数学代写|运筹学作业代写operational research代考|MA3212

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|运筹学作业代写operational research代考|Conic kernel based bridge estimator

We want to solve problem Equation 1.24 by CQP to benefit the advantage of convex optimization mentioned above. Therefore, the problem Equation $1.24$ should be written as a standard form of CQP which is a well structured convex optimization problem. For this reason we will consider two special cases of the kernel based bridge problem, Equation 1.24 given in Equation $1.25$ and Equation 1.27.

Let firstly tackle kernel based ridge problem in Equation $1.25$ for obtaining conic kernel based bridge estimator of $\theta, \hat{\theta}{C K B B}^R$. The problem in Equation $1.25$ can be formulated as a CQP problem based on an appropriate selection of bound T. Thus, the problem in Equation $1.25$ can be written as follow: $$\text { minimize }\theta|A \theta-a|_2^2 \text { subject to }|\theta|_2^2 \leq T \text {. }$$
The biggest contribution of $\mathrm{CQP}$ to the solution of the problem in Equation 1.33 is that the smoothing parameter $\varphi$ does not need to be calculated separately. In other words, the solution of the problem in Equation $1.33$ by CQP creates an alternative solution to determine $\varphi$. In this sense, bound $T$ should be found as a result of a careful learning process, with the help of model-free or model-based methods [16]. The problem in Equation $1.34$ involves the least-squares objective function $|A \theta-a|_2^2$ and the inequality constraint function $-|\theta|_2^2+T$ that should be non-negative for feasibility. Then, optimization problem in Equation $1.34$ is equivalently written as follows:
$$\text { minimize }{t, \theta} \text { subject to }|A \theta-a|_2^2, t \geq 0 \text { subject }|\theta|_2^2 \leq T$$ or, equivalently again $$\text { minimize }{t, \theta} \text { subject to }|A \theta-a|_2^2 \leq t^2, t \geq 0|\theta|_2 \leq \sqrt{T} \text {. }$$
If optimization problem in Equation $1.36$ is compared with the standard form of $\mathrm{CQP}$, it is observed that it is a $\mathrm{CQP}$ programme with
$$\begin{gathered} c=\left(1,0_{q d}^T\right)^T, \phi=\left(t, \theta^T\right)^T, D_1=\left(0_n, A\right), d_1=a, p_1=(1,0, \ldots, 0)^T, q_1=0 \ D_2=\left(0_{q d}, I_n, d_2=0_{q d}, p_2=0_{q d+1} \text {, and } q_2=-\sqrt{T} .\right. \end{gathered}$$
The dual problem of the problem Equation $1.35$ is written
\begin{aligned} & \text { mazimize }\left(a^T, 0\right) \omega_1+\left(0_{q d}^T,-\sqrt{T}\right) \omega_2 \ & \text { such that }\left[\begin{array}{cc} 0_n^T & 1 \ \Lambda_{n \times q d}^T & 0_{\bar{q} d} \end{array}\right] \omega_1+\left[\begin{array}{cc} 0_{q d}^T & 0 \ I_{q d \times q d} & o_{q d} \end{array}\right] \omega_2=\left[\begin{array}{c} 1 \ 0_{q d} \end{array}\right], \ & \omega_1 \in L^{n+1}, \omega_2 \in L^{q d+1} . \end{aligned}

数学代写|运筹学作业代写operational research代考|C-KBBE for case (α = 1)

As stated before, the kernel based bridge estimator is equivalent to the kernel based Lasso in case $(\alpha=1)$ and it can be found by the solution of problem in Equation 1.27, that is, a non-smooth optimization problem. The most common methods used to solve such problems are Quadratic Programming, iterated ridge regression [17] and methods that use sub-gradient strategies [31]. Quadratic Programming needs $2^{q d}$ constraint functions for solving the problem in Equation 1.27. Unfortunately, this situation makes it very difficult to perform iterations over all the constraints generated by this expansion for non-trivial values of $r p$. The methods involving sub-gradient strategies are very difficult for large scale problems, as they may need so many coordinate updates that they become unpractical. Therefore, we want to use conic optimization for solving problem in Equation $1.27$ to avoid the disadvantages of mentioned two methods. However, the problem cannot be written as a CQP problem since the objective function is nondifferentiable in $L_1$-regularization given in Equation 1.27. So, we cannot find conic kernel based bridge estimator of $\theta$, say $\hat{\theta}{C K B B}^{L a s s o}$. For this reason, it should be considered a differentiable approximation method to $L_1$-regularization. The iterated ridge regression (IRR) [17] method provides a differentiable approximation to $L_1$-regularization that contains a non-differentiable objective function and it updates multiple variables at each iteration. Therefore, we consider the IRR to $L_1$-regularization in Equation $1.27$ and we will solve it by CQP. IRR method is based on the following approximation: $$\left|\theta_l^j\right| \approx \frac{\theta_l^{j 2}}{\left|\theta_l^{j k}\right|},$$ where $\theta_l^{j k}$ is the value from the previous iteration $k$. Substituting this approximation into the unconstrained formulation in Equation $1.27$, we can obtain an expression similar to the least-squares estimation with an $L_2$-penalty (ridge regression) as follows: $$\operatorname{minimize}\theta \quad|A \theta-a|_2^2+\varphi \theta^T R \theta,$$ where $R$ is $(q d \times q d)$-dimensional diagonal matrix whose diagonal element are consist of $\left|\theta_1^{j k}\right|^{-1}$, that is, $R=\operatorname{diag}\left(\left|\theta_1^{l k}\right|^{-1}, \ldots,\left|\theta_d^{l k}\right|^{-1}, \ldots,\left|\theta_1^{q k}\right|^{-1}, \ldots,\left|\theta_d^{q k}\right|^{-1}\right):=$ $\operatorname{diag}\left(\left|\theta^k\right|^{-1}\right)$

运筹学代考

数学代写|运筹学作业代写operational research代考|Conic kernel based bridge estimator

$$\operatorname{minimize} \theta|A \theta-a|2^2 \text { subject to }|\theta|_2^2 \leq T .$$ 最大的贡献CQP公式 $1.33$ 中问题的解决方案是平滑参数 $\varphi$ 不需要单独计算。换句话说，方程式中问题的 解1.33通过 CQP 创建替代解决方案以确定 $\varphi$. 在这个意义上，束缚 $T$ 应该在无模型或基于模型的方法 [16] 的帮助下，作为仔细学习过程的结果发现。方程式中的问题 $1.34$ 涉及最小二乘目标函数 $|A \theta-a|_2^2$ 和不等 式约束函数 $-|\theta|_2^2+T$ 这对于可行性应该是非负的。那么，方程中的优化问题 $1.34$ 等价地写成： $$\text { minimize } t, \theta \text { subject to }|A \theta-a|_2^2, t \geq 0 \text { subject }|\theta|_2^2 \leq T$$ 或者，同样地 $$\text { minimize } t, \theta \text { subject to }|A \theta-a|_2^2 \leq t^2, t \geq 0|\theta|_2 \leq \sqrt{T} \text {. }$$ 如果方程式中的优化问题 $1.36$ 与标准形式比较 $\mathrm{CQP}$ ，观察到它是一个CQP程序与 $$c=\left(1,0{q d}^T\right)^T, \phi=\left(t, \theta^T\right)^T, D_1=\left(0_n, A\right), d_1=a, p_1=(1,0, \ldots, 0)^T, q_1=0 D_2=\left(0_{q d},\right.$$

$$\text { mazimize }\left(a^T, 0\right) \omega_1+\left(0_{q d}^T,-\sqrt{T}\right) \omega_2 \quad \text { such that }\left[\begin{array}{ccc} 0_n^T & 1 \Lambda_{n \times q d}^T & 0_{\bar{q} d} \end{array}\right] \omega_1+\left[0_{q d}^T\right.$$

数学代写|运筹学作业代写operational research代考|C-KBBE for case (α = 1)

$$\left|\theta_l^j\right| \approx \frac{\theta_l^{j 2}}{\left|\theta_l^{j k}\right|},$$

$$\operatorname{minimize} \theta \quad|A \theta-a|_2^2+\varphi \theta^T R \theta,$$

$$R=\operatorname{diag}\left(\left|\theta_1^{l k}\right|^{-1}, \ldots,\left|\theta_d^{l k}\right|^{-1}, \ldots,\left|\theta_1^{q k}\right|^{-1}, \ldots,\left|\theta_d^{q k}\right|^{-1}\right):=\operatorname{diag}\left(\left|\theta^k\right|^{-1}\right)$$

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MATLAB代写

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