动力系统/优化与控制学术速递[1.10]
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math.DS动力系统，共计4篇
math.OC优化与控制，共计13篇
1.math.DS动力系统:
【1】 Exponential multiple mixing for commuting automorphisms of a nilmanifold
标题：零流形交换自同构的指数多重混合
链接：https://arxiv.org/abs/2201.02556
备注：13 pages
摘要：Let $l\in \mathbb{N}_{\geq 1}$ and $\alpha : \mathbb{Z}^l\rightarrow
\text{Aut}(\mathscr{N})$ be an action of $\mathbb{Z}^l$ by automorphisms on a
compact nilmanifold $\mathscr{N}$. We assume the action of every $\alpha(z)$ is
ergodic for $z\in \mathbb{Z}^l\smallsetminus\{0\}$ and show that $\alpha$
satisfies exponential $n$-mixing for any integer $n\geq 2$. This extends
results of Gorodnik and Spatzier [Acta Math., 215 (2015)].
【2】 Projective Embedding of Dynamical Systems: uniform mean field equations
标题：动力系统的射影嵌入：一致平均场方程
链接：https://arxiv.org/abs/2201.02355
备注：45 pages; one column; 10 figures;
摘要：We study embeddings of continuous dynamical systems in larger dimensions via
projector operators. We call this technique PEDS, projective embedding of
dynamical systems, as the stable fixed point of the dynamics are recovered via
projection from the higher dimensional space. In this paper we provide a
general definition and prove that for a particular type of projector operator
of rank-1, the uniform mean field projector, the equations of motion become a
mean field approximation of the dynamical system. While in general the
embedding depends on a specified variable ordering, the same is not true for
the uniform mean field projector. In addition, we prove that the original
stable fixed points remain stable fixed points of the dynamics, saddle points
remain saddle, but unstable fixed points become saddles.
【3】 Long time and large crowd dynamics of discrete Cucker-Smale alignment models
标题：离散Cucker-Smer线形模型的长时间大人群动力学
链接：https://arxiv.org/abs/2201.02281
摘要：We provide a bird's eye view on developments in analyzing the long time,
large crowd behavior of Cucker-Smale alignment dynamics. We consider a class of
(fully-)discrete models, paying particular attention to general alignment
protocols in which agents, with possibly time-dependent masses, are driven by a
large class of heavy-tailed communication kernels. The presence of
time-dependent masses allows, in particular, non-symmetric communication. While
revisiting known results in the literature, we also shed new light of various
aspects on the long time flocking/swarming behavior, driven by the decay of
energy fluctuations and heavy-tailed connectivity. We also discuss the large
crowd dynamics in terms of the hydrodynamic description of Euler alignment
models.
【4】 An Input-to-State Safety Approach to Anomaly-Resilient Parabolic PDEs: Application to Cyber-Physical Battery Modules
标题：异常弹性抛物型偏微分方程的输入到状态安全方法：在网络物理电池模块中的应用
链接：https://arxiv.org/abs/2201.02239
摘要：Distributed Parameter Cyber-Physical Systems (DPCPSs), modelled by Partial
Differential Equations (PDEs), are increasingly vulnerable to anomalies such as
physical faults as well as cyber-attacks. This motivates the need for
strategies towards anomaly-resilient control of these systems. Although anomaly
detection and diagnostics in PDE systems have received considerable attention
in existing literature, fault-tolerant or anomaly-resilient control for PDEs
remains relatively under-explored. However, given the vulnerabilities of these
systems against anomalies, it is essential that the control systems possess
resilience against these disruptions. In this context, we explore a Practical
Input-to-Safety (pISSf) based control design approach for a class of DPCPSs
modelled by linear Parabolic PDEs. Specifically, we develop a design framework
for anomaly-resilient control for this class of system with both safety and
stability guarantees based on control Lyapunov functional and control barrier
functional. To illustrate our methodology, we apply our strategy to design a
thermal-anomaly resilient boundary coolant control system for a cyber-physical
battery module. Several simulation studies are done to show the efficacy of our
method under anomalies such as mechanical battery degradation and cyber-attack
mediated overdischarge.
2.math.OC优化与控制:
【1】 QN Optimization with Hessian Sample
标题：基于黑森样本的QN优化
链接：https://arxiv.org/abs/2201.02608
摘要：This article explores how to effectively incorporate curvature information
generated using SIMD-parallel forward-mode Algorithmic Differentiation (AD)
into unconstrained Quasi-Newton (QN) minimization of a smooth objective
function, $f$. Specifically, forward-mode AD can be used to generate block
Hessian samples $Y=\nabla^2 f(x)\,S$ whenever the gradient is evaluated. Block
QN algorithms then update approximate inverse Hessians, $H_k \approx \nabla^2
f(x_k)$, with these Hessian samples. Whereas standard line-search based BFGS
algorithms carefully filter and correct secant-based approximate curvature
information to maintain positive definite approximations, our algorithms
directly incorporate Hessian samples to update indefinite inverse Hessian
approximations without filtering. The sampled directions supplement the
standard QN two-dimensional trust-region sub-problem to generate a moderate
dimensional subproblem which can exploit negative curvature. The resulting
quadratically-constrained quadratic program is solved accurately with a
generalized eigenvalue algorithm and the step advanced using standard trust
region step acceptance and radius adjustments. The article aims to avoid serial
bottlenecks, exploit accurate positive and negative curvature information, and
conduct a preliminary evaluation of selection strategies for $S$.
【2】 Machine-learning-based arc selection for constrained shortest path problems in column generation
标题：基于机器学习的列生成约束最短路径问题的圆弧选择
链接：https://arxiv.org/abs/2201.02535
摘要：Column generation is an iterative method used to solve a variety of
optimization problems. It decomposes the problem into two parts: a master
problem, and one or more pricing problems (PP). The total computing time taken
by the method is divided between these two parts. In routing or scheduling
applications, the problems are mostly defined on a network, and the PP is
usually an NP-hard shortest path problem with resource constraints. In this
work, we propose a new heuristic pricing algorithm based on machine learning.
By taking advantage of the data collected during previous executions, the
objective is to reduce the size of the network and accelerate the PP, keeping
only the arcs that have a high chance to be part of the linear relaxation
solution. The method has been applied to two specific problems: the vehicle and
crew scheduling problem in public transit and the vehicle routing problem with
time windows. Reductions in computational time of up to 40% can be obtained.
【3】 An efficient and easy-to-extend Matlab code of the Moving Morphable Component (MMC) method for three-dimensional topology optimization
标题：一种高效、易扩展的移动可变形分量(MMC)三维拓扑优化Matlab代码
链接：https://arxiv.org/abs/2201.02491
摘要：Explicit topology optimization methods have received ever-increasing interest
in recent years. In particular, a 188-line Matlab code of the two-dimensional
(2D) Moving Morphable Component (MMC)-based topology optimization method was
released by Zhang et al. (Struct Multidiscip Optim 53(6):1243-1260, 2016). The
present work aims to propose an efficient and easy-to-extend 256-line Matlab
code of the MMC method for three-dimensional (3D) topology optimization
implementing some new numerical techniques. To be specific, by virtue of the
function aggregation technique, accurate sensitivity analysis, which is also
easy-to-extend to other problems, is achieved. Besides, based on an efficient
loading path identification algorithm, the degrees of freedoms (DOFs) not
belonging to the loading path are removed in finite element analysis (FEA),
which significantly accelerates the optimization process. As a result, compared
to the corresponding 188-line 2D code, the performance of the optimization
results, the computational efficiency of FEA, and the convergence rate and the
robustness of optimization process are greatly improved. For the sake of
completeness, a refined 218-line Matlab code implementing the 2D-MMC method is
also provided.
【4】 Indefinite least squares with a quadratic constraint
标题：带二次约束的不定最小二乘法
链接：https://arxiv.org/abs/2201.02442
备注：32 pages. Comments are welcomed!
摘要：An abstract indefinite least squares problem with a quadratic constraint is
considered. This is a quadratic programming problem with one quadratic equality
constraint, where neither the objective nor the constraint are convex
functions. Necessary and sufficient conditions are found for the existence of
solutions.
【5】 Linear pencils and quadratic programming problems with a quadratic constraint
标题：带二次约束的线性铅笔和二次规划问题
链接：https://arxiv.org/abs/2201.02439
备注：24 pages. Comments are welcomed!
摘要：Given bounded selfadjoint operators $A$ and $B$ acting on a Hilbert space
$\mathcal{H}$, consider the linear pencil $P(\lambda)=A+\lambda B$,
$\lambda\in\mathbb{R}$. The set of parameters $\lambda$ such that $P(\lambda)$
is a positive (semi)definite operator is characterized. These results are
applied to solving a quadratic programming problem with an equality quadratic
constraint (or a QP1EQC problem).
【6】 An Adaptive Penalty Method for Inequality Constrained Minimization Problems
标题：不等式约束极小化问题的一种自适应惩罚方法
链接：https://arxiv.org/abs/2201.02425
备注：None
摘要：The primal-dual active set method is observed to be the limit of a sequence
of penalty formulations. Using this perspective, we propose a penalty method
that adaptively becomes the active set method as the residual of the iterate
decreases. The adaptive penalty method (APM) therewith combines the main
advantages of both methods, namely the ease of implementation of penalty
methods and the exact imposition of inequality constraints inherent to the
active set method. The scheme can be considered a quasi-Newton method in which
the Jacobian is approximated using a penalty parameter. This spatially varying
parameter is chosen at each iteration by solving an auxiliary problem.
【7】 Model-Free Nonlinear Feedback Optimization
标题：无模型非线性反馈优化
链接：https://arxiv.org/abs/2201.02395
摘要：Feedback optimization is a control paradigm that enables physical systems to
autonomously reach efficient operating points. Its central idea is to
interconnect optimization iterations in closed-loop with the physical plant.
Since iterative gradient-based methods are extensively used to achieve
optimality, feedback optimization controllers typically require the knowledge
of the steady-state sensitivity of the plant, which may not be easily
accessible in some applications. In contrast, in this paper we develop a
model-free feedback controller for efficient steady-state operation of general
dynamical systems. The proposed design consists in updating control inputs via
gradient estimates constructed from evaluations of the nonconvex objective at
the current input and at the measured output. We study the dynamic
interconnection of the proposed iterative controller with a stable nonlinear
discrete-time plant. For this setup, we characterize the optimality and the
stability of the closed-loop behavior as functions of the problem dimension,
the number of iterations, and the rate of convergence of the physical plant. To
handle general constraints that affect multiple inputs, we enhance the
controller with Frank-Wolfe type updates.
【8】 Control problem for quadratic parabolic differential equations with sensor sets of finite volume or anisotropically decaying density
标题：具有有限体积或各向异性衰减传感器组的二次抛物型微分方程的控制问题
链接：https://arxiv.org/abs/2201.02370
备注：35 pages
摘要：We prove observability and null-controllability for quadratic parabolic
differential equations. The sensor set is allowed to have finite volume if the
generator has trivial singular space $S$. In the case of generators with
singular space $S \neq \{ 0 \}$ the sensor set is permitted to decay in
directions determined by $S$. The proof is based on dissipation estimates for
the quadratic differential operator with respect to spectral projections of
partial harmonic oscillators and corresponding uncertainty relations.
【9】 Stochastic Saddle Point Problems with Decision-Dependent Distributions
标题：决策相关分布的随机鞍点问题
链接：https://arxiv.org/abs/2201.02313
摘要：This paper focuses on stochastic saddle point problems with
decision-dependent distributions in both the static and time-varying settings.
These are problems whose objective is the expected value of a stochastic payoff
function, where random variables are drawn from a distribution induced by a
distributional map. For general distributional maps, the problem of finding
saddle points is in general computationally burdensome, even if the
distribution is known. To enable a tractable solution approach, we introduce
the notion of equilibrium points -- which are saddle points for the stationary
stochastic minimax problem that they induce -- and provide conditions for their
existence and uniqueness. We demonstrate that the distance between the two
classes of solutions is bounded provided that the objective has a
strongly-convex-strongly-concave payoff and Lipschitz continuous distributional
map. We develop deterministic and stochastic primal-dual algorithms and
demonstrate their convergence to the equilibrium point. In particular, by
modeling errors emerging from a stochastic gradient estimator as sub-Weibull
random variables, we provide error bounds in expectation and in high
probability that hold for each iteration; moreover, we show convergence to a
neighborhood in expectation and almost surely. Finally, we investigate a
condition on the distributional map -- which we call opposing mixture dominance
-- that ensures the objective is strongly-convex-strongly-concave. Under this
assumption, we show that primal-dual algorithms converge to the saddle points
in a similar fashion.
【10】 Electric Vehicle Routing Problem with Spatio-temporal Varying Electricity Price and Incentive-aware Customers
标题：具有时空变化电价和激励顾客的电动汽车路径问题
链接：https://arxiv.org/abs/2201.02311
备注：Submitted to IEEE TSG. arXiv admin note: substantial text overlap with arXiv:2110.06441
摘要：This paper investigates the optimization problem of a fleet of electric
vehicles (EVs) serving a set of time-specified customers, where the operator
needs to optimize routing and charging problem jointly for each EV. In
particular, regarding to the spatio-temporal varying electricity price, we
consider incentive-aware customers and propose that the operator offers
monetary incentives to exchange time flexibility of customers. In this manner,
a win-win situation is achievable since time flexibility enables the fleet
operator to obtain a routing and charging schedule with lower cost, whilst the
customers receives monetary compensation. Specifically, we first devise a
bi-level model whereby the fleet operator optimizes the routing and charging
schedule jointly with a monetary incentive to reimburse the delivery time
flexibility experienced by the customers. At the same time, the customers
choose the optimal time flexibility by minimizing its own cost. Second, we
tackle the complexity resulting from the bi-level and nonlinear problem with an
equivalent transformation method. Eventually, we reformulate the problem as a
single-level optimization problem, which later is solved by proposed Benders
dual decomposition method holding a faster convergence rate than the
generalized Benders decomposition method. To evaluate the effectiveness of our
framework and proposed Benders dual decomposition algorithm, we carry out
extensive numerical experiments using VRP-REP data from Belgium.
【11】 Local and Global Convergence of General Burer-Monteiro Tensor Optimizations
标题：广义布里-蒙泰罗张量优化问题的局部收敛性和全局收敛性
链接：https://arxiv.org/abs/2201.02298
摘要：Tensor optimization is crucial to massive machine learning and signal
processing tasks. In this paper, we consider tensor optimization with a convex
and well-conditioned objective function and reformulate it into a nonconvex
optimization using the Burer-Monteiro type parameterization. We analyze the
local convergence of applying vanilla gradient descent to the factored
formulation and establish a local regularity condition under mild assumptions.
We also provide a linear convergence analysis of the gradient descent algorithm
started in a neighborhood of the true tensor factors. Complementary to the
local analysis, this work also characterizes the global geometry of the best
rank-one tensor approximation problem and demonstrates that for orthogonally
decomposable tensors the problem has no spurious local minima and all saddle
points are strict except for the one at zero which is a third-order saddle
point.
【12】 Sparse PCA on fixed-rank matrices
标题：固定秩矩阵上的稀疏PCA
链接：https://arxiv.org/abs/2201.02487
备注：None
摘要：Sparse PCA is the optimization problem obtained from PCA by adding a sparsity
constraint on the principal components. Sparse PCA is NP-hard and hard to
approximate even in the single-component case. In this paper we settle the
computational complexity of sparse PCA with respect to the rank of the
covariance matrix. We show that, if the rank of the covariance matrix is a
fixed value, then there is an algorithm that solves sparse PCA to global
optimality, whose running time is polynomial in the number of features. We also
prove a similar result for the version of sparse PCA which requires the
principal components to have disjoint supports.
【13】 An Input-to-State Safety Approach to Anomaly-Resilient Parabolic PDEs: Application to Cyber-Physical Battery Modules
标题：异常弹性抛物型偏微分方程的输入到状态安全方法：在网络物理电池模块中的应用
链接：https://arxiv.org/abs/2201.02239
摘要：Distributed Parameter Cyber-Physical Systems (DPCPSs), modelled by Partial
Differential Equations (PDEs), are increasingly vulnerable to anomalies such as
physical faults as well as cyber-attacks. This motivates the need for
strategies towards anomaly-resilient control of these systems. Although anomaly
detection and diagnostics in PDE systems have received considerable attention
in existing literature, fault-tolerant or anomaly-resilient control for PDEs
remains relatively under-explored. However, given the vulnerabilities of these
systems against anomalies, it is essential that the control systems possess
resilience against these disruptions. In this context, we explore a Practical
Input-to-Safety (pISSf) based control design approach for a class of DPCPSs
modelled by linear Parabolic PDEs. Specifically, we develop a design framework
for anomaly-resilient control for this class of system with both safety and
stability guarantees based on control Lyapunov functional and control barrier
functional. To illustrate our methodology, we apply our strategy to design a
thermal-anomaly resilient boundary coolant control system for a cyber-physical
battery module. Several simulation studies are done to show the efficacy of our
method under anomalies such as mechanical battery degradation and cyber-attack
mediated overdischarge.
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