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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|凸优化作业代写Convex Optimization代考|The Implementation of One-Step Optimality

The idea of the algorithm is to tighten iteratively the Lipschitz lower bounds for the non-dominated solutions, and to indicate the subintervals of $[a, b]$ of dominated decisions which can be excluded from the further search. At the $n+1$ iteration a subinterval $\left[x_{o i}, x_{o i+1}\right]$ is selected and subdivided by the point $x_{n+1}$ where a new objective vector $\mathbf{f}\left(x_{n+1}\right)$ is computed. Let us assume that an interval is selected, and a point of subdivision should be defined. Similarly as in the previous section, the standard interval $[0, v]$ is considered to simplify the notation. The values of the objective functions at the endpoints of the interval are supposed to be known and denoted by $\mathbf{f}(0)=\mathbf{f}1=\left(y_1, z_1\right)^T, \mathbf{f}(v)=\mathbf{f}_2=\left(y_2, z_2\right)^T$. We start the analysis from the case where $\mathbf{f}_1$ and $\mathbf{f}_2$ do not dominate each other; without loss of generality, we assume that $$y_1 \leq y_2, z_2 \leq z_1,$$ and denote $\delta y=y_2-y_1, \delta z=z_1-z_2$. The error of approximation of the local Pareto front is bounded by the tolerance $\Delta\left(\mathbf{f}_1, \mathbf{f}_2, 0, v\right)$. The optimal decision theory suggests to choose the point $\hat{t}$ for the current computation of the vector of objectives by minimizing the maximum of two forecasted tolerances corresponding to the subintervals, obtained by the subdivision by the chosen point. According to the worst-case paradigm, the tolerances are forecasted assuming the most unfavorable values of the objective functions at the chosen point, and the point $\hat{t}$ is defined as follows: $$\hat{t}=\arg \min {0 \leq t \leq v} \max _{\mathbf{w} \in \mathbf{W}(t)} \max \left(\Delta\left(\mathbf{f}_1, \mathbf{w}, 0, t\right), \Delta\left(\mathbf{w}, \mathbf{f}_2, t, v\right)\right),$$

where $\mathbf{w}=\left(w_1, w_2\right)^T$, and $\mathbf{W}(t)$ is a two-dimensional interval defined by the lower and upper Lipschitz bounds for the function values $\mathbf{f}(t)$.

Theorem 6.2 Let the inequality $\delta y \leq \delta z$ be satisfied besides (6.28). Then $\hat{t}=\frac{v}{2}$ is the point of the current computation of the objective functions, since it is a minimizer of (6.29).

Proof To find $\hat{t}$, the expression, supposed to minimize with respect to $t$ in (6.29), should be evaluated for various potential values of the objective functions from the intervals $\mathbf{W}(t)$ which are defined by the lower bounds (6.20), and the upper bounds for the objective functions
$$\begin{array}{r} h_1(t)=y_1+t, 0 \leq t \leq \tau_1, h_1(t)=y_2+(v-t), \tau_1 \leq t \leq v, \ h_2(t)=z_1+t, 0 \leq t \leq \tau_2, h_2(t)=z_2-(v-t), \tau_2 \leq t \leq v, \ \tau_1=\frac{v}{2}-\frac{y_1-y_2}{2}, \tau_2=\frac{v}{2}-\frac{z_1-z_2}{2}, \end{array}$$
which are defined similarly as in (6.20).

## 数学代写|凸优化作业代写Convex Optimization代考|Numerical Experiments

The performance of the one-step worst-case optimal algorithm is demonstrated below by solving several typical test problems. The first multi-objective test considered consists of two (slightly modified) Rastrigin functions (Rastr) which are widely used (see, e.g., [216]) for testing single-objective global minimization algorithms:
\begin{aligned} &f_1(x)=(x-0.5)^2-\cos (18(x-0.5)), \ &f_2(x)=(x+0.5)^2-\cos (18(x+0.5)),-1 \leq x \leq 1 . \end{aligned}
The Lipschitz constant of both objective functions is equal to 21 .

The second problem used (1.2) is referred as Fo\&Fle. We present below its definition for a one-dimensional decision variable
\begin{aligned} f_1(x) &=1-\exp \left(-(x-1)^2\right) \ f_2(x) &=1-\exp \left(-(x+1)^2\right) \ -4 & \leq x \leq 4 \end{aligned}
The Lipschitz constant of both objective functions is equal to 1. Problem (6.47) presents a specific challenge from the point of view of global minimization. The functions $f_1(x)$ and $f_2(x)$ in (6.47) are similar to the most difficult objective function, the response surface of which is almost constant over a large part of the feasible decision region and has an unknown number of sharp spikes. The discrepancy between the model of objective functions used to substantiate the algorithm, and the actual objective functions can negatively influence the efficiency of the algorithm.
The third one-dimensional test problem by Schaffer (see [42, pp. 339-340]) is defined by the following formulas:
\begin{aligned} &f_1(x)= \begin{cases}-x, & \text { if } x \leq 1 \ x-2, & \text { if } 14\end{cases} \ &f_2(x)=(x-5)^2,-1 \leq x \leq 8 \end{aligned}

## 数学代写|凸优化作业代写凸优化代考|一步优化的实现

$$\begin{array}{r} h_1(t)=y_1+t, 0 \leq t \leq \tau_1, h_1(t)=y_2+(v-t), \tau_1 \leq t \leq v, \ h_2(t)=z_1+t, 0 \leq t \leq \tau_2, h_2(t)=z_2-(v-t), \tau_2 \leq t \leq v, \ \tau_1=\frac{v}{2}-\frac{y_1-y_2}{2}, \tau_2=\frac{v}{2}-\frac{z_1-z_2}{2}, \end{array}$$
(由(6.20)定义类似)求出目标函数的各种潜在值。

## 数学代写|凸优化作业代写凸面优化代考|数值实验

.

\begin{aligned} &f_1(x)=(x-0.5)^2-\cos (18(x-0.5)), \ &f_2(x)=(x+0.5)^2-\cos (18(x+0.5)),-1 \leq x \leq 1 . \end{aligned}

\begin{aligned} f_1(x) &=1-\exp \left(-(x-1)^2\right) \ f_2(x) &=1-\exp \left(-(x+1)^2\right) \ -4 & \leq x \leq 4 \end{aligned}

\begin{aligned} &f_1(x)= \begin{cases}-x, & \text { if } x \leq 1 \ x-2, & \text { if } 14\end{cases} \ &f_2(x)=(x-5)^2,-1 \leq x \leq 8 \end{aligned}

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师