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CTR Prediction Model
Published in Peng Liu, Wang Chao, Computational Advertising, 2020
For such a high-dimensional problem, we can adopt the conjugate gradient method [80] to solve the subproblem of the trust region. When the objective function is the quadratic positive definite function fx+∇fxTs+12sTHs, the conjugate gradient method can reach convergence after n times (feature dimensionality) of iterations, thus avoiding the storage and calculation of the Hessian matrix. Unlike the conjugate gradient method for unconstrained optimization, the constraint condition herein ∥s∥≤δk shall be satisfied. Considering that the displacement in the subalgorithm is progressively increasing [80], when a displacement is beyond the trust region, it can be returned to the boundary of the trust region along the original search direction.
Random Measures in Infinite-Dimensional Dynamics
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Palle E.T. Jorgensen, Feng Tian
A process {Xg | g ∈ G} is called Gaussian if each random variable Xg is Gaussian, i.e., its distribution is Gaussian. For Gaussian processes we only need two moments. So if we normalize, setting the mean equal to 0, then the process is determined by the covariance function. In general the covariance function is a function on G × G, or on a subset, but if the process is stationary, the covariance function will in fact be a positive definite function defined on G, or a subset of G. Examples include Brownian motion, Brownian bridge, and the Ornstein-Uhlenbeck (OU) process etc, all Gaussian or Itō integrals. See Figures A.1 through A.3.
Analysis on Locally Compact Groups
Published in Hugo D. Junghenn, Principles of Analysis, 2018
Using the Euclidean inner product, we may write this condition as Ac|c≥0 $ \left(A c \,\boldsymbol{|} \, c\right) \ge 0 $ , where c=(c1,…,cn) $ c= (c_1,\ldots , c_n) $ and A=[ajk]n×n $ A = [a_{jk}]_{n\times n} $ , ajk:=ϕ(xk-1xj) $ a_{jk} := \phi (x_k^{-1}x_j) $ . Thus ϕ $ \phi $ is a positive definite function iff A is a positive definite matrix.
Zero-sum differential games-based fast adaptive robust optimal sliding mode control design for uncertain missile autopilot with constrained input
Published in International Journal of Control, 2022
Yuxin Gao, Chunsheng Liu, Sen Jiang, Shaojie Zhang
Considering zero-sum differential games theory, the performance index function is given by where , and function matrix is symmetric positive definite. is a positive definite function. For sake of simplifying the writing, and are replaced by V and d, respectively, in the following statement. To satisfy the constrained-input condition in system (6), is given as where is positive and symmetric define matrices, is the bound of input. After some mathematical transformation, (18) is rewritten as where and .
Strongly perturbed sliding mode adaptive control of vehicle active suspension system considering actuator nonlinearity
Published in Vehicle System Dynamics, 2022
Shuang Liu, Tian Zheng, Dingxuan Zhao, Ruolan Hao, Mengke Yang
The sliding mode adaptive controller is designed for the 6-order system, and the total system is a 10-order system, so the zero-dynamic system consists of four states. In order to obtain the zero dynamics of the system, assume s1 = s2 = e9 = e10 = 0 to obtain: The zero dynamic equation of the system can be obtained by substituting in (36) into Formula (37): where Define positive definite function , where is a positive definite matrix, then: It is easy to show that the real part of the eigenvalue of A is negative. Define , where is a positive definite matrix. Again because: where is an adjustable positive value, the following inequality can be obtained: Choosing an appropriate matrix and adjustable parameter , so that is a positive number.
A numerical approach to hybrid nonlinear optimal control
Published in International Journal of Control, 2021
Esmaeil Sharifi, Christopher J. Damaren
Applying the Galerkin spectral method directly to the continuous-time portion of the hybrid HJB equation, a continuous-time set of differential equations is derived in this section to compute the time-varying optimal control gains between impulsive instants. Defining , (5) can be written as (Lewis et al., 2012): where specifies the value function (the optimum value of the performance index), is a positive-definite function called the continuous-time state penalty function, and denotes a symmetric positive-definite matrix called the continuous-time control penalty matrix. Minimising (21) with respect to , the continuous-time optimal control law in terms of is found as follows: