• ### 最小妒忌公平性区位问题：数学模型与案例分析

分类： 地球科学 >> 地理学 分类： 数学 >> 控制和优化 提交时间： 2023-03-20

摘要：区位模型广泛应用于公共健康、义务教育、应急管理、物流配送等领域，然而空间公平导向区位模型研究进展缓慢。常见的空间公平度量指标包括距离极值、方差、变异系数、Gini系数，以及2SFCA可达性指标等。本文通过案例实验，总结了最小化方差区位模型的特征及局限，并将最小化方差目标函数改进为最小化“妒忌”目标函数，将其整合在容量约束P中值模型(CPMP)和容量约束设施区位模型(CFLP)中。新模型克服了最小方差目标函数会扭曲效率指标的局限，能够显著地改进设施区位空间公平性指标，在公共服务设施布局规划中具有应用价值。

• ### A Prediction Framework for Turning Period Structures in COVID-19 Epidemic and Its Application to Practical Emergency Risk Management

分类： 数学 >> 控制和优化 提交时间： 2023-02-01

The aim of this paper is first to establish a general prediction framework for turning (period) term structures in COVID-19 epidemic related to the implementation of emergency risk man#2;agement in the practice, which allows us to conduct the reliable estimation for the peak period based on the new concept of “Turning Period” (instead of the traditional one with the focus on “Turning Point”) for infectious disease spreading such as the COVID-19 epidemic appeared early in year 2020. By a fact that emergency risk management is necessarily to implement emergency plans quickly, the identification of the Turning Period is a key element to emergency planning as it needs to provide a time line for effective actions and solutions to combat a pandemic by reducing as much unexpected risk as soon as possible. As applications, the paper also discusses how this “Turning Term (Period) Structure” is used to predict the peak phase for COVID-19 epidemic in Wuhan from January/2020 to early March/2020. Our study shows that the predication framework established in this paper is capa#2;ble to provide the trajectory of COVID-19 cases dynamics for a few weeks starting from Feb.10/2020 to early March/2020, from which we successfully predicted that the turning period of COVID-19 epi#2;demic in Wuhan would arrive within one week after Feb.14/2020, as verified by the true observation in the practice. The method established in this paper for the prediction of “Turning Term (Period) Structures”, and associated criteria for the Turning Term Structure of COVID-19 epidemic is expected to be a useful and powerful tool to implement the so-called “dynamic zero-COVID-19 policy” ongoing basis in the practice.

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• ### Uniformly exponentially stable approximation for the transmission line with varying structure parameters

分类： 数学 >> 控制和优化 提交时间： 2023-01-27

摘要：

• ### Uniform exponential stability approximations of semi-discretization schemes for two hybrid systems

分类： 数学 >> 控制和优化 提交时间： 2023-01-27

摘要：This paper deals with the uniform exponential stabilities (UESs) of two hybrid control systems consisting of wave equation and a second-order ordinary differential equation. Linear feedback law and local viscosity, and nonlinear feedback law and  interior anti-damping are considered, respectively. Firstly, the hybrid system is reduced to a first order port-Hamiltonian system with dynamical boundary conditions and the resulting systems  are then discretized by  average central-difference scheme.  Secondly,  the UES of the discrete system is obtained without  prior knowledge on the exponential stability of continuous system.  The frequency domain characterization of UES for a family of  contractive semigroups and discrete multiplier method are utilized to verify main results, respectively. Finally,  the convergence analysis of the numerical approximation scheme is performed by the Trotter-Kato Theorem. Most interestingly,  the exponential stability of the  continuous system is derived by the convergence of energy and UES and this is a new idea to investigate the exponential stability of some complicate systems. The effectiveness of the numerical approximating  scheme is verified by numerical simulation.

• ### Uniformly exponentially stable approximations for Timoshenko beams

分类： 数学 >> 控制和优化 提交时间： 2023-01-27

摘要：In this note, Timoshenko beams with interior damping and boundary damping are studied from  the viewpoints of  control theory and numerical approximation. Especially, the uniform exponential stabilities of the beams are studied. The meaning of uniform exponential stability in this paper is two-fold: The first one is in the classical sense and also is concisely called exponential stability by many authors; The second one is  that the semi-discretization systems, which  are derived from an exponentially stable continuous beam by some semi-discretization schemes,  are uniformly exponentially stable with respect to the discretized parameter.  To investigate uniform exponential stability of continuous  and discrete systems, five completely different methods,  which are  stability theory of port-Hamiltonian system,  direct method  of Lyapunov functional, perturbation theory of $C_{0}$-semigroup, spectral analysis of unbounded operator and frequency  standard of exponential stability for contractive semigroup, are involved.  Especially, a new method, which is based on the frequency domain characteristics of uniform exponential stability of  $C_{0}$-semigroup of contractions, is established  to verify the uniform exponential stability of semi-discretization systems derived from coupled system.  The effectiveness of the numerical approximating  algorithms is verified by numerical simulations.

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• ### Uniform Exponential Stability and Control Convergence of Semi-discrete Scheme for a Timoshenko Beam

分类： 数学 >> 控制和优化 提交时间： 2023-01-27

摘要：

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• ### Frequency Energy Multiplier Approach to Uniform Exponential Stability Analysis of Semi-discrete Scheme for a Schrodinger Equation under Boundary Feedback

分类： 数学 >> 控制和优化 提交时间： 2023-01-27

摘要：In this paper, we investigate the  uniform exponential stability of a semi-discrete scheme for  a  Schr "{o}dinger equation under boundary feedback stabilizing control in the natural state space $L^2(0,1)$. This study is significant since  a time domain energy multiplier that allows proving the exponential stability of this continuous Schr "{o}dinger system has not yet found, thus leading to a major mathematical challenge to semi-discretization of the PDE, an open problem for a long time.  Although the powerful frequency domain energy multiplier approach has been used in proving  exponential stability for PDEs since 1980s, its use to the emph{uniform} exponential  stability of the semi-discrete scheme for PDEs has not been reported yet. The difficulty associated with the uniformity is that due to the parameter of the step size, it involves a family of   operators in different state spaces that need to be considered simultaneously. Based on the Huang-Pr "{u}ss frequency domain criterion for uniform exponential stability of a family of $C_0$-semigroups in Hilbert spaces, we solve this problem for the first time by proving the uniform boundedness for all the resolvents of these operators on the imaginary axis. The proof almost exactly follows the procedure for the exponential stability of the continuous counterpart, highlighting
the advantage of this discretization method.

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• ### Integral input-to-state stabilization in different norms for a class of linear parabolic PDEs

分类： 数学 >> 控制和优化 提交时间： 2022-12-12

In this paper, we study the problem of  integral input-to-state stabilization in different norms for  parabolic PDEs with integrable inputs. More precisely, we apply the method of backstepping to design a boundary  control law for certain linear   parabolic PDEs with  destabilizing terms and $L^r$-inputs, and establish  the integral  input-to-state stability   in  the spatial $L^p$-norm and $W^{1,p}$-norm, respectively, for the closed-loop system, whenever $p in 1,+ infty$ and $r in p,+ infty$.  In order to deal with singularities in the case of $p in 1,2)$, we employ  the approximative Lyapunov method to analyze  the stability in different norms. Concerning with the appearance of external inputs, we apply the   method of functional analysis and the theory of series to prove the unique existence and regularity of solution to the closed-loop system.

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• ### 计及灵活资源日内调节离散性的日前两阶段分布鲁棒机组组合——基于L1范数Wasserstein模糊集的稀疏建模与求解方法

分类： 动力与电气工程 >> 电气工程 分类： 数学 >> 控制和优化 提交时间： 2022-08-07

摘要：快速启停发电机组等具备强调节能力的设备是保证高比例可再生能源电力系统经济可靠运行的重要灵活资源.通过在电力系统的日前发电计划优化或现货市场出清中构建风电的不确定性模型、快速启停机组日内含离散决策量的运行模型, 本文基于Wasserstein概率分布模糊集建立了含混合整数追索问题的日前两阶段分布鲁棒机组组合优化模型. 为高效计算所建立模型的最优解, 本文提出了两个可行的算法框架. 其中第一个算法用有限多个离散的事件来逼近并最终等效刻画风电出力的连续支撑集, 另一个算法则利用了由子程序辨识得到的风电出力极端概率分布来更新日前机组开停机方案. 两个算法都依赖于经典的嵌套列和约束生成法（the nested column-and-constraint generation method, nested G&CG）. 本文的理论和计算分析表明, 得益于L1范数Wasserstein距离的稀疏特性, 连续支撑集可以由相对较少的离散事件等效表征,而极限分布同样具有稀疏性. 稀疏性带来的缩减效应使得两个算法均能够高效处理含混合整数追索的两阶段分布鲁棒优化问题. 本文的数值实验表明, 精确考虑快速启停机组的日内离散行为有益于得到更具鲁棒性和经济性的日前机组开停机方案; 此外, 分布鲁棒优化在样本外测试中能够可靠和经济地应对风电出力的不确定性.

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• ### 考虑风功率预测条件误差和时空关联性的分布鲁棒机组组合模型及算法

分类： 动力与电气工程 >> 电气工程 分类： 数学 >> 控制和优化 提交时间： 2022-08-07

摘要：本文研究含随机风电的机组组合问题, 基于分布鲁棒优化提出了考虑风功率预测的条件误差和误差时空相关性的建模及优化方法. 首先, 为了量化计算风功率的预测误差, 用Lasso（最小绝对值收敛和选择算子）训练得到鲁棒的条件误差估计器；为了获取风功率预测误差的时空关联信息, 通过无偏估计从历史数据得到误差的协方差矩阵. 用所得的条件误差和协方差矩阵构造了改进的风功率预测误差的概率分布模糊集. 其次, 在多面体支撑集上建立了风功率预测误差的随机量, 构建了两阶段分布鲁棒机组组合模型, 并提出了与该模型等效的混合整数半正定规划模型. 再次, 提出了从第二阶段半正定优化问题辨识极限分布的方法, 并提出了利用极限分布求解两阶段分布鲁棒机组组合问题的高效割平面算法. 最后, 进行数值实验, 验证了所提模型在应对风功率预测误差的时空关联性方面的能力和优势, 验证了模型决策方案的经济性和鲁棒性.

• ### 深度学习损失函数地貌分析研究进展

分类： 数学 >> 控制和优化 提交时间： 2021-11-29

摘要：在机器学习和数学优化研究领域, 深度学习优化问题易优性的数学解释极具挑战性. 损失函数存在高维、非凸、不光滑等特质性, 然而也能通过梯度下降法搜索到全局最优值. 损失函数地貌分析成为揭示深度学习优化问题易优性本质的重要研究方向. 为促进可解释、可信的深度学 习在更关键领域的应用, 本文回顾了损失函数地貌特征（局部极小点的数量和空间分布、最优点之间的连通性、临界点的最优性）、梯度下降法收敛性、以及损失函数地貌可视化等方面的研究进展和挑战.

• ### 均等分区问题模型、算法及应用

分类： 地球科学 >> 地理学 分类： 数学 >> 控制和优化 提交时间： 2021-02-24

摘要：分区问题广泛应用于地理、经济、政治、商业、公共服务等领域。均等分区问题是其中一类问题，通常要求分区人口数量或任务量均等、几何形状紧凑和空间连续，应用于选区、销售区和巡逻区的划分。本文针对均等分区问题提出了一个混合整型线性规划模型，并设计了一个基于迭代局部搜索(ILS)的混合算法。该算法从三个方面扩展ILS：群解搜索、VND搜索及SPP模型改进。选择5个区域对模型和算法进行测试，结果表明：数学模型能够求解空间单元数量为324的案例；混合算法优化性能优异，鲁棒性强，计算效率较高。所提出的均等分区问题适用于政治分区等经典问题，在新冠疫情应急服务等领域也具有应用潜力。

• ### DEED: A general quantization scheme for saving bits in communication

分类： 数学 >> 控制和优化 提交时间： 2020-06-16

摘要：Quantization is a popular technique to reduce communication in distributed optimization. Motivated by the classical work on inexact gradient descent (GD) \cite{bertsekas2000gradient}, we provide a general convergence analysis framework for inexact GD that is tailored for quantization schemes. We also propose a quantization scheme Double Encoding and Error Diminishing (DEED). DEED can achieve small communication complexity in three settings: frequent-communication large-memory, frequent-communication small-memory, and infrequent-communication (e.g. federated learning). More specifically, in the frequent-communication large-memory setting, DEED can be easily combined with Nesterov's method, so that the total number of bits required is $\tilde{O}( \sqrt{\kappa} \log 1/\epsilon )$, where $\tilde{O}$ hides numerical constant and $\log \kappa$ factors. In the frequent-communication small-memory setting, DEED combined with SGD only requires $\tilde{O}( \kappa \log 1/\epsilon)$ number of bits in the interpolation regime. In the infrequent communication setting, DEED combined with Federated averaging requires a smaller total number of bits than Federated Averaging. All these algorithms converge at the same rate as their non-quantized versions, while using a smaller number of bits.

• ### Analytical expressions of copositivity for 4th order symmetric tensors and applications

分类： 数学 >> 控制和优化 提交时间： 2019-08-30

摘要： In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are 4th order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of 4th order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for 4th order 2 dimensional symmetric tensors. For 4th order 3 dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) cpositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is showed for 4th order 2 dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for 4th order 2 dimensional symmetric tensors. Finally, we apply these results to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.

• ### B tensors and tensor complementarity problems

分类： 数学 >> 控制和优化 提交时间： 2017-07-25

摘要：In this paper, one of our main purposes is to prove the boundedness of solution set of tensor complementarity problem with B tensor such that the specific bounds only depend on the structural properties of tensor. To achieve this purpose, firstly, we present that each B tensor is strictly semi-positive and each B$_0$ tensor is semi-positive. Subsequencely, the strictly lower and upper bounds of different operator norms are given for two positively homogeneous operators defined by B tensor. Finally, with the help of the upper bounds of different operator norms, we show the strcitly lower bound of solution set of tensor complementarity problem with B tensor. Furthermore, the upper bounds of spectral radius and $E$-spectral radius of B (B$_0$) tensor are obtained, respectively, which achieves our another objective. In particular, such the upper bounds only depend on the principal diagonal entries of tensors.

• ### Conic optimization and complementarity problems V2

分类： 数学 >> 控制和优化 提交时间： 2016-12-16

摘要：Although the Karush-Kuhn-Tucker conditions suggest a connection between a conic optimization problem and a complementarity problem, it is difficult to find an accessible explicit form of this relationship in the literature. This note will present such a relationship.

• ### Positive operators of Extended Lorentz cones

分类： 数学 >> 控制和优化 提交时间： 2016-08-30

摘要：In this paper necessary conditions and sufficient conditions are given for a linear operator to be a positive operators of an Extended Lorentz cone. Similarities and differences with the positive operators of Lorentz cones are investigated.

• ### An efficient m-step Levenberg-Marquardt method for nonlinear equations

分类： 数学 >> 控制和优化 分类： 数学 >> 计算数学 提交时间： 2016-07-11

摘要：In this paper, we construct and analyze an efficient m-step Levenberg-Marquardt method for nonlinear equations. The main advantage of this method is that the m-step LM method could save more Jacobian calculations with frozen $(J_k^TJ_k+\lambda_kI)^{-1}J_k^T$ at every iteration. Under the local error bound condition which is weaker than nonsingularity, the m-step LM method has been proved to have $(m+1)$th convergence order. The global convergence has also been given by trust region technique. Numerical results show that the m-step LM method is efficient and could save many calculations of the Jacobian especially for large scale problems.