# 数学代写|凸优化作业代写Convex Optimization代考|ELEN90026

## 数学代写|凸优化作业代写Convex Optimization代考|Experiments with the -Algorithm

A version of the bi-objective $\pi$-algorithm has been implemented as described in Section 7.4.2. A product of two arctangents was used for $\pi(\cdot)$. Then the $n+1$ step of the $\pi$-algorithm is defined as the following optimization problem
$$\mathbf{x}{n+1}=\arg \max {\mathbf{x} \in \mathbf{A}} \arctan \left(\frac{y_1^{\prime}-m_1(\mathbf{x})}{s_1(\mathbf{x})}+\frac{\pi}{2}\right) \cdot \arctan \left(\frac{y_2^n-m_2(\mathbf{x})}{s_2(\mathbf{x})}+\frac{\pi}{2}\right),$$
where the information collécted at previous steps is takeen intó account when computing $m_i(\mathbf{x})=m_i\left(\mathbf{x} \mid \mathbf{x}_j, \mathbf{y}_j, j=1, \ldots, n\right)$ and $s_i(\mathbf{x})=s_i\left(\mathbf{x} \mid \mathbf{x}_j, \mathbf{y}_j, j=1, \ldots, n\right)$. The maximization in (7.26) was performed by a simple version of multistart: from the best of 1000 points, generated randomly with uniform distribution over the feasible region, a local descent was performed using the codes from the MATLAB Optimization Toolbox. By this implementation, we wanted to check whether the function $\arctan (\cdot) \cdot \arctan (\cdot)$ chosen rather arbitrarily could be as good as the Gaussian cumulative distribution function for constructing statistical model-based multi-objective optimization algorithms. The experimentation with this version of the algorithm can be helpful also in selecting the most appropriate statistical model for a further development where two alternatives seem competitive: a Gaussian random field versus a statistical model, based on the assumptions of subjective probability [216].

Some experiments have been done for the comparison of the $\pi$-algorithm with the multi-objective P-algorithm described in Section 7.3. The optimization results by RUS from Section 7.5.3 are included to highlight the properties of the selected test problems. The results obtained by a multi-objective genetic algorithm (the MATLAB implementation in [80]) are also provided for the comparison. Two examples are presented and commented; we think that extensive competitive testing would be premature, as argued in Section 7.5.1.

Since the considered approach is oriented to expensive problems, we are interested in the quality of the result obtained computing a modest number of the values of objectives. Following the concept of experimentation above, a termination condition of all the considered algorithms was defined by the maximum number of computations of the objective function values, equal to 100 . The parameters of the statistical model, needed by the $\pi$-algorithm, have been estimated using a sample of $\mathbf{f}(\mathbf{x})$ values, chosen similarly to the experiments with the P-algorithm: the sites for the first 50 computations of $\mathbf{f}(\mathbf{x})$ were chosen randomly with a uniform distribution over the feasible region; the obtained data were used not only for estimating parameters but also in planning of the next 50 observations according to $(7.26)$

## 数学代写|凸优化作业代写Convex Optimization代考|Statistical Inference About the Minimum of a Function

Let $f(\mathbf{x})$ be a function given on a feasible region $\mathbf{A}$ and let $\mathbf{x}1, \ldots, \mathbf{x}_n$, be identically distributed points in A. Note that in the present section the estimation of the minimum of a scalar-valued function is considered. Further, we provide estimates for $\mathrm{m}=\min {\mathbf{x} \in \mathrm{A}} f(\mathbf{x})$, and show how to construct confidence intervals for $\mathrm{m}$.

From the sample $\left{\mathbf{x}1, \ldots, \mathbf{x}_n\right}$ we pass to the sample $\mathbf{Y}=\left{y_1, \ldots, y_n\right}$ consisting of the values $y_j=f\left(\mathbf{x}_j\right)$ of the objective function $f(\mathbf{x})$ at the points $\mathbf{x}_j(j=1, \ldots, n)$. The sample $\mathbf{Y}$ is independent, and its underlying cumulative distribution function is given by $$G(t)=\operatorname{Pr}{\mathbf{x} \in \mathbf{A}: f(\mathbf{x}){f(\mathbf{x})j \in \mathbf{A}. Denote by \eta a random variable which has cumulative distribution function G(t), and by y{1, n} \leq \ldots \leq y_{n, n} the order statistics corresponding to the sample \mathbf{Y}. The parameter \mathrm{m}=\min _{\mathbf{x} \in \mathbf{A}} f(\mathbf{x}) is at the same time the lower endpoint of the random variable \eta, i.e., \mathrm{m}= essinf \eta. That is, \mathrm{m} is such that G(\mathrm{~m})=0 and G(\mathrm{~m}+\varepsilon)>0 for any \varepsilon>0. For a very wide class of functions f(\mathbf{x}) and distributions P, the cumulative distribution function G(\cdot) can be shown to have the following representation for t \simeq \mathrm{m} :$$
G(t)=c(t-\mathrm{m})^\alpha+\mathrm{o}\left((t-\mathrm{m})^\alpha\right), t \downarrow \mathrm{m} .
$$## 数学代写|凸优化作业代写Convex Optimization代考|Experiments with the -Algorithm 双目标的一个版本圆周率-algorithm 已按照第 7.4.2 节中的描述实现。两个反正切的乘积被用于圆周率(⋅). 然后n+1的步骤圆周率-算法定义为如下优化问题 Xn+1=参数⁡最大限度X∈一个反正切⁡(是1′−米1(X)s1(X)+圆周率2)⋅反正切⁡(是2n−米2(X)s2(X)+圆周率2), 在计算时考虑在前面步骤中收集的信息米一世(X)=米一世(X∣Xj,是j,j=1,…,n)和s一世(X)=s一世(X∣Xj,是j,j=1,…,n). (7.26) 中的最大化是通过一个简单版本的 multistart 执行的：从 1000 个点中最好的一个，在可行区域上以均匀分布随机生成，使用 MATLAB 优化工具箱中的代码执行局部下降。通过这个实现，我们想检查函数是否反正切⁡(⋅)⋅反正切⁡(⋅)任意选择可以与构建基于统计模型的多目标优化算法的高斯累积分布函数一样好。使用该版本的算法进行实验也有助于选择最合适的统计模型以进行进一步发展，其中两种选择似乎具有竞争力：基于主观概率假设的高斯随机场与统计模型 [216]。 已经做了一些实验来比较圆周率-算法与第 7.3 节中描述的多目标 P 算法。包含第 7.5.3 节中 RUS 的优化结果以突出所选测试问题的属性。还提供了通过多目标遗传算法（[80] 中的 MATLAB 实现）获得的结果用于比较。提出并评论了两个例子；正如第 7.5.1 节所述，我们认为进行广泛的竞争性测试还为时过早。 由于所考虑的方法面向代价高昂的问题，因此我们对计算适度数量的目标值所获得的结果的质量感兴趣。遵循上述实验的概念，所有考虑的算法的终止条件由目标函数值的最大计算次数定义，等于 100。统计模型所需的参数圆周率-算法，已使用样本估计F(X)值，选择类似于使用 P 算法的实验：前 50 次计算的站点F(X)随机选择，在可行区域内均匀分布；获得的数据不仅用于估计参数，还用于规划接下来的 50 个观测值(7.26) ## 数学代写|凸优化作业代写Convex Optimization代考|Statistical Inference About the Minimum of a Function 让F(X)是在可行域上给定的函数一个然后让X1,…,Xn, 是 A 中相同分布的点。请注意，在本节中，考虑了对标量值函数的最小值的估计。此外，我们提供估计米=分钟X∈一个F(X)，并展示如何构建置信区间米. 从样本\left{\mathbf{x}1, \ldots, \mathbf{x}_n\right}\left{\mathbf{x}1, \ldots, \mathbf{x}_n\right}我们传递给样本\mathbf{Y}=\left{y_1, \ldots, y_n\right}\mathbf{Y}=\left{y_1, \ldots, y_n\right}由值组成是j=F(Xj)目标函数F(X)在点Xj(j=1,…,n). 样本是是独立的，其基础累积分布函数由$$ G(t)=\operatorname{Pr}{\mathbf{x}\in \mathbf{A} 给出： f(\mathbf{x}){f(\ ) mathbf{x})j \in \mathbf{A}.D和n○吨和b是和一个r一个nd○米在一个r一世一个bl和在H一世CHH一个sC在米在l一个吨一世在和d一世s吨r一世b在吨一世○nF在nC吨一世○nG(t),一个ndb是y{1, n} \leq \ldots \leq y_{n, n}吨H和○rd和rs吨一个吨一世s吨一世CsC○rr和sp○nd一世nG吨○吨H和s一个米pl和\mathbf{Y}.吨H和p一个r一个米和吨和r\mathrm{m}=\min_{\mathbf{x}\in \mathbf{A}} f(\mathbf{x})一世s一个吨吨H和s一个米和吨一世米和吨H和l○在和r和ndp○一世n吨○F吨H和r一个nd○米在一个r一世一个bl和和,一世.和.,\mathrm{m}=和ss一世nF和.吨H一个吨一世s,\数学{m}一世ss在CH吨H一个吨G(\mathrm{~m})=0一个ndG(\mathrm{~m}+\varepsilon)>0F○r一个n是\伐普西隆>0\$。

G(吨)=C(吨−米)一个+○((吨−米)一个),吨↓米.

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