Subjects: Mathematics >> Control and Optimization. submitted time 2023-12-04
Abstract： In this paper, we mainly dicuss the analytic conditions of copositivity of a class of 4th order 3-dimensional symmetric tensors. For a 4th order 3-dimensional symmetric tensor with its entries $1$ or $-1$, an analytic necessary and sufficient condition is given for its strict copositivity with the help of the properties of strictly semi-positive tensors. And by means of usual maxi-min theory, a necessary and sufficient condition is established for copositivity of such a tensor also. Applying these conclusions to a general 4th order 3-dimensional symmetric tensor, the analytic conditions are successfully obtained for verifying the (strict) copositivity, and these conditions can be very easily parsed and validated.
Peer Review Status:
Awaiting Review
Subjects: Geosciences >> Geography Subjects: Mathematics >> Control and Optimization. submitted time 2023-05-13
Abstract： Location problems have been widely applied to service planning of public health, compulsory education, emergency management, and delivery logistics. However, the mainstream location models are usually to optimize the efficiency objectives such as travel cost, facility cost and the number of customs to be served, rather than the equality objectives. A few location models aim to optimize one of the equality measures, such as the variance of distances, the deviation of distances, the Gini coefficient between the travel distances, and the variance of spatial accessibility indexes. However, the facility locations, capacities and their service areas can be easily distorted by most equality-oriented objective functions. In this paper, a spatially envy objective function for service equality is proposed to overcome the shortcomings of commonly used equality functions. The envy value of customers at a location is determined by their travel distance that beyond a predefined distance. The envy function can be added to mainstream location models in a weighted manner. As a result, the capacitated p-median problem (CPMP) is enhanced as CPMP-envy. The original and improved models were tested on three large instances. Case experiments show that the equality measures, such as maximum travel distance, variance of distances, coefficient of variation, and Gini coefficient between travel distances, can be substantially improved by minimizing the weighted sum of spatial envy and travel cost. It is argued that the envy indicator has theoretical and practical potentials in facility planning towards spatial equality of public service.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2023-02-01
Abstract：
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.
Subjects: Mathematics >> Control and Optimization. submitted time 2023-01-27
Abstract：
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2023-01-27
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2023-01-27
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2023-01-27
Abstract：
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2023-01-27
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2022-12-12
Abstract：
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.
Peer Review Status:Awaiting Review
Subjects: Dynamic and Electric Engineering >> Electrical Engineering Subjects: Mathematics >> Control and Optimization. submitted time 2022-08-07
Xiaodong Zheng
Baorong Zhou
Xiuli Wang
Bo Zeng
Jizhong Zhu
Haoyong Chen
Peizheng Xuan
Waisheng Zheng
Abstract： Quick-start generation units are critical devices and flexible resources to ensure a high penetration level of renewable energy in power systems. By considering the wind uncertainty, and both binary and continuous decisions of quick-start units within the intraday dispatch, we develop a Wasserstein-metric-based distributionally robust optimization model for the day-ahead network-constrained unit commitment (NCUC) problem with mixed integer recourse. We propose two feasible frameworks for solving the optimization problem. One approximates the continuous support of random wind power with finitely many events, the other leverages the extremal distributions instead. Both solution frameworks rely on the classic nested column-and-constraint generation (C&CG) method. It is shown that due to the sparsity of L1-norm Wasserstein metric, the continuous support of wind power generation could be represented by a discrete one with a small number of events, and the extremal distributions rendered are sparse as well. With this reduction, the distributionally robust NCUC model with complicated mixed-integer recourse problems can be efficiently handled by both solution frameworks. Numerical studies are carried out, demonstrating that the model considering quick-start generation units ensures unit commitment (UC) schedules to be more robust and cost effective, and the distributionally robust optimization method captures the wind uncertainty well in terms of out-of-sample tests.
Subjects: Dynamic and Electric Engineering >> Electrical Engineering Subjects: Mathematics >> Control and Optimization. submitted time 2022-08-07
Abstract： In this paper, a study of the day-ahead unit commitment problem with stochastic wind power generation is presented, which considers conditional and correlated wind power forecast errors through a distributionally robust optimization approach. Firstly, to capture the characteristics of random wind power forecast errors, the least absolute shrinkage and selection operator (Lasso) is utilized to develop a robust conditional error estimator, while an unbiased estimator is used to obtain the covariance matrix. The conditional error and the covariance matrix are then used to construct an enhanced ambiguity set. Secondly, we develop an equivalent mixed integer semidefinite programming (MISDP) formulation of the two-stage distributionally robust unit commitment model with a polyhedral support of random variables. Further, to efficiently solve this problem, a novel cutting plane algorithm that makes use of the extremal distributions identified from the second-stage semidefinite programming (SDP) problems is introduced. Finally, numerical case studies show the advantage of the proposed model in capturing the spatiotemporal correlation in wind power generation, as well as the economic efficiency and robustness of dispatch decisions.
Subjects: Mathematics >> Control and Optimization. submitted time 2021-11-29
Abstract： In the filed of machine learning and mathematical optimization, it is a challenge to mathematically explain optimality of loss function for deep learning. Loss function is high-dimensional, non-convex, and non-smooth. It was, however, observed that gradient descent could reach zero training loss of this highly non-convex function. Loss landscape analysis is critical to reveal reasons why deep networks are easily optimizable. We reviewed the advance on loss landscape analysis, such as landscape features (number and spatial distribution of local minima, connectivity between global optima, and global optimality of critical points), convergence of gradient descent, and visualization of loss landscape. This survey aimed to promote interpretable and reliable deep learning in critical applications. "
Peer Review Status:Awaiting Review
Subjects: Geosciences >> Geography Subjects: Mathematics >> Control and Optimization. submitted time 2021-02-24
Abstract： Districting problems have been widely applied in geography, economics, environmental science, politics, business, public service and many other areas. The equal districting problem (EDP) arises in applications such as political redistricting, police patrol area delineation, sales territory design and some service area design. The important criteria for these problems are district equality, contiguity and compactness. A mixed integer linear programming (MILP) model and a hybrid algorithm are proposed for the EDP. The hybrid algorithm is designed by extending iterative local search (ILS) algorithm with three schemes: population-based ILS, variable neighborhood descent (VND) local search, and set partitioning. The performance of the algorithm was tested on five areas. Experimentation showed that the instances could be solved effectively and efficiently. The potential applications of the EDP in emergency services are also discussed.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2020-06-16
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2019-08-30
Abstract： " 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. "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2017-07-25
Abstract：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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2016-12-16
Abstract：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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2016-08-30
Abstract：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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. Subjects: Mathematics >> Computational Mathematics. submitted time 2016-07-11
Abstract：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.
Peer Review Status:Awaiting Review
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