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This paper investigates the problem of stability and state-feedback control design for linear parameter-varying systems with time-varying delays. The uncertain parameters are assumed to belong to a polytope with bounded known variation rates. The new conditions are based on the Lyapunov theory and are expressed through Linear Matrix Inequalities. An alternative parameter-dependent Lyapunov-Krasovskii functional is employed and its time-derivative is handled using recent integral inequalities for quadratic functions proposed in the literature. As main results, a novel sufficient stability condition for delay-dependent systems as well as a new sufficient condition to design gain-scheduled state-feedback controllers are stated. In the new proposed methodology, the Lyapunov matrices and the system matrices are put separated making it suitable for supporting in a new way the design of the stabilization controller. An example, based on a model of a real-world problem, is provided to illustrate the effectiveness of the proposed method.

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

We introduce a family of weighted conjugate-gradient-type methods, for strictly convex quadratic functions, whose parameters are determined by a minimization model based on a convex combination of the objective function and its gradient norm. This family includes the classical linear conjugate gradient method and the recently published delayed weighted gradient method as the extreme cases of the convex combination. The inner cases produce a merit function that offers a compromise between function-value reduction and stationarity which is convenient for real applications. We show that each one of the infinitely many members of the family exhibits q-linear convergence to the unique solution. Moreover, each one of them enjoys finite termination and an optimality property related to the combined merit function. In particular, we prove that if the n × n Hessian of the quadratic function has p < n different eigenvalues, then each member of the family obtains the unique global minimizer in exactly p iterations. Numerical results are presented that demonstrate that the proposed family is promising and exhibits a fast convergence behavior which motivates the use of preconditioning strategies, as well as its extension to the numerical solution of general unconstrained optimization problems.

Gradient method is a simple optimization approach using minus gradient of the objective function as a search direction. Its efficiency highly relies on the choices of the stepsize. In this paper, the convergence behavior of a class of gradient methods, where the stepsize has an important property introduced in (Dai in Optimization 52:395–415, 2003), is analyzed. Our analysis is focused on minimization on strictly convex quadratic functions. We establish the R-linear convergence and derive an estimate for the R-factor. Specifically, if the stepsize can be expressed as a collection of Rayleigh quotient of the inverse Hessian matrix, we are able to show that these methods converge R-linearly and their R-factors are bounded above by $$1-\frac{1}{\varkappa }$$ , where $$\varkappa$$ is the associated condition number. Preliminary numerical results demonstrate the tightness of our estimate of the R-factor.

This article concerns the zero-Hopf bifurcation of a quadratic polynomial differential system in $\mathbb {R}^{4}$ . By using the averaging theory of third order, we provide that at most 25 limit cycles can bifurcate from one singularity with eigenvalues of the form ± bi, 0 and 0.

In this paper, we focus on the synchronization of fractional-order coupled neural networks (FCNNs). First, by taking information on activation functions into account, we construct a convex Lur’e–Postnikov Lyapunov function. Based on the convex Lyapunov function and a general convex quadratic function, we derive a novel Mittag-Leffler synchronization criterion for the FCNNs with symmetrical coupled matrix in the form of linear matrix inequalities (LMIs). Then we present a robust Mittag-Leffler synchronization criterion for the FCNNs with uncertain parameters. These two Mittag-Leffler synchronization criteria can be solved easily by LMI tools in Matlab. Moreover, we present a novel Lyapunov synchronization criterion for the FCNNs with unsymmetrical coupled matrix in the form of LMIs, which can be easily solved by YALMIP tools in Matlab. The feasibilities of the criteria obtained in this paper are shown by four numerical examples.

Made available in DSpace on 2021-07-14T10:21:14Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-06-25. Added 1 bitstream(s) on 2021-07-14T11:27:56Z : No. of bitstreams: 1 S0100-69162021000300319.pdf: 915553 bytes, checksum: 032aa323077110649b223df7fe9ee77a (MD5) In order to estimate the response of biometric variables in different irrigation depths in radish crop, as well as their relations in the development of the crop, a fuzzy mathematical analysis was carried out from irrigation with depths of different percentages of the crop evapotranspiration (ETc), using Gaussian pertinence functions for the input variable and triangular for the biometric output variables. Validations were performed using neural network models, smoothing splines and polynomial regression. The relation among the biometric variables was measured applying the Pearson correlation coefficient. The results showed that the fuzzy modeling presented superiority in the crop development estimate over the quadratic polynomial regression model, neural network and smoothing splines, because it achieved an average reduction of errors among the biometric variables, of 7.8% 94.6% and 9.2% for the RMSE in the respective models, as well as a better adjustment of the data with average R2 of the variables. The modeling with neural network showed inadequate agronomic behavior in data representation. Regarding biometric variables, the length and diameter of the tuberous root are inversely correlated, and the fresh phytomass of the tuberous root is correlated only with the fresh phytomass of the root. São Paulo State University, School of Agriculture São Paulo State University, School of Sciences and Engineering São Paulo State University, School of Agriculture São Paulo State University, School of Sciences and Engineering

The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete description about the existence or nonexistence of solution in terms of the (strict) copositivity of the matrix involved in the integrand. In addition, we characterize optimality via the Hamiltonian function.

The extremely useful superelastic behavior of NiTi has been poorly explored because of the limited number of models that can describe the complete hysteretic behavior of NiTi, including a superelastic condition that strongly depends on the applied stress. This paper presents the development of a stress-dependent phenomenological model of NiTi by modifying the existing generalized Prandtl–Ishlinskii (GPI) model. The parameters of the envelop function of the GPI model’s play operator are reformulated as quadratic functions of the applied stress. The stress-dependent GPI model can satisfactorily predict the output strain of a NiTi #6 wire under temperature and stress variation.

This paper is concerned with the motor skill learning from human demonstrations using the framework of dynamic regressive models (DRMs). The DRM-based motion planner is preferred as it generates the end-effector trajectory dynamically based on the current state of the end-effector. Within existing frameworks, a single DRM can learn a single motion profile. In addition, such learned DRMs from the data may not be stable. This paper addresses both these issues in a comprehensive manner. In this paper a single DRM has been used to encode human demonstrations involving multitask profiles and multiple task-equilibriums which is novel. We have introduced the idea that the learned DRM will generate human-like stable motion if the energy dissipation rate (EDR) of the generated trajectory follows that of the human demonstration. Thus, the DRM structure has been modified by adding a continuous guiding signal which can be called as the control signal. This signal has been derived using control theoretic principle to ensure asymptotic stability while maintaining the EDR equivalent to that of the human demonstration. The asymptotic stability of the learned DRM has been established by involving a nonmonotonic Lyapunov function consisting of first derivative of a quadratic function and the energy function associated with the DRM. The proposed framework can be learned using many existing regression techniques in this paper Gaussian mixture regression , locally weighted projection regression , and support vector regression techniques have been used successfully. During the pick and place tasks, human demonstrations involving multiple task profiles and multiple task-equilibriums are generated using a 7 DOF commercial robot manipulator. Experimental validations show that the DRMs learned using these three regression schemes are able to guide the robot along the multitask profiles in a stable manner.

We study the global picture of the metric entropy on the space of expanding Blaschke products. We first construct a smooth path in the space tending to a parabolic Blaschke product. We prove that the metric entropy on this path tends to 0 as the path tends to this parabolic Blaschke product. It turns out that the limiting parabolic Blaschke product on the unit circle is conjugate to the famous Boole map on the real line. Thus we can give a new explanation of Boole's formula discovered more than one hundred and fifty years ago. We modify the first smooth path to get a second smooth path in the space of expanding Blaschke products. The second smooth path tends to a totally degenerate map. We see that the first and second smooth paths have completely different asymptotic behaviors near the boundary of the space of expanding Blaschke products. However, they represent the same smooth path in the space of all smooth conjugacy classes of expanding Blaschke products. We use this to give a complete description of the global graph of the metric entropy on the space of expanding Blaschke products. We prove that the global graph looks like a bell. It is the first result to show a global picture of the metric entropy on a space of hyperbolic dynamical systems. We apply our results to the measure-theoretic entropy of a quadratic polynomial with respect to its Gibbs measure on its Julia set. We prove that the measure-theoretic entropy on the main cardioid of the Mandelbrot set is a real analytic function and asymptotically zero near the boundary.

In this article, the repetitive finite-length linear discrete-time singular system is formulated as an input-output equation by virtue of the lifted-vector method and a gain-optimized P-type iterative learning control profile is architected by sequentially arguing the learning-gain vector in minimizing the addition of the quadratic norm of the tracking-error vector and the weighed quadratic norm of the compensation vector. By virtue of the elementary permutation matrix and the property of the quadratic function, the optimized-gain vector is solved and explicitly expressed by the system Markov matrix and the iteration-wise tracking error. Then the linearly monotonic convergence of the tracking error is derived under the assumption that the initial state of the dynamic subsystem is resettable. Furthermore, for the circumstance that the system parameters uncertainties exist, the quasi scheme is established by replacing the exact system Markov matrix with the approximated one in the optimized gain. Rigorous analysis conveys that the proposed gain-optimized scheme is robust to the system internal disturbance within a suitable range. The validity and effectiveness are demonstrated numerically.

We theoretically analyze the information geometric optimization (IGO), which is a unified framework of stochastic search algorithms for black-box optimization. The IGO framework has two parameters: 1) the learning rate and 2) the sample size, and they influence the behavior of the algorithm. We investigate the strategy parameters of the IGO with the family of isotropic Gaussian distributions on a general convex quadratic function. Compared to the previous theoretical works, where an infinite sample size is assumed and the deterministic algorithm dynamics is studied, we investigate the expected improvement of the algorithm with a finite sample size. The analysis finds that the relative decrease rates of the distance from the distribution mean to the landscape optimum and the distribution standard deviation must be the same, which we observe in practice, while the analysis based on an infinite sample size failed to obtain. We derive these rates explicitly as a function of the eigenvalues of the Hessian of the objective function and the strategy parameters. We also derive the stable value of the ratio of the square distance to the optimum over the distribution variance, as well as the conditions that the stable value exists. These theoretical values coincide with our numerical simulations.

This study presents a framework for the enhancement of parameter estimation issues by computing the time average of several quantum observables measured under the evolution of the Hamiltonian system. These time-cum-quantum averages are expressed as a quadratic function of the orthogonal projections on the eigen subspaces of the system. Estimation of the parameters is done using the approximate time-independent perturbation theory assuming the other parameters to be small with respect to one of the parameters that is normalised to unity in the presence of noise. The study also proves that the derived expression for the mean square parameter estimation error attains the quantum limit. When the measurement noise is classical and for general positive operator-valued measure successive time, the authors derive an expression for the quantum Cramer–Rao lower bound.

This article presents the exact evaluation of the physical optics (PO) integral in time-domain. The surface of a perfect electrically conducting (PEC) scatterer is modeled by quadratic triangles. Using the Radon transform interpretation, the PO integral is reduced to a line integral, which is formed by the intersection of the quadratic surface and the plane defined by the Dirac delta function. It is shown that the resulting line is a quadratic curve, e.g., an ellipse, in barycentric coordinates of the quadratic triangle, since the incident wave is a plane wave and the scattered fields are observed at the far-field. To evaluate the PO integral: 1) the PO integral is represented in barycentric coordinates and the type of the intersecting curve is determined; 2) an appropriate coordinate transformation is applied, e.g., elliptic coordinates for the ellipse; and then 3) the Gauss–Legendre quadrature rule (GLQR) is applied. It is shown that a suitable GLQR order yields the PO integral exactly. The validity and accuracy of the proposed method are demonstrated via numerical examples.

In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’ actions, but with delays; and (b) the second one, which we term the noncooperative scenario, where the players have access only to their own payoff values, again with delay. Both approaches are based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal excitation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In order to compensate distinct delays in the inputs of the players, we have employed predictor feedback. We apply a small-gain analysis as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results for the unknown quadratic payoffs to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.

Funding Information: Roberto Andreani was financially supported by FAPESP (Projects 2013/05475-7 and 2017/18308-2) and CNPq (Project 301888/2017-5). Marcos Raydan was financially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Project UIDB/MAT/00297/2020 (Centro de Matemática e Aplicações). Roberto Andreani would like to thank the Operations Research Group at CMA (Centro de Matemática e Aplicações), FCT, NOVA University of Lisbon, Portugal, for the hospitality during a two-week visit in December 2019. Funding Information: We would like to thank two anonymous referees for their comments and suggestions that helped us to improve the final version of this paper. Roberto Andreani was financially supported by FAPESP (Projects 2013/05475-7 and 2017/18308-2) and CNPq (Project 301888/2017-5). Marcos Raydan was financially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Project UIDB/MAT/00297/2020 (Centro de Matemática e Aplicações). Roberto Andreani would like to thank the Operations Research Group at CMA (Centro de Matemática e Aplicações), FCT, NOVA University of Lisbon, Portugal, for the hospitality during a two-week visit in December 2019. Publisher Copyright: © 2020, Springer Science+Business Media, LLC, part of Springer Nature. The delayed weighted gradient method, recently introduced in Oviedo-Leon (Comput Optim Appl 74:729–746, 2019), is a low-cost gradient-type method that exhibits a surprisingly and perhaps unexpected fast convergence behavior that competes favorably with the well-known conjugate gradient method for the minimization of convex quadratic functions. In this work, we establish several orthogonality properties that add understanding to the practical behavior of the method, including its finite termination. We show that if the n× n real Hessian matrix of the quadratic function has only p< n distinct eigenvalues, then the method terminates in p iterations. We also establish an optimality condition, concerning the gradient norm, that motivates the future use of this novel scheme when low precision is required for the minimization of non-quadratic functions. authorsversion published

The phenomenon of wave set-up may substantially contribute to the formation of devastating coastal flooding in certain coastal areas. We study the appearance and properties of empirical probability density distributions of the occurrence of different set-up heights on an approximately 80 km long section of coastline near Tallinn in the Gulf of Finland, eastern Baltic Sea. The study area is often attacked by high waves propagating from various directions, and the typical approach angle of high waves varies considerably along the shore. The distributions in question are approximated by an exponential distribution with a quadratic polynomial as the exponent. Even though different segments of the study area have substantially different wave regimes, the leading term of this polynomial is usually small (between −0.005 and 0.005) and varies insignificantly along the study area. Consequently, the distribution of wave set-up heights substantially deviates from a Rayleigh or Weibull distribution (that usually reflect the distribution of different wave heights). In about three-quarters of the occasions, it is fairly well approximated by a standard exponential distribution. In about 25 % of the coastal segments, it qualitatively matches a Wald (inverse Gaussian) distribution. The Kolmogorov–Smirnov test (D value) indicates that the inverse Gaussian distribution systematically better matches the empirical probability distributions of set-up heights than the Weibull, exponential, or Gaussian distributions.

In this paper, the spherical quasi-convexity of quadratic functions on spherically subdual convex sets is studied. Sufficient conditions for spherical quasi-convexity on spherically subdual convex sets are presented. A partial characterization of spherical quasi-convexity on spherical Lorentz sets is given, and some examples are provided.

This paper addresses the analysis of covariance matrix self-adaptive Evolution Strategies (CMSA-ES) on a subclass of quadratic functions subject to additive Gaussian noise: the noisy ellipsoid model. To this end, it is demonstrated that the dynamical systems approach from the context of isotropic mutations can be transferred to ES that also control the covariance matrix. Theoretical findings such as the component-wise quadratic progress rate or the self-adaptation response function can thus be reused for the CMSA-ES analysis. By deriving the steady state quantities approached on the noisy ellipsoid model for constant population size, a detailed description of the asymptotic CMSA-ES behavior is obtained. By providing self-adaptive ES with a population control mechanism, despite noise disturbances, the algorithm is able to realize continuing progress towards the optimum. Regarding the population control CMSA-ES (pcCMSA-ES), the analytical findings allow to specify its asymptotic long-term behavior and to identify influencing parameters. The finally obtained convergence rate matches the theoretical lower bound of all comparison-based direct search algorithms.

This work is based on the observation that the first Poincaré–Lyapunov constant is a quadratic function of the coefficients of the two-dimensional vector field at a Hopf bifurcation. From a given parameter point, we find the distance to the “Hopf quadric.” This distance provides a measure of the criticality of the Hopf bifurcation. The viability of the approach is demonstrated through numerical examples.

Energy consumption and construction period in the construction process have been the focus of experts and researchers in the industry. Income by energy consumption and construction period is the long-term goal of industry managers. However, random events on the construction site are unpredictable, which have a random impact on energy consumption, construction period and its benefits. Therefore, a multi-objective stochastic optimization algorithm is designed to analyze the energy consumption, construction period and its benefits in the construction process, considering the random events. By introducing the idea of Lyapunov drift plus penalty, the construction process is simulated as a network transmission model, starting from the begin of the construction, and a series of construction sections are used as transmission. In this way, a virtual construction queue is constructed, Lyapunov quadratic function is defined, drift function is constructed, and the greedy objective function is punished to the minimum. Through the simulation process, the practicability of the algorithm is verified under the creation of several random events. The simulation accuracy and benefit value discrimination of the algorithm are higher comparing with the situation before optimization.

We propose and experimental demonstrate a Dynamic Threshold Identification Method (DTIM) based on the Raman distributed fiber sensor (RDFS) system for detecting the pipeline leak. The DTIM can detect the position of pipeline leak according to distributed temperature measured by the RDFS. The DTIM includes three steps, which are the space segmentation detection, $3\sigma $ criterion and identification of historical data. The experimental results show that the proposed RDFS can detect and accurately locate the leak position by using the DTIM, and the positioning accuracy is 1 m. In addition, the temperature change rate of the leak position performs a quadratic function relationship with the leak rate, and the leak rate can be estimated by this method. It provides an effective distributed monitoring method for the leak monitoring of pipeline.

In this paper, a new optimal nonlinear controller is proposed for different classes of nonlinear systems using black hole (BH) optimization method. The different classes presented by these nonlinear systems include, time-varying nonlinear system with external disturbance, non-affine nonlinear strict-feedback system with full state constraints and uncertain nonlinear system. First, the nonlinear systems are analyzed using Lyapunov quadratic function. Second, the nonlinear controller is designed to guarantee the stability of the system after using a model reference to achieve the required performance. Then, the Integral Square Error performance index is used as a cost function to minimize the error between the reference output and the actual output of the proposed systems to zero. Afterward, the BH optimization method is applied as a new powerful optimization method to find the optimal parameters for the proposed nonlinear controller. Finally, the simulation results that the asymptotic stability and optimal performance are achieved with a lowest possible control action. In addition, it shows the efficiency of the proposed nonlinear controller for compensating different nonlinear systems problems.

It is proved that any convex viscosity solution of det ⁡ D 2 u = 1 outside a bounded domain of the half space is asymptotic to a quadratic polynomial at infinity under reasonable assumptions, where the asymptotic rate is the same as the Poisson kernel of the half space. Consequently, it follows the Liouville type theorem on Monge-Ampere equation in the half space. Meanwhile, it is established the existence theorem for the Dirichlet problem with prescribed asymptotic behavior at infinity.

In this paper, we propose a fast and stable evaluation method for the analytic derivatives of splines generated by the 7-direction quartic box-spline. We can maintain the spline structure by determining the derivative functions that can be represented as finite differences of box-splines defined by the sub-directions. Thus, the evaluation overhead can be reduced. We demonstrate that the first and second derivative functions are composed of only three cubic and six quadratic polynomial formulas, respectively, owing to their symmetries. Moreover, for each derivative order, all of the required functions possess change-of-variables relation with each other. Therefore, additional formulas are not required. As a result, for a given point, we only need to evaluate one quartic, three cubic, and six quadratic polynomial formulas to evaluate its spline value, gradient, and Hessian, respectively. This reduction in cases is especially advantageous for graphics processing unit (GPU) kernels, where conditional statements significantly degrade performance. To verify our technique, we implemented a real-time curvature-based GPU isosurface raycaster. Compared with other implementations, our method (i) achieves superior accuracy, (ii) is more than four times faster, and (iii) requires less than 15% of memory.

Considering periodic environmental changes and random disturbance, we explore the dynamical behaviors of a stochastic competitive system with impulsive and periodic parameters in this paper. Firstly, by use of extreme-value theory of quadratic function and constructing suitable functional, we study the existence of periodic Markovian process. Secondly, by comparison theory of the stochastic differential equation, we study the extinction and permanence in the mean of all species. Thirdly, applying an important lemma, we investigate the stochastic persistence of this system. Finally, some numerical simulations are given to illustrate the main results.

In this paper, a receiving scheme for intermediate frequency (IF) signals enhanced by stochastic resonance (SR) is proposed. The proposed scheme mitigates the reception failure of these signals, which can occur in radio and communication systems under extremely low signal-to-noise ratio (SNR). The SR mechanism for enhancing sinusoidal signals is analyzed. An analytic solution with time parameters of the Fokker-Planck Equation (FPE) is obtained by introducing the decision time from the non-autonomous FPE into an autonomous one. A quadratic polynomial receiving structure for sinusoidal signals enhanced by SR is proposed by comparing the characteristics of energy detection and matched filter detection. And the polynomial coefficients of the quadratic system are obtained by maximizing the deflection. Based on the idea of “the average of N samples”and the assumption of Gaussian distribution approximation under the law of large numbers, a quadratic polynomial receiving scheme for sinusoidal signals enhanced by SR is proposed. The conclusions are as below: 1) when the noise intensity is constant, the smaller the correlation time, the bigger the local SNR around the IF frequency due to the better performance of the low-pass filter; 2) The error bit ratio of the quadratic polynomial receiver is less than 1 × 10-2 when N = 20 and the SNR is above -14 dB, which can be applied to the military emergency communication under extremely low SNR. Experiment verifies the theory.

Interior point methods have attracted most of the attention in the recent decades for solving large scale convex quadratic programming problems. In this paper we take a different route as we present an augmented Lagrangian method for convex quadratic programming based on recent developments for nonlinear programming. In our approach, box constraints are penalized while equality constraints are kept within the subproblems. The motivation for this approach is that Newton’s method can be efficient for minimizing a piecewise quadratic function. Moreover, since augmented Lagrangian methods do not rely on proximity to the central path, some of the inherent difficulties in interior point methods can be avoided. In addition, a good starting point can be easily exploited, which can be relevant for solving subproblems arising from sequential quadratic programming, in sensitivity analysis and in branch and bound techniques. We prove well-definedness and finite convergence of the method proposed. Numerical experiments on separable strictly convex quadratic problems formulated from the Netlib collection show that our method can be competitive with interior point methods, in particular when a good initial point is available and a second-order Lagrange multiplier update is used.

In different spheres of science and technology almost any appearing complicated problem of the optimal selection is multicriterion, since in course of searching for the best alternative we have to take into consideration a number of different requirements and some of them may even contradict each other. Meanwhile in practice linear and quadratic functions are the most frequently applied as criterion ones. They enable us to describe processes under investigation rather adequately and to apply known and studied algorithms for solving the problems. The first methods used for solving multicriterion problems of optimization were the methods based on the approach of reducing an input problem to a one-criterion one. However, the given procedure in most cases leads to dramatic changes of properties of an input problem and thus to unjustified replacing of a multicriterion model of the problem by a one-criterion model. We also should remember about calculation difficulties caused by the loss of properties of an input problem and about the impossibility of applying known algorithms in order to solve corresponding one-criterion problems. So developing methods for solving vector problems of optimization, in which original properties of criterion and limitation functions are not lost, remains relevant. In terms of one-criterion optimization a number of algorithms for extremum searching are elaborated using the dual approach. This issue is also of great importance to multicriterion optimization problems. The article deals with convex quadratic problems of lexicographic optimization within the set prescribed by a linear inequality system and with the issue of drawing up dual problems to them. Dual problems to an original one are constructed by means of Lagrange’s reflection where Lagrange multipliers are vector variables and the value set of each of them is a set of vectors of the space, the dimensionality of which is equal to a quantity of partial criteria and which is prescribed by some lexicographic order. Necessary and sufficient conditions of the existence and optimality of lexicographic solutions of the derived problem are established. We suggest an approach that enables the reducing of the solving of a lexicographic optimization problem to the sequence of inequality and equation systems in lexicographic order. The scalarization scheme analogue and applying of properties of Lagrange’s reflection constructed for a vector function and limitations underlie the approach. There is also promising possibility of constructing calculation algorithms for solving a lexicographic problem of quadratic optimization which are based on the dual approach.

A Newton-type method is proposed for numerical minimization of convex piecewise quadratic functions, and its convergence is analyzed. Previously, a similar method was successfully applied to optimization problems arising in mesh generation. It is shown that the method is applicable to computing the projection of a given point onto the set of nonnegative solutions of a system of linear equations and to determining the distance between two convex polyhedra. The performance of the method is tested on a set of problems from the NETLIB repository.

This paper investigates Lyapunov and external stability of Caputo fractional order switching systems. It proves an estimation for the Caputo fractional order derivative of an arbitrary quadratic function whose variable is a function of time with the same differential properties as those given by the most recent smoothness result, and then establishes criteria on Lyapunov stability that bridge over existing difficulties. On the other hand, the equivalence between Caputo fractional order switching control systems and their diffusive realizations is shown. This together with a Lyapunov-like function proven to be well defined shows their advantages in solving the problem of external stability.

When studying systems with delay, a particular attention is paid to such important property as their stability. In investigation of stability of systems of ODEs by the direct Lyapunov method, quadratic Lyapunov functions are widely used. For a system with delay, we suggest to use as Lyapunov functions some quadratic Lyapunov functions constructed for a system of ODEs and satisfying some restrictions on the first derivative. For such systems we find conditions which make possible to use such quadratic functions as Lyapunov functions.

Reactor core power mapping can be regarded as an interpolation problem of scattered data which is common in many scientific and engineering fields, and the Radial Basis Function (RBF) method provides an excellent tool for interpolating multidimensional scattered data and thus an appropriate approach for power mapping. By using the dual form of the RBF interpolation expression, this study shows that the Ordinary Kriging (OK) method authors utilized before can be regarded as an instance of RBF framework. Moreover, Gaussian function, multiquadric function, inverse quadratic function and inverse multiquadric function have been implemented in RBF interpolation framework to carry out core power mapping. Measurements from DayaBay Unit 1 PWR are used to verify the accuracies of these four RBF functions, and comparisons are made among these four methods and the OK method. The reconstructed assembly power distribution results and the calculation speeds show that the RBF interpolation framework is suitable for on-line core power distribution monitoring.

We consider the asymptotic behavior of a family of gradient methods, which include the steepest descent and minimal gradient methods as special instances. It is proved that each method in the family will asymptotically zigzag between two directions. Asymptotic convergence results of the objective value, gradient norm, and stepsize are presented as well. To accelerate the family of gradient methods, we further exploit spectral properties of stepsizes to break the zigzagging pattern. In particular, a new stepsize is derived by imposing finite termination on minimizing two-dimensional strictly convex quadratic function. It is shown that, for the general quadratic function, the proposed stepsize asymptotically converges to the reciprocal of the largest eigenvalue of the Hessian. Furthermore, based on this spectral property, we propose a periodic gradient method by incorporating the Barzilai-Borwein method. Numerical comparisons with some recent successful gradient methods show that our new method is very promising., Comment: 28 pages, 7 figures

This chapter presents a continuous-time implementation of a reset control system that has a linear flow map, a linear jump map, and a quadratic function that describes the jump set. The implementation is a homogeneous differential inclusion that depends only on the data of the reset system and matches the flows of the reset system in the flow set. Assuming that the reset control system admits a strongly convex Lyapunov function that establishes stability of its origin, the continuous-time implementation has the origin globally exponentially stable. In particular, the continuous-time implementation eliminates purely discrete-time solutions of the reset system that do not converge. The behavior of the continuous-time implementation is illustrated through multiple examples.

Consider the random quadratic form $T_n=\sum_{1 \leq u < v \leq n} a_{uv} X_u X_v$, where $((a_{uv}))_{1 \leq u, v \leq n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal, and $X_1,$ $X_2, \ldots, X_n$ are i.i.d. $\mathrm{Ber}(p_n)$. In this paper, we prove various characterization theorems about the limiting distribution of $T_n$, in the sparse regime, where $0 < p_n \ll 1$ such that $\mathbb E(T_n)=O(1).$ The main result is a decomposition theorem showing that distributional limits of $T_n$ is the sum of three components: a mixture which consists of a quadratic function of independent Poisson variables; a linear Poisson mixture, where the mean of the mixture is itself a (possibly infinite) linear combination of independent Poisson random variables; and another independent Poisson component. This is accompanied with a universality result which allows us to replace the Bernoulli distribution with a large class of other discrete distributions. Another consequence of the general theorem is a necessary and sufficient condition for Poisson convergence, where an interesting second moment phenomenon emerges., 48 pages, 1 figure

The (1+1)-evolution strategy (ES) with success-based step-size adaptation is analyzed on a general convex quadratic function and its monotone transformation, that is, $f(x) = g((x - x^*)^\mathrm{T} H (x - x^*))$, where $g:\mathbb{R}\to\mathbb{R}$ is a strictly increasing function, $H$ is a positive-definite symmetric matrix, and $x^* \in \mathbb{R}^d$ is the optimal solution of $f$. The convergence rate, that is, the decrease rate of the distance from a search point $m_t$ to the optimal solution $x^*$, is proven to be in $O(\exp( - L / \mathrm{Tr}(H) ))$, where $L$ is the smallest eigenvalue of $H$ and $\mathrm{Tr}(H)$ is the trace of $H$. This result generalizes the known rate of $O(\exp(- 1/d ))$ for the case of $H = I_{d}$ ($I_d$ is the identity matrix of dimension $d$) and $O(\exp(- 1/ (d\cdot\xi) ))$ for the case of $H = \mathrm{diag}(\xi \cdot I_{d/2}, I_{d/2})$. To the best of our knowledge, this is the first study in which the convergence rate of the (1+1)-ES is derived explicitly and rigorously on a general convex quadratic function, which depicts the impact of the distribution of the eigenvalues in the Hessian $H$ on the optimization and not only the impact of the condition number of $H$., Comment: 17 pages

For the incompressible Euler equations the pressure formally scales as a quadratic function of velocity. We provide several optimal regularity estimates on the pressure by using regularity of velocity in various Sobolev, Besov and Hardy spaces. Our proof exploits the incompressibility condition in an essential way and is deeply connected with the classic Div-Curl lemma which we also generalise as a fractional Leibniz rule in Hardy spaces. To showcase the sharpness of results, we construct a class of counterexamples at several end-points., 23 pages

We propose a non-model based strategy for locally stable convergence to Nash equilibria in quadratic noncooperative (duopoly) games with player actions subject to diffusion (heat) PDE dynamics with distinct diffusion coefficients and each player having access only to his own payoff value. The proposed approach employs extremum seeking, with sinusoidal perturbation signals employed to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In our previous work, we solved Nash equilibrium seeking problems with input delays. This is the first instance of noncooperative games being tackled in a model-free fashion in the presence of heat PDE dynamics. In order to compensate distinct diffusion processes in the inputs of the two players, we employ boundary control with averaging-based estimates. We apply a small-gain analysis for the resulting Input-to-State Stable (ISS) parabolic PDE-ODE loop as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the heat PDEs, in order to obtain local convergence results to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and illustrate the theoretical results numerically on an example of a two-player game under heat PDEs.

In this paper, the distribution power loss of DC microgrids comprising both line loss and converter loss is modelled as a quadratic function of current allocation coefficients, which is a convex function with constraints. On basis of convex optimization theory, the optimal current allocation coefficients are on-line calculated by Lagrange multiplier method. Then, an adaptive virtual resistance droop control (AVRDC) is proposed to achieve the given current-sharing rate. In the proposed control, the current-sharing error generates additional adaptive virtual resistance. It is known that the sum of virtual resistance and line resistance should be in inverse proportion to current distribution coefficient. By sampling data of two control periods, the line resistance can be identified in real time. In this way, the real-time efficiency optimization control of DC microgrid can be realized, regardless of line resistance variations. The effectiveness of the proposed control strategy is validated by both simulation and experimental results.

In this paper, sequential Gaussian simulation (SGS) and number–size (N–S) fractal modeling were used for copper geochemical anomaly mapping in the western part (training area) of Kuhsiah-e-Urmak area, Iran. Then, according to the generated anomaly model in the training area, mineral potential mapping (MPM) was performed for the entire study area based on a well-fitted regression model as a data-driven method. In order to select the best model, six multivariate regression models including two linear, two quadratic, and two cubic functions were examined. For developing a mineral potential map of the study area, the geochemical anomaly raster map of the training area was utilized to create six models based on the values of geo-data sets. According to the results of $$R^{2}$$,$$R_{\text{adj}}^{2}$$, and $${\text{EF}}$$, the fourth model, generated using a quadratic function, was found to be the superior model compared with the rest of regression models defined in this paper. Based on the mathematical formula derived for the superior model, the geo-exploration data sets were synthesized to generate a potential map showing favorable areas for prospecting copper. The potential map generated was verified by the results of lithogeochemical sampling conducted in the training area and field observations in the entire study area.

The Lagrangian finite volume cell-centered hydrodynamic method introduces dissipation into the calculation by solving a multidirectional approximate Riemann problem. The amount of dissipation created is a function of the jump in the velocity and stress at the node. These jumps in velocity and stress can be reduced by using higher-order reconstructions that are constructed by fitting cell-average values in neighboring cells. One challenge is that a large stencil is required to create high-order reconstructions. To address this challenge, a new two-step reconstruction process is proposed to build a quadratic polynomial by only communicating with face-neighboring cells. The two-step reconstruction method is applied to scalar, vector, and tensor fields. Another challenge is that limiters must be used to reduce the higher order reconstructions towards piecewise constant fields near discontinuities to prevent artificial new extrema. We address this second challenge and propose a new hierarchical limiter that uses a discrete Mach number as a smoothness indicator. The inclusion of the Mach number is essential for minimizing dissipation errors. The new limiter is used with the reconstructions of pressure, velocity, and deviatoric stress. The accuracy and robustness of the new two-step reconstruction method with the new limiter is demonstrated by simulating a suite of 3D Cartesian test problems covering both gas and solid dynamics.

History matching is a very challenging task because tremendous amount of simulations runs are required to achieve an acceptable accuracy target. In this paper, we develop and evaluate an assisted history matching (AHM) workflow using proxy-based Markov chain Monte Carlo (MCMC) algorithm, which is fully-automated and efficient for application in unconventional reservoirs. We use a synthetic simple case to compare the accuracy between four types of proxies: quadratic polynomial, cubic polynomial, k-nearest neighboring (KNN), and kriging. Also, we investigate the accuracy of proxy in different contexts such as different measurement errors and different quantity of initial simulation points. The results show that kriging proxy is more accurate than KNN proxy and cubic proxy. The quadratic proxy was the least accurate in our evaluations. However, if larger measurement error is introduced, the distinction between accuracy of proxy of the four proxies becomes less clear in spite of their different computational costs. The appropriate number of initial simulation points is another key to improve the AHM workflow efficiency. Incorporating these findings, we implement the AHM workflow on a synthetic field case and successfully perform history matching and uncertainty quantification.

A multi-objective optimization of corrugated tube with loose-fit twisted tape (CLT) is conducted to obtain the optimal performance, using Response Surface Methodology (RSM) and Non-dominated Sorting Genetic Algorithm (NSGA-II). A series of simulation runs are performed to optimize the design variables on the objective functions. The corrugation parameters are fixed at constant values. The design variables include the Reynolds number (Re), twisted ratio (Y), and clearance ratio (CR). The objective functions involving the Nusselt number ratio (Nu/Nus), friction factor ratio (f/fs), and overall heat transfer performance (η) are calculated from numerical simulations. Regression models for Nu/Nus, f/fs, and η are tested, the statistical importance of each is assessed by means of the analysis of variance (ANOVA), and the models are expressed in quadratic polynomial forms. The multi-objective NSGA II is adopted to obtain the Pareto-optimal fronts. The results show that the most significant factor is the linear term of Y for Nuc/Nus and fc/fs, while that for η is the linear term of Re. Pareto optimal solution can be obtained with the optimum Nuc/Nus = 2.484 and fc/fs = 3.324. The corresponding values of the design variables are Re = 4000.08, Y = 2.79, and CR = 0.20. The optimal design variables against Re are also obtained, which possesses practical significances for designing heat exchanger.

The quadrature rule of Hammer and Stroud (1956) for cubic polynomials has been shown to be exact for a larger space of functions, namely the C 1 cubic Clough–Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle Kosinka and Bartoň (2018). We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of C 1 quadratic Powell–Sabin 6-split macro-triangles. We show that the 3-node Gaussian quadrature(s) for quadratics can be generalised to the C 1 quadratic Powell–Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in Kosinka and Bartoň (2018). The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three. For the inner split-point at the barycentre, we prove that the two 3-node quadratic polynomial quadratures of Hammer and Stroud exactly integrate also the C 1 quadratic Powell–Sabin spline space if and only if the edge split-points are at their respective edge midpoints. For other positions of the inner and edge split-points we provide numerical examples showing that three nodes suffice to integrate the space exactly, but a full classification and a closed-form solution in the generic case remain elusive.

Point positioning and direction of arrival (DOA) localization require separate estimators, depending on whether the source is near or distant from the sensors. The use of modified polar representation (MPR) for the source location enables the integration of the two estimators together and eliminates the knowledge needed if the source is in the near-field or far-field. The previous work on MPR only provides an iterative implementation of the maximum likelihood estimator (MLE) initialized by a coarse semidefinite relaxation (SDR) solution. This paper proposes a formulation for time difference of arrival (TDOA) localization of a 2-D or 3-D source in MPR that would lead to a closed-form solution through the minimization of a quadratic function with a quadratic constraint. Two techniques, the successive unconstrained minimization (SUM) and the generalized trust region subproblem (GTRS), are applied to solve the optimization. Detailed analysis in the first order for mean-square and in the second order for estimation bias is performed for both methods. The theoretical results illustrate that the closed-form solution from each of the methods provides the Cramer–Rao lower bound performance for Gaussian noise. The GTRS solution has better accuracy than the SUM solution when the source signal is arriving at an azimuth or elevation angle close to zero, 90 $^\circ$ , or 180 $^\circ$ or when the measurement noise level is large. They yield comparable performance with the MLE albeit having less complexity and avoiding possible divergence behavior. They outperform the closed-form solutions from the literature in angle estimation and have smaller bias.

In the military field, equipment development has increasingly focused on the system-of-systems (SoS) oriented combat. To answer the question “how to judge the success of a mission in SoS combat”, this paper proposes a concept of “Mission Success Space (MSS)”. After designing MSS and computing Mission Success Rate (MSR), an evaluation method of effectiveness is defined to quantify the contribution of a combat aircraft to a combat SoS by comparing MSRs of the SoS with and without this aircraft. During MSS design procedure, Mission Success Function (MSF) plays an important role since it gives key parts where the success criterion is changed. In the simulation case, a quadratic function and an S-curve function are used as MSFs to verify the method proposed. In addition, based on the Gaussian fitting results, a new function using inverse design is studied by finding bilateral quantiles of the fitting curves. Finally, the MSS of inverse design is chosen as the best for this case after analyzing the contribution results of these three functions.

To address the shortcomings of the existing linear power flow calculation methods for distribution networks with low versatility and calculation accuracy, a linear power flow calculation method considering distributed generation (DG) is proposed. According to the characteristics of distribution networks and using the Taylor series expansion technology, the nonlinear terms of the basic power flow equation are linearized. The reactive power generated by the non-PV type DG is simplified to a quadratic function of the voltage amplitude. Then, combined with linear approximation and the simplified DG calculation model, the basic power flow equation is simplified into a linear form. Based on the linear power flow equation, a linear power flow calculation method for distribution networks with DG is proposed. The proposed method, which considers various types of DG, has a strong versatility. Finally, the calculation results of typical test systems show that compared with the existing linear power flow calculation methods, the proposed method has a higher calculation accuracy and more versatility and can be used for rapid analysis of distribution networks.