Relaxed Stability Criteria for Time-Delay Systems:A Novel Quadratic Function Convex Approximation Approach

This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.

Heat transport of hybrid nanomaterial in an annulus with quadratic Boussinesq approximation

The convective heat transfer of hybrid nanoliquids within a concentric annulus has wide engineering applications such as chemical industries, solar collectors, gas turbines, heat exchangers, nuclear reactors, and electronic component cooling due to their high heat transport rate. Hence, in this study, the characteristics of the heat transport mechanism in an annulus filled with the Ag-MgO/H_2O hybrid nanoliquid under the influence of quadratic thermal radiation and quadratic convection are analyzed. The nonuniform heat source/sink and induced magnetic field mechanisms are used to govern the basic equations concerning the transport of the composite nanoliquid. The dependency of the Nusselt number on the effective parameters(thermal radiation, nonlinear convection,and temperature-dependent heat source/sink parameter) is examined through sensitivity analyses based on the response surface methodology(RSM) and the face-centered central composite design(CCD). The heat transport of the composite nanoliquid for the spacerelated heat source/sink is observed to be higher than that for the temperature-related heat source/sink. The mechanisms of quadratic convection and quadratic thermal radiation are favorable for the momentum of the nanoliquid. The heat transport rate is more sensitive towards quadratic thermal radiation.

Stochastic level-value approximation for quadratic integer convex programming

We propose a stochastic level value approximation method for a quadratic integer convex minimizing problem in this paper. This method applies an importance sampling technique, and make use of the cross-entropy method to update the sample density functions. We also prove the asymptotic convergence of this algorithm, and report some numerical results to illuminate its effectiveness.

EXISTENCE AND LOCAL BEHAVIOR OF NONDIAGONAL BIVARIATE QUADRATIC FUNCTION APPROXIMATION

This paper analyses the local behavior of the simple off-diagonal bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin.It is shown that the simple off-diagonal bivariate quadratic Hermite-Padé form always defines a bivariate quadratic function and that this function is analytic in a neighbourhood of the origin.Numerical examples compare the obtained results with the approximation power of diagonal Chisholm approximant and Taylor polynomial approximant.

The Existence and Local Behavior of the Bivariate Quadratic Function Approximation

This paper analysis the local behavior of the bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin. It is shown that the bivariate quadratic Hermite-Padé form always defines a bivariate quadratic function and that this function is analytic in a neighborhood of the origin.

An Overview of Sequential Approximation in Topology Optimization of Continuum Structure

This paper offers an extensive overview of the utilization of sequential approximate optimization approaches in the context of numerically simulated large-scale continuum structures.These structures,commonly encountered in engineering applications,often involve complex objective and constraint functions that cannot be readily expressed as explicit functions of the design variables.As a result,sequential approximation techniques have emerged as the preferred strategy for addressing a wide array of topology optimization challenges.Over the past several decades,topology optimization methods have been advanced remarkably and successfully applied to solve engineering problems incorporating diverse physical backgrounds.In comparison to the large-scale equation solution,sensitivity analysis,graphics post-processing,etc.,the progress of the sequential approximation functions and their corresponding optimizersmake sluggish progress.Researchers,particularly novices,pay special attention to their difficulties with a particular problem.Thus,this paper provides an overview of sequential approximation functions,related literature on topology optimization methods,and their applications.Starting from optimality criteria and sequential linear programming,the other sequential approximate optimizations are introduced by employing Taylor expansion and intervening variables.In addition,recent advancements have led to the emergence of approaches such as Augmented Lagrange,sequential approximate integer,and non-gradient approximation are also introduced.By highlighting real-world applications and case studies,the paper not only demonstrates the practical relevance of these methods but also underscores the need for continued exploration in this area.Furthermore,to provide a comprehensive overview,this paper offers several novel developments that aim to illuminate potential directions for future research.

Stochastic resonance in a single-mode laser driven by quadratic pump noise and amplitude-modulated signal

This paper investigates the phenomenon of stochastic resonance in a single-mode laser driven by quadratic pump noise and amplitude-modulated signal. A new linear approximation approach is advanced to calculate the signal-to-noise ratio. In the linear approximation only the drift term is linearized, the multiplicative noise term is unchangeable. It is found that there appears not only the standard form of stochastic resonance but also the broad sense of stochastic resonance, especially stochastic multiresonance appears in the curve of signal-to-noise ratio as a function of coupling strength A between the real and imaginary parts of the pump noise.

A branch-and-bound algorithm for multi-dimensional quadratic 0-1 knapsack problems

In this paper, a branch-and-bound method for solving multi-dimensional quadratic 0-1 knapsack problems was studied. The method was based on the Lagrangian relaxation and the surrogate constraint technique for finding feasible solutions. The Lagrangian relaxations were solved with the maximum-flow algorithm and the Lagrangian bounds was determined with the outer approximation method. Computational results show the efficiency of the proposed method for multi-dimensional quadratic 0-1 knapsack problems.

Wavelet-Based Fractal Function Approximation

In this paper, we study on the application of radical B-spline wavelet scaling function in fractal function approximation system. The paper proposes a wavelet-based fractal function approximation algorithm in which the coefficients can be determined by solving a convex quadraticprogramming problem. And the experiment result shows that the approximation error of this algorithm is smaller than that of the polynomial-based fractal function approximation. This newalgorithm exploits the consistency between fractal and scaling function in multi-scale and multiresolution, has a better approximation effect and high potential in data compression, especially inimage compression.

WAVELET KERNEL SUPPORT VECTOR MACHINES FOR SPARSE APPROXIMATION

Wavelet, a powerful tool for signal processing, can be used to approximate the target func-tion. For enhancing the sparse property of wavelet approximation, a new algorithm was proposed by using wavelet kernel Support Vector Machines (SVM), which can converge to minimum error with bet-ter sparsity. Here, wavelet functions would be firstly used to construct the admitted kernel for SVM according to Mercy theory; then new SVM with this kernel can be used to approximate the target fun-citon with better sparsity than wavelet approxiamtion itself. The results obtained by our simulation ex-periment show the feasibility and validity of wavelet kernel support vector machines.

A quadratic programming method for optimal degree reduction of Bézier curves with G^1-continuity

This paper presents a quadratic programming method for optimal multi-degree reduction of B6zier curves with G^1-continuity. The L2 and I2 measures of distances between the two curves are used as the objective functions. The two additional parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applications of degree reduction of Bezier curves with their parameterizations close to arc-length parameterizations are also discussed.

互换期权的近似计算方法在利率模型Quadratic Gaussian ++上的应用

对Quadratic Gaussian++这样的复杂利率模型,其互换期权价值的高速计算既是本身定价的需要,也是校正利率模型参数的需要。本文应用互换期权价值计算的近似方法,通过测度变换和Taylor展开,将复杂的期望值计算转化为正态分布变量的二次多项式形式的期望值计算,既获得了足够实用的计算精度(小于3.5%),又达到高速计算的要求(比直接数值积分快了约200倍),能够在实际计算中发挥作用。

METHOD BASED ON DUAL-QUADRATIC PROGRAMMING FOR FRAME STRUCTURAL OPTIMIZATION WITH LARGE SCALE

The optimality criteria （OC） method and mathematical programming （MP） were combined to found the sectional optimization model of frame structures. Different methods were adopted to deal with the different constraints. The stress constraints as local constraints were approached by zero-order approximation and transformed into movable sectional lower limits with the full stress criterion. The displacement constraints as global constraints were transformed into explicit expressions with the unit virtual load method. Thus an approximate explicit model for the sectional optimization of frame structures was built with stress and displacement constraints. To improve the resolution efficiency, the dual-quadratic programming was adopted to transform the original optimization model into a dual problem according to the dual theory and solved iteratively in its dual space. A method called approximate scaling step was adopted to reduce computations and smooth the iterative process. Negative constraints were deleted to reduce the size of the optimization model. With MSC/Nastran software as structural solver and MSC/Patran software as developing platform, the sectional optimization software of frame structures was accomplished, considering stress and displacement constraints. The examples show that the efficiency and accuracy are improved.

二次及三次Hermite-Padé逼近中的几个问题

给出了二次Hermite-Pad啨逼近的对偶性.证明了三次Hermite-Pad啨逼近的局部唯一性,并对其逼近阶进行估计.

Approximation bounds for quadratic maximization and max-cut problems with semidefinite programming relaxation

In this paper,we consider a class of quadratic maximization problems.For a subclass of the problems,we show that the SDP relaxation approach yields an approximation solution with the worst-case performance ratio at leastα=0.87856….In fact,the estimated worst-case performance ratio is dependent on the data of the problem withαbeing a uniform lower bound.In light of this new bound,we show that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at leastα+δ_d if every weight is strictly positive,whereδ_d>0 is a constant depending on the problem dimension and data.

Semidefinite Relaxation for Two Mixed Binary Quadratically Constrained Quadratic Programs:Algorithms and Approximation Bounds

This paper develops new semidefinite programming(SDP)relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs and analyzes their approximation performance.The first class of problems finds two minimum norm vectors in N-dimensional real or complex Euclidean space,such that M out of 2M concave quadratic constraints are satisfied.By employing a special randomized rounding procedure,we show that the ratio between the norm of the optimal solution of this model and its SDP relaxation is upper bounded by 54πM2 in the real case and by 24√Mπin the complex case.The second class of problems finds a series of minimum norm vectors subject to a set of quadratic constraints and cardinality constraints with both binary and continuous variables.We show that in this case the approximation ratio is also bounded and independent of problem dimension for both the real and the complex cases.

凸近似避障及采样区优化的智能车辆轨迹规划

针对结构化道路下作匀速运动的智能车辆避障轨迹规划问题,提出一种基于凸近似避障原理及采样区域优化的智能车辆轨迹规划方法。引入凸近似避障原理,得到轨迹可行域范围;将采样区域分为静态采样区、动态采样区两部分,并根据障碍物运动状态,另外划分动态、静态障碍物采样区;采用“动态规划(DP)+二次规划(QP)”思想求解轨迹:利用五次多项式对采样点依次连接,建立动态规划代价函数并筛选得到粗略轨迹;通过二次规划及约束条件的构造,对粗略轨迹进行平滑,最终得到最优轨迹。仿真结果表明:对于静态、低速、动态三种障碍物,该车能够有效地得到平滑轨迹并避开障碍物。

基于二次型迭代逼近法的电力系统电压鞍结分岔点识别

为实现负荷增长过程中电力系统鞍结分岔点(SNB)的快速准确识别,提出一种直接计算电力系统电压崩溃点的二次型迭代逼近方法,基于系统中PQ节点输出的PV曲线为近似二次型的特点,在节点功率平衡方程中引入负荷增长参数,运用复合函数求导法则就功率方程进行两次求导,理论推导节点电压对负荷参数的一阶、二阶导数表达式,由此确定PV曲线二项式,依靠顶点坐标确定电力系统鞍结分岔点的初始位置,经多次迭代收敛逼近电压崩溃点。所提方法避免了连续潮流法的多次潮流计算,可显著降低计算量。以IEEE 14,IEEE 118节点系统进行仿真验证,证明了该方法的有效性,相较增补P’Q节点法及戴维南等值法,二次型迭代逼近法具有较高的计算效率和鲁棒性。

A Quadratically Approximate Framework for Constrained Optimization,Global and Local Convergence

This paper presents a quadratically approximate algorithm framework （QAAF） for solving general constrained optimization problems, which solves, at each iteration, a subproblem with quadratic objective function and quadratic equality together with inequality constraints. The global convergence of the algorithm framework is presented under the Mangasarian-Fromovitz constraint qualification （MFCQ）, and the conditions for superlinear and quadratic convergence of the algorithm framework are given under the MFCQ, the constant rank constraint qualification （CRCQ） as well as the strong second-order sufficiency conditions （SSOSC）. As an incidental result, the definition of an approximate KKT point is brought forward, and the global convergence of a sequence of approximate KKT points is analysed.

WONG-ZAKAI APPROXIMATIONS FOR STOCHASTIC VOLTERRA EQUATIONS

In this paper,we shall prove a Wong-Zakai approximation for stochastic Volterra equations under appropriate assumptions.We may apply it to a class of stochastic differential equations with the kernel of fractional Brownian motion with Hurst parameter H∈(1/2,1)and subfractional Brownian motion with Hurst parameter H∈(1/2,1).As far as we know,this is the first result on stochastic Volterra equations in this topic.