This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussio...This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussion on the convergence rates of regular solutions.展开更多
It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth for ill-posed problems,so fractional Tikhonov methods have been introduced to remedy this shortcom...It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth for ill-posed problems,so fractional Tikhonov methods have been introduced to remedy this shortcoming.And Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining apartial Arnoldi decomposition of the given matrix.In this paper,we propose a new method to compute an approximate solution of large scale linear discrete ill-posed problems which applies projection fractional Tikhonov regularization in Krylov subspace via Arnoldi process.The projection fractional Tikhonov regularization combines the fractional matrices and orthogonal projection operators.A suitable value of the regularization parameter is determined by the discrepancy principle.Numerical examples with application to image restoration are carried out to examine that the performance of the method.展开更多
Tikhonov regularization is a powerful tool for solving linear discrete ill-posed problems.However,effective methods for dealing with large-scale ill-posed problems are still lacking.The Kaczmarz method is an effective...Tikhonov regularization is a powerful tool for solving linear discrete ill-posed problems.However,effective methods for dealing with large-scale ill-posed problems are still lacking.The Kaczmarz method is an effective iterative projection algorithm for solving large linear equations due to its simplicity.We propose a regularized randomized extended Kaczmarz(RREK)algorithm for solving large discrete ill-posed problems via combining the Tikhonov regularization and the randomized Kaczmarz method.The convergence of the algorithm is proved.Numerical experiments illustrate that the proposed algorithm has higher accuracy and better image restoration quality compared with the existing randomized extended Kaczmarz(REK)method.展开更多
The generalized Tikhonov regularization method is one of the most classical methods for the solution of linear systems of equations that arise from the discretization of linear ill-posed problems.However,the approxima...The generalized Tikhonov regularization method is one of the most classical methods for the solution of linear systems of equations that arise from the discretization of linear ill-posed problems.However,the approximate solution obtained by the Tikhonov regularization method in general form may lack many details of the exact solution.Combining the fractional Tikhonov method with the preconditioned technique,and using the discrepancy principle for determining the regularization parameter,we present a preconditioned projected fractional Tikhonov regularization method for solving discrete ill-posed problems.Numerical experiments illustrate that the proposed algorithm has higher accuracy compared with the existing classical regularization methods.展开更多
We build a computer program to reconstruct convex bodies using even L_(p)surface area measures for p≥1.Firstly,we transform the minimization problem Pi,which is equivalent to solving the even L_(p)Minkowski problem,i...We build a computer program to reconstruct convex bodies using even L_(p)surface area measures for p≥1.Firstly,we transform the minimization problem Pi,which is equivalent to solving the even L_(p)Minkowski problem,into a convex optimization problem P4 with a finite number of constraints.This transformation makes it suitable for computational resolution.Then,we prove that the approximate solutions obtained by solving the problem P4 converge to the theoretical solution when N and k are sufficiently large.Finally,based on the convex optimization problem P_(4),we provide an algorithm for reconstructing convex bodies from even L_(p)surface area measures,and present several examples implemented using MATLAB.展开更多
In order to improve the reconstruction performance for ill-posed emission tomographic problems with limited projections, a generalized interpolation method is proposed in this paper, in which the virtual lines of proj...In order to improve the reconstruction performance for ill-posed emission tomographic problems with limited projections, a generalized interpolation method is proposed in this paper, in which the virtual lines of projection are fabricated from, but not linearly dependent on, the measured projections. The method is called the virtual projection（VP） method.Also, an iterative correction method for the integral lengths is proposed to reduce the error brought about by the virtual lines of projection. The combination of the two methods is called the iterative virtual projection（IVP） method. Based on a scheme of equilateral triangle plane meshes and a six asymmetrically arranged detection system, numerical simulations and experimental verification are conducted. Simulation results obtained by using a non-negative linear least squares method,without any other constraints or regularization, demonstrate that the VP method can gradually reduce the reconstruction error and converges to the desired one by fabricating additional effective projections. When the mean square deviation of normal error superimposed on the simulated measured projections is smaller than 0.03, i.e., the signal-to-noise ratio（SNR）for the measured projections is higher than 30.4, the IVP method can further reduce the reconstruction error reached by the VP method apparently. In addition, as the regularization matrix in the Tikhonov regularization method is updated by an iterative correction process similar to the IVP method presented in this paper, or the Tikhonov regularization method is used in the IVP method, good improvement is achieved.展开更多
This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contain...This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibilky method and regularized semigroups. Finally, an example is given.展开更多
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 c...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.展开更多
Purpose:A text generation based multidisciplinary problem identification method is proposed,which does not rely on a large amount of data annotation.Design/methodology/approach:The proposed method first identifies the...Purpose:A text generation based multidisciplinary problem identification method is proposed,which does not rely on a large amount of data annotation.Design/methodology/approach:The proposed method first identifies the research objective types and disciplinary labels of papers using a text classification technique;second,it generates abstractive titles for each paper based on abstract and research objective types using a generative pre-trained language model;third,it extracts problem phrases from generated titles according to regular expression rules;fourth,it creates problem relation networks and identifies the same problems by exploiting a weighted community detection algorithm;finally,it identifies multidisciplinary problems based on the disciplinary labels of papers.Findings:Experiments in the“Carbon Peaking and Carbon Neutrality”field show that the proposed method can effectively identify multidisciplinary research problems.The disciplinary distribution of the identified problems is consistent with our understanding of multidisciplinary collaboration in the field.Research limitations:It is necessary to use the proposed method in other multidisciplinary fields to validate its effectiveness.Practical implications:Multidisciplinary problem identification helps to gather multidisciplinary forces to solve complex real-world problems for the governments,fund valuable multidisciplinary problems for research management authorities,and borrow ideas from other disciplines for researchers.Originality/value:This approach proposes a novel multidisciplinary problem identification method based on text generation,which identifies multidisciplinary problems based on generative abstractive titles of papers without data annotation required by standard sequence labeling techniques.展开更多
A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gr...A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption,which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method.Under some additional conditions,the method presented has a superlinear convergence rate,which can be regarded as an extension and supplement of BFGS-type methods with the projection technique.Finally,the effectiveness and application prospects of the proposed method are verified by numerical experiments.展开更多
This paper studies the non-homogeneous generalized Riemann-Hilbert(RH)problems involving two unknown functions.Using the uniformization theorem,such problems are transformed into the case of homogeneous type.By the th...This paper studies the non-homogeneous generalized Riemann-Hilbert(RH)problems involving two unknown functions.Using the uniformization theorem,such problems are transformed into the case of homogeneous type.By the theory of classical boundary value problems,we adopt a novel method to obtain the sectionally analytic solutions of problems in strip domains,and analyze the conditions of solvability and properties of solutions in various domains.展开更多
A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations.Its key feature is the selection of bicubic B-splines...A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations.Its key feature is the selection of bicubic B-splines as trial functions to approximate the state and costate variables in two space dimensions.A Crank-Nicolson difference scheme is constructed for time discretization.The resulting numerical solutions belong to C2in space,and the order of the coefficient matrix is low.Moreover,the Bogner-Fox-Schmit element is considered for comparison.Two numerical experiments demonstrate the feasibility and effectiveness of the proposed method.展开更多
In this paper,the L_(p)chord Minkowski problem is concerned.Based on the results shown in[20],we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvatu...In this paper,the L_(p)chord Minkowski problem is concerned.Based on the results shown in[20],we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvature flow for p>−n with p≠0.展开更多
Recent studies employing deep learning to solve the traveling salesman problem(TSP)have mainly focused on learning construction heuristics.Such methods can improve TSP solutions,but still depend on additional programs...Recent studies employing deep learning to solve the traveling salesman problem(TSP)have mainly focused on learning construction heuristics.Such methods can improve TSP solutions,but still depend on additional programs.However,methods that focus on learning improvement heuristics to iteratively refine solutions remain insufficient.Traditional improvement heuristics are guided by a manually designed search strategy and may only achieve limited improvements.This paper proposes a novel framework for learning improvement heuristics,which automatically discovers better improvement policies for heuristics to iteratively solve the TSP.Our framework first designs a new architecture based on a transformer model to make the policy network parameterized,which introduces an action-dropout layer to prevent action selection from overfitting.It then proposes a deep reinforcement learning approach integrating a simulated annealing mechanism(named RL-SA)to learn the pairwise selected policy,aiming to improve the 2-opt algorithm's performance.The RL-SA leverages the whale optimization algorithm to generate initial solutions for better sampling efficiency and uses the Gaussian perturbation strategy to tackle the sparse reward problem of reinforcement learning.The experiment results show that the proposed approach is significantly superior to the state-of-the-art learning-based methods,and further reduces the gap between learning-based methods and highly optimized solvers in the benchmark datasets.Moreover,our pre-trained model M can be applied to guide the SA algorithm(named M-SA(ours)),which performs better than existing deep models in small-,medium-,and large-scale TSPLIB datasets.Additionally,the M-SA(ours)achieves excellent generalization performance in a real-world dataset on global liner shipping routes,with the optimization percentages in distance reduction ranging from3.52%to 17.99%.展开更多
The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary ...The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.展开更多
Cloud Computing Assisted Instruction shows incomparable advantages over the traditional language teaching, but meanwhile, it exists some major problems, for instance, the information technology is omnipotent, informat...Cloud Computing Assisted Instruction shows incomparable advantages over the traditional language teaching, but meanwhile, it exists some major problems, for instance, the information technology is omnipotent, information input is too excessive and teachers' role is considerably weakened. This article attempts to analyze the problems and promote language teaching reform base on Cloud Computing Assisted Instruction.展开更多
Listening is the foundation of four basic skills.It is one of the most difficult parts for students.There are a lot of problems in English listening comprehension.How to solve the problems and increase students listen...Listening is the foundation of four basic skills.It is one of the most difficult parts for students.There are a lot of problems in English listening comprehension.How to solve the problems and increase students listening ability? This paper analyses students' problems in English listening and advances some corresponding solutions and strategies.展开更多
International transmission of Chinese medical culture is of great importance for the worldwide spread of traditional Chinese culture, which faces many problems, like the serious impact from western medicine and other ...International transmission of Chinese medical culture is of great importance for the worldwide spread of traditional Chinese culture, which faces many problems, like the serious impact from western medicine and other traditional medicines, low professional quality of its practitioners, language barriers, etc. It's urgent to propose some practical solutions.展开更多
By establishing equivalent fixed point theorem, the boundary value problems of p Laplace equations with finite time delay are studied. It’s the first time that the functional differential equation is discussed w...By establishing equivalent fixed point theorem, the boundary value problems of p Laplace equations with finite time delay are studied. It’s the first time that the functional differential equation is discussed with p Laplacian. The topological degree and fixed point theorem on cone are used to prove the existence of solution and positive solution. The conditions are all easy to check.展开更多
The existence of solutions for singular nonlinear two point boundary value problems subject to Sturm Liouville boundary conditions with p Laplacian operators is studied by the method of upper and lower solution...The existence of solutions for singular nonlinear two point boundary value problems subject to Sturm Liouville boundary conditions with p Laplacian operators is studied by the method of upper and lower solutions. The proof is based on an application of Schauder’s fixed point theorem to a modified problem whose solutions are that of the original one. At the same time, Arzela Ascoli theorem is used to prove that the defined operator N is a compact map.展开更多
文摘This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussion on the convergence rates of regular solutions.
基金supported by the National Natural Science Foundations of China(Nos.11571171and 61473148)
文摘It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth for ill-posed problems,so fractional Tikhonov methods have been introduced to remedy this shortcoming.And Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining apartial Arnoldi decomposition of the given matrix.In this paper,we propose a new method to compute an approximate solution of large scale linear discrete ill-posed problems which applies projection fractional Tikhonov regularization in Krylov subspace via Arnoldi process.The projection fractional Tikhonov regularization combines the fractional matrices and orthogonal projection operators.A suitable value of the regularization parameter is determined by the discrepancy principle.Numerical examples with application to image restoration are carried out to examine that the performance of the method.
基金supported by the National Natural Science Foundations of China(Nos.11571171,62073161,and 61473148)。
文摘Tikhonov regularization is a powerful tool for solving linear discrete ill-posed problems.However,effective methods for dealing with large-scale ill-posed problems are still lacking.The Kaczmarz method is an effective iterative projection algorithm for solving large linear equations due to its simplicity.We propose a regularized randomized extended Kaczmarz(RREK)algorithm for solving large discrete ill-posed problems via combining the Tikhonov regularization and the randomized Kaczmarz method.The convergence of the algorithm is proved.Numerical experiments illustrate that the proposed algorithm has higher accuracy and better image restoration quality compared with the existing randomized extended Kaczmarz(REK)method.
基金supported in part by the National Natural Science Foundation of China(No.62073161)the Fundamental Research Funds 2019“Artificial Intelligence+Special Project”of Nanjing University of Aeronautics and Astronautics(No.2019009)
文摘The generalized Tikhonov regularization method is one of the most classical methods for the solution of linear systems of equations that arise from the discretization of linear ill-posed problems.However,the approximate solution obtained by the Tikhonov regularization method in general form may lack many details of the exact solution.Combining the fractional Tikhonov method with the preconditioned technique,and using the discrepancy principle for determining the regularization parameter,we present a preconditioned projected fractional Tikhonov regularization method for solving discrete ill-posed problems.Numerical experiments illustrate that the proposed algorithm has higher accuracy compared with the existing classical regularization methods.
文摘We build a computer program to reconstruct convex bodies using even L_(p)surface area measures for p≥1.Firstly,we transform the minimization problem Pi,which is equivalent to solving the even L_(p)Minkowski problem,into a convex optimization problem P4 with a finite number of constraints.This transformation makes it suitable for computational resolution.Then,we prove that the approximate solutions obtained by solving the problem P4 converge to the theoretical solution when N and k are sufficiently large.Finally,based on the convex optimization problem P_(4),we provide an algorithm for reconstructing convex bodies from even L_(p)surface area measures,and present several examples implemented using MATLAB.
基金Project supported by the China National Funds for Distinguished Young Scientists of National Natural Science Foundation of China(Grant No.51025622)the National Natural Science Foundation of China(Grant No.51406095)the 100 Top Talents Program of Tsinghua University,Beijing,China(2011)
文摘In order to improve the reconstruction performance for ill-posed emission tomographic problems with limited projections, a generalized interpolation method is proposed in this paper, in which the virtual lines of projection are fabricated from, but not linearly dependent on, the measured projections. The method is called the virtual projection（VP） method.Also, an iterative correction method for the integral lengths is proposed to reduce the error brought about by the virtual lines of projection. The combination of the two methods is called the iterative virtual projection（IVP） method. Based on a scheme of equilateral triangle plane meshes and a six asymmetrically arranged detection system, numerical simulations and experimental verification are conducted. Simulation results obtained by using a non-negative linear least squares method,without any other constraints or regularization, demonstrate that the VP method can gradually reduce the reconstruction error and converges to the desired one by fabricating additional effective projections. When the mean square deviation of normal error superimposed on the simulated measured projections is smaller than 0.03, i.e., the signal-to-noise ratio（SNR）for the measured projections is higher than 30.4, the IVP method can further reduce the reconstruction error reached by the VP method apparently. In addition, as the regularization matrix in the Tikhonov regularization method is updated by an iterative correction process similar to the IVP method presented in this paper, or the Tikhonov regularization method is used in the IVP method, good improvement is achieved.
基金This project was supported by TRAPOYT, the Key Project of Chinese Ministry of Education(104126) the NNSF of China(10371046)
文摘This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibilky method and regularized semigroups. Finally, an example is given.
基金supported by the National Natural Science Foundation of China(Grant No.92365206)the support of the China Postdoctoral Science Foundation(Certificate Number:2023M740272)+1 种基金supported by the National Natural Science Foundation of China(Grant No.12247168)China Postdoctoral Science Foundation(Certificate Number:2022TQ0036)。
文摘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.
基金supported by the General Projects of ISTIC Innovation Foundation“Problem innovation solution mining based on text generation model”(MS2024-03).
文摘Purpose:A text generation based multidisciplinary problem identification method is proposed,which does not rely on a large amount of data annotation.Design/methodology/approach:The proposed method first identifies the research objective types and disciplinary labels of papers using a text classification technique;second,it generates abstractive titles for each paper based on abstract and research objective types using a generative pre-trained language model;third,it extracts problem phrases from generated titles according to regular expression rules;fourth,it creates problem relation networks and identifies the same problems by exploiting a weighted community detection algorithm;finally,it identifies multidisciplinary problems based on the disciplinary labels of papers.Findings:Experiments in the“Carbon Peaking and Carbon Neutrality”field show that the proposed method can effectively identify multidisciplinary research problems.The disciplinary distribution of the identified problems is consistent with our understanding of multidisciplinary collaboration in the field.Research limitations:It is necessary to use the proposed method in other multidisciplinary fields to validate its effectiveness.Practical implications:Multidisciplinary problem identification helps to gather multidisciplinary forces to solve complex real-world problems for the governments,fund valuable multidisciplinary problems for research management authorities,and borrow ideas from other disciplines for researchers.Originality/value:This approach proposes a novel multidisciplinary problem identification method based on text generation,which identifies multidisciplinary problems based on generative abstractive titles of papers without data annotation required by standard sequence labeling techniques.
基金supported by the Guangxi Science and Technology base and Talent Project(AD22080047)the National Natural Science Foundation of Guangxi Province(2023GXNFSBA 026063)+1 种基金the Innovation Funds of Chinese University(2021BCF03001)the special foundation for Guangxi Ba Gui Scholars.
文摘A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems.The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption,which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method.Under some additional conditions,the method presented has a superlinear convergence rate,which can be regarded as an extension and supplement of BFGS-type methods with the projection technique.Finally,the effectiveness and application prospects of the proposed method are verified by numerical experiments.
基金Supported by National Natural Science Foundation of China(Grant No.11971015).
文摘This paper studies the non-homogeneous generalized Riemann-Hilbert(RH)problems involving two unknown functions.Using the uniformization theorem,such problems are transformed into the case of homogeneous type.By the theory of classical boundary value problems,we adopt a novel method to obtain the sectionally analytic solutions of problems in strip domains,and analyze the conditions of solvability and properties of solutions in various domains.
基金supported by the National Natural Science Foundation of China(11871312,12131014)the Natural Science Foundation of Shandong Province,China(ZR2023MA086)。
文摘A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations.Its key feature is the selection of bicubic B-splines as trial functions to approximate the state and costate variables in two space dimensions.A Crank-Nicolson difference scheme is constructed for time discretization.The resulting numerical solutions belong to C2in space,and the order of the coefficient matrix is low.Moreover,the Bogner-Fox-Schmit element is considered for comparison.Two numerical experiments demonstrate the feasibility and effectiveness of the proposed method.
基金supported by the National Natural Science Foundation of China(12171144,12231006,12122106).
文摘In this paper,the L_(p)chord Minkowski problem is concerned.Based on the results shown in[20],we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvature flow for p>−n with p≠0.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.72101046 and 61672128)。
文摘Recent studies employing deep learning to solve the traveling salesman problem(TSP)have mainly focused on learning construction heuristics.Such methods can improve TSP solutions,but still depend on additional programs.However,methods that focus on learning improvement heuristics to iteratively refine solutions remain insufficient.Traditional improvement heuristics are guided by a manually designed search strategy and may only achieve limited improvements.This paper proposes a novel framework for learning improvement heuristics,which automatically discovers better improvement policies for heuristics to iteratively solve the TSP.Our framework first designs a new architecture based on a transformer model to make the policy network parameterized,which introduces an action-dropout layer to prevent action selection from overfitting.It then proposes a deep reinforcement learning approach integrating a simulated annealing mechanism(named RL-SA)to learn the pairwise selected policy,aiming to improve the 2-opt algorithm's performance.The RL-SA leverages the whale optimization algorithm to generate initial solutions for better sampling efficiency and uses the Gaussian perturbation strategy to tackle the sparse reward problem of reinforcement learning.The experiment results show that the proposed approach is significantly superior to the state-of-the-art learning-based methods,and further reduces the gap between learning-based methods and highly optimized solvers in the benchmark datasets.Moreover,our pre-trained model M can be applied to guide the SA algorithm(named M-SA(ours)),which performs better than existing deep models in small-,medium-,and large-scale TSPLIB datasets.Additionally,the M-SA(ours)achieves excellent generalization performance in a real-world dataset on global liner shipping routes,with the optimization percentages in distance reduction ranging from3.52%to 17.99%.
文摘The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.
文摘Cloud Computing Assisted Instruction shows incomparable advantages over the traditional language teaching, but meanwhile, it exists some major problems, for instance, the information technology is omnipotent, information input is too excessive and teachers' role is considerably weakened. This article attempts to analyze the problems and promote language teaching reform base on Cloud Computing Assisted Instruction.
文摘Listening is the foundation of four basic skills.It is one of the most difficult parts for students.There are a lot of problems in English listening comprehension.How to solve the problems and increase students listening ability? This paper analyses students' problems in English listening and advances some corresponding solutions and strategies.
基金sponsored by academic programs of Hubei provincial department of education(15Q112)
文摘International transmission of Chinese medical culture is of great importance for the worldwide spread of traditional Chinese culture, which faces many problems, like the serious impact from western medicine and other traditional medicines, low professional quality of its practitioners, language barriers, etc. It's urgent to propose some practical solutions.
文摘By establishing equivalent fixed point theorem, the boundary value problems of p Laplace equations with finite time delay are studied. It’s the first time that the functional differential equation is discussed with p Laplacian. The topological degree and fixed point theorem on cone are used to prove the existence of solution and positive solution. The conditions are all easy to check.
文摘The existence of solutions for singular nonlinear two point boundary value problems subject to Sturm Liouville boundary conditions with p Laplacian operators is studied by the method of upper and lower solutions. The proof is based on an application of Schauder’s fixed point theorem to a modified problem whose solutions are that of the original one. At the same time, Arzela Ascoli theorem is used to prove that the defined operator N is a compact map.
