin-polarized ground states with spin-polaron quasiparticle excitations. In addition, an applied displacement field induces a metal-insulator transition driven by tuning between Γ- and K-valley moiré bands. Our results demonstrate control over the spin and valley character of the correlated ground and excited states in this system.

Learn Branching Policies with an Attention-based Model

The branch-and-bound method is an exact tree search method in mixed integer linear programming. Recently, learning branching policies become a popular topic, but most existing machine learning methods only consider problems with similar structures, and pay little attention to heterogeneous problems. The parameterized search tree method demonstrates promising performance on heterogeneous problems, but there is still room for improvement when it generalizes to other problems. In this paper, we propose an attention-based branching framework, which can exploit information among candidate variables. In this way, we can take full advantage of the properties exploited from the parameterized branch-and-bound search tree, resulting in a better branching policy. Our proposed model is evaluated on multiple mixed integer linear programming instances and attains remarkable performance within a given time limit.

Using multiple reference vectors and objective scaling in the Feasibility Pump

The Feasibility Pump (FP) is one of the best-known primal heuristics for mixed-integer programming (MIP): more than 15 papers suggested various modifications of all of its steps. So far, no variant considered information across multiple iterations, but all instead maintained the principle to optimize towards a single reference integer point. In this paper, we evaluate the usage of multiple reference vectors in all stages of the FP algorithm. In particular, we use LP-feasible vectors obtained during the main loop to tighten the variable domains before entering the computationally expensive enumeration stage, a procedure we refer to as mRENS. Moreover, we consider multiple integer reference vectors to explore further optimizing directions and introduce alternative objective scaling terms to balance the contributions of the distance functions and the original MIP objective.Our computational experiments demonstrate that the new method can improve performance on general MIP test sets. In detail, our modifications provide a 29.3% solution quality improvement and 4.0% running time improvement in an embedded setting, needing 16.0% fewer iterations over a large test set of MIP instances. In addition, the method's success rate increases considerably within the first few iterations. In a standalone setting, we also observe a moderate performance improvement, which makes our version of FP suitable for the two main use-cases of the algorithm.

Nonlinear Hybrid Model Predictive Control for building energy systems

This paper presents a nonlinear hybrid Model Predictive Control (MPC) approach for building energy systems based on Modelica. The MPC approach takes into account two characteristics that are very common for building energy systems: nonlinearities (inherent in the building envelope and Heating, Ventilation and Air Conditioning (HVAC) systems) and discontinuities (in the form of on/ off operation, discrete operation states and operation modes). The hybrid MPC approach integrates both continuous and discrete optimization variables into the control concept and thus is capable of controlling building energy systems with binary or integer decision variables, switching dynamics or logic if-then-else constraints. By employing a time-variant linearization approach, nonlinear Modelica optimization problems are approximated with high accuracy and transformed into a linearized state-space representation. Based on the linearization output, a linearized optimization problem is generated automatically in every MPC iteration, which is extensible by various integer characteristics and is accessible for a wide range of mixed-integer solvers. A simulation study on a nonlinear Modelica building energy system demonstrates the control quality of the proposed toolchain revealing a small linearization error and successful integration of multiple integer characteristics. The benefits of the approach are manifested by comparing its performance with different reference control strategies.

Optimal Deployment of Flood Emergency Response Materials under Stochastic Demands

This study aims to develop a planning model for the deployment of flood emergency response materials under stochastic demands. For ease of modeling, a deterministic demand model is first proposed, from which the stochastic model is further extended. These models are used to determine the allocation of materials the transport truck routes and distribution of the stockpiled materials to emergency sites. The models are formulated as multiple integer network flow problems with special side constraints which are characterized as NP-hard in terms of optimization. To solve the complicated stochastic model efficiently, we also develop a solution algorithm based on a problem decomposition technique coupled with a variable fixing technique. Finally, to preliminarily evaluate the proposed models and solution algorithm, we perform numerical tests using data related to a Taiwan flood emergency response system. The test results are good, indicating the potential usefulness of the models and solution algorithm in practice.

Geometric progression and relative strength index applied to FX hedging

This research aims at profit maximization and loss minimization in any FX market trading. A geometric progression (geometric sequence), is said to be a non-zero number progression or sequence in which each term following the first is obtained via multiplying the prior by the common ratio, which is a predetermined non-zero value. This method seeks to open an opposite position to an existing initial position in order to hedge that initial position in the event that the market moves against our trade. There are a number of mathematical models to develop new hedging strategies for Forex trading. Due to the apparent high level of unpredictability in price movement, forecasting the future of a stock price is a challenging endeavor. The research would like to study one with geometric progression in particular. In other words, every number can be entered as multiple integer of a different number. Thus, 1, 2, 4, 8,..., 2n is a step forward. Smart traders never take more risks than their capital, however, the otherwise is viewed and considered here.

Planning strategy of fast-charging stations in coupled transportation and distribution systems considering human health impact

The proliferation of the electric vehicles field brings attention to fast-charging station planning. Recently, some studies found electric vehicles that receive fast-charging service may also bring much air pollution due to fossil-fuel fired power generation. Hence, depending on the power generation mix, electric vehicles may have adverse environmental impacts. The current studies of fast-charging station planning only focus on power system planning and often ignore other parts of the transportation system such as internal combustion engine vehicles and gasoline stations. Moreover, most the current studies lack a comprehensive planning strategy for the fast-charging station that involves elements above simultaneously. In this context, an interdisciplinary study is implemented on the coordinated planning strategy of the fast-charging station in coupled transportation and power distribution systems. The proposed strategy with a multiple objective integer model can measure the impact of air pollutions from electric vehicles on air quality and human health in different areas. Considering the traffic flow assignment, the driving range logic constraints are used. The branch flow model is applied to power distribution systems. To measure the impacts of air pollutions on air quality and human health, the air pollution from the Road traffic model, the Gaussian Plum Model and the C-R function are adopted. Finally, numerical experiments are conducted to verify the proposed strategy and the mathematical model. The fast-charging station location and size can be determined by the mathematical model. The transition of gasoline station to fast-charging station is suggested according to the model as well.

A multi-objective optimization strategy of steam power system to achieve standard emission and optimal economic by NSGA-Ⅱ

In this work, the models of desulfurization and denitrification are added to solve the problem of SO2 and NOX standardized discharge. The design and optimization strategy of steam power system (SPS) considering contaminant emissions reduction technology is proposed to achieve the trade-off between economic and environmental goals. Detailed superstructure networks of desulfurization based on wet limestone flue gas desulfurization and denitrification based on selective catalytic reduction were established and embedded in the SPS model. Then, based on this combined superstructure model, a mathematical formulation of multiple objective mixed integer nonlinear programming describing the SPS coupled with desulfurization and denitrification was established. The steam flow rate, outlet enthalpy, the consumption of the turbine power of the direct drive equipment and the electricity generated by the turbine, the flow rate and efficiency of desulfurization and denitrification are chosen as the optimization variables. The operating conditions and equipment parameters of the global system are optimized. Finally, the second-generation non-dominated sorting genetic algorithm (NSGA-II) was applied to obtain the Pareto optimization curve, exploring trade-offs between economic and environmental goals. Two case studies are used to assess the applicability and performance of the optimization formulation and solution algorithm.

Design and implementation of parallel MSD integer divider in ternary optical computer

Division is one of the four basic operations. It is a very important operation in the field of numerical calculation, such as numerical calculation, large integer decomposition and so on. How to divide quickly has been a hot issue in electronic computers, embedded systems and other new computing systems. It is of great significance to give full play to the advantages of ternary optical computer processors, such as a large number of bits, reconfigurable operation functions, bit allocation, etc. by designing the efficient parallel MSD number divider to improve the division operation efficiency of large data and promote the application of ternary optical computer in the field of numerical calculation. In this paper, the symbol decision algorithm for MSD numbers is proposed first time, and based on the SRT algorithm, the scheme for implementing a parallel division of multiple MSD integers using a parallel carry free SJ-MSD adder and an MSD comparator, which is called as parallel MSD integer division, is designed first time. For the corresponding parallel MSD integer divider, the machine cycles required to run multiple MSD integer division in parallel is the same as the machine cycles required for the operation of a single MSD integer division. Experiments show that the parallel SRT integer division scheme is feasible. It will effectively improve the efficiency of large data processing and accelerate TOC to enter practical applications such as numerical computation.