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数学代写|凸优化作业代写Convex Optimization代考|Upper and Lower Estimates for the Pareto Front

In this section, we visualize the behavior of the function
$$
L_{\mathbf{A}}(\mathbf{w})=\min _{\mathbf{x} \in \mathbf{A}} g(\mathbf{x}, \mathbf{w})
$$
and the estimates of this function which are based on the use of (8.6) and (8.7). Two test problems are used below. The first test problem is ideal for the application of the statistical method described in Section 8.2. The second test function represents/models problems aimed by the bi-objective optimization method.

Visualization of the function $L_{\mathbf{A}}(\mathbf{w})$ corresponding to more than two objectives is difficult, and we thus assume $m=2$; that is, $\mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_7(\mathbf{x})\right)$. In this case, we set $\mathbf{w}=(w, 1-w)$ and consider the function $L_{\mathbf{A}}(\mathbf{w})=L_{\mathbf{A}}(w)$ depending on one variable only, $w \in[0,1]$. We also assume that $d=2$ and $\mathbf{x}=\left(x_1, x_2\right) \in \mathbf{A}=[0,1] \times[0,1]$.
As the first test problem, we consider (1.3) where both objectives are quadratic functions. The sets of Pareto optimal solutions and Pareto optimal decisions are presented in Figure 1.1.

The second multi-objective problem (1.6) is composed of two Shekel functions which are frequently used for testing of single-objective global optimization algorithms. The sets of Pareto optimal solutions and Pareto optimal decisions are presented in Figure 1.3.

In Figures $8.1$ and $8.2$, we show the following estimates of $L_{\mathbf{A}}(w)$, for different $w \in[0,1]:$
(a): $y_{1, n}$, the minimal order statistic corresponding to the sample $\left{y_j=g\left(\mathbf{x}j, w\right) ; j=\right.$ $1, \ldots, n}$ (b): $\widehat{\mathrm{m}}{n, k}$ constructed by the formula (8.6);
(c): $y_{1, n}-\left(y_{k, n}-y_{1, n}\right) / c_{k, 8}$, the lower end of the confidence interval (8.7).
In Figure $8.3$ we illustrate the precision of these estimates for $L_{\mathbf{S}}(w)$ where $\mathbf{S}$ is a subset of $\mathbf{A}$ defined in the capture.

The sample size $n$ is chosen to be $n=300$, the number $k$ of order statistics used is $k=4$ and $\delta=0.05$. Any increase of $n$ (as well a slight increase of $k$ ) leads to an improvement of the precision of the estimates. However, to observe any visible effect of the improvement one needs to significantly increase $n$, see [239] and especially [238] for related discussions.

For each $w$, the minimal order statistic $y_{1, n}$ is an upper bound for the value of the minimum $L_{\mathbf{A}}(w)=\min {\mathbf{x} \in \mathbf{A}} g(\mathbf{x}, w)$ so that it is not a very good estimator. Similarly, $y{1, n}-\left(y_{k, n}-y_{1, n}\right) / c_{k, 8}$, the lower end of the confidence interval (8.7), is not a good estimator of $L_{\mathbf{A}}(\mathbf{w})$ as by the definition it is an upper bound $L_{\mathbf{A}}(\mathbf{w})$ in the majority of cases. The estimator $\widehat{\mathrm{m}}{n, k}$ is always in-between the above two bounds, so that we always have $y{1, n} \leq \widehat{\mathrm{m}}{n, k} \leq y{1, n}-\left(y_{k, n}-y_{1, n}\right) / c_{k, 8}$ (this can be proved theoretically). We have found that the estimator $\widehat{\mathrm{m}}_{n, k}$ is rather precise in the chosen test problems.

数学代写|凸优化作业代写Convex Optimization代考|Branch and Probability Bound Methods

For a single-objective optimization, branch and bound optimization methods are widely known. They are frequently based on the assumption that the objective function $f(\mathbf{x})$ satisfies the Lipschitz condition; see Section 4.2. These methods consist of several iterations, each includes the three following stages:

(i) branching of the optimization set into a tree of subsets,
(ii) making decisions about the prospectiveness of the subsets for further search, and
(iii) selection of the subsets that are recognized as prospective for further branching.
To make a decision at stage (8.5) prior information about $f(\mathbf{x})$ and values of $f(\mathbf{x})$ at some points in $\mathbf{A}$ are used, deterministic lower bounds (often called “underestimates”) for the infimum of $f(\mathbf{x})$ on the subsets of $\mathbf{A}$ are constructed, and those subsets $\mathbf{S} \subset \mathbf{A}$ are rejected for which the lower bound for $\mathrm{m}S=\inf {\mathbf{x} \in \mathbf{S}} f(\mathbf{x})$ does not exceed an upper bound $\hat{f}^$ for $\mathbf{m}=\min {\mathbf{x} \in \mathbf{A}} f(\mathbf{x})$. (The minimum among evaluated values of $f(\mathbf{x})$ in $\mathbf{A}$ is a natural upper bound $\hat{f}^$ for $\mathrm{m}\iota$ )

The branch and bound techniques are among the best deterministic techniques developed for single-objective global optimization. These techniques are naturally extensible to multi-objective case as shown in Chapter 5 . In the case of singleobjective optimization, deterministic branch and bound techniques have been generalized in [238] and [237] to the case where the bounds are stochastic rather than deterministic, and are constructed on the base of statistical inferences about the minimal value of the objective function. The corresponding methods are called branch and probability bound methods. In these methods, statistical procedures for testing the hypothesis $H_0: M_S \leq \hat{f}^*$ are applied to make a decision concerning the prospectiveness of a set $\mathbf{S} \subset \mathbf{A}$ at stage (ii). Rejection of the hypothesis $H_0$ corresponds to the decision that the global minimum $\mathrm{m}=\min _{\mathbf{x} \in \mathrm{A}} f(\mathbf{x})$ cannot be reached in $\mathbf{S}$. Unlike the deterministic decision rules such rejection may be false. This may result that the global maximizer is lost. However, an asymptotic level for the probability of the false rejection can be controlled and it will be fixed.

数学代写|凸优化作业代写Convex Optimization代考|MATH4071

数学代写|凸优化作业代写凸优化代考| Pareto Front的上下估计


在本节中,我们可视化函数
$$
L_{\mathbf{A}}(\mathbf{w})=\min _{\mathbf{x} \in \mathbf{A}} g(\mathbf{x}, \mathbf{w})
$$
的行为,以及基于(8.6)和(8.7)的该函数的估计。下面使用两个测试问题。第一个测试问题非常适合应用第8.2节中描述的统计方法。第二个测试函数表示/建模双目标优化方法所针对的问题


函数$L_{\mathbf{A}}(\mathbf{w})$对应于两个以上的目标的可视化是困难的,因此我们假设$m=2$;网址是$\mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_7(\mathbf{x})\right)$。在本例中,我们设置$\mathbf{w}=(w, 1-w)$并考虑仅依赖于一个变量$w \in[0,1]$的函数$L_{\mathbf{A}}(\mathbf{w})=L_{\mathbf{A}}(w)$。我们还假设$d=2$和$\mathbf{x}=\left(x_1, x_2\right) \in \mathbf{A}=[0,1] \times[0,1]$ .
作为第一个测试问题,我们考虑(1.3),其中两个目标都是二次函数。帕累托最优解集和帕累托最优决策集如图1.1所示


第二个多目标问题(1.6)由两个经常用于测试单目标全局优化算法的谢克尔函数组成。帕累托最优解集和帕累托最优决策集如图1.3所示


在图$8.1$和$8.2$中,我们显示了$L_{\mathbf{A}}(w)$的以下估计,对于不同的$w \in[0,1]:$
(a): $y_{1, n}$,对应于样本$\left{y_j=g\left(\mathbf{x}j, w\right) ; j=\right.$$1, \ldots, n}$ (b): $\widehat{\mathrm{m}}{n, k}$的最小序统计量,由公式(8.6)构造;
(c): $y_{1, n}-\left(y_{k, n}-y_{1, n}\right) / c_{k, 8}$,置信区间(8.7)的下端。
在图$8.3$中,我们说明了$L_{\mathbf{S}}(w)$的这些估计的精度,其中$\mathbf{S}$是捕获中定义的$\mathbf{A}$的子集

选择样本容量$n$为$n=300$,使用的订单统计数$k$为$k=4$和$\delta=0.05$。$n$的任何增加(以及$k$的轻微增加)都会导致估计精度的提高。然而,要观察任何明显的改进效果,就需要显著增加$n$,参见[239],特别是[238]进行相关讨论。


对于每个$w$,最小阶统计量$y_{1, n}$是最小值$L_{\mathbf{A}}(w)=\min {\mathbf{x} \in \mathbf{A}} g(\mathbf{x}, w)$的上界,因此它不是一个很好的估计量。类似地,$y{1, n}-\left(y_{k, n}-y_{1, n}\right) / c_{k, 8}$,置信区间(8.7)的低端,不是$L_{\mathbf{A}}(\mathbf{w})$的一个很好的估计值,因为根据定义,它在大多数情况下是$L_{\mathbf{A}}(\mathbf{w})$的上限。估计量$\widehat{\mathrm{m}}{n, k}$总是在上面两个边界之间,所以我们总是有$y{1, n} \leq \widehat{\mathrm{m}}{n, k} \leq y{1, n}-\left(y_{k, n}-y_{1, n}\right) / c_{k, 8}$(这可以从理论上证明)。我们发现,在所选的测试问题中,估计量$\widehat{\mathrm{m}}_{n, k}$是相当精确的

数学代写|凸优化作业代写凸优化代考|分支和概率绑定方法

.


对于单目标优化,分支和定界优化方法是众所周知的。它们通常基于这样的假设:目标函数$f(\mathbf{x})$满足Lipschitz条件;见4.2节。这些方法由几个迭代组成,每个迭代包括以下三个阶段

(i)将优化集分支成一棵子集树,
(ii)对进一步搜索的子集的前景做出决定,
(iii)选择被认为有前景进行进一步分支的子集。
在(8.5)阶段作出决定 $f(\mathbf{x})$ 和价值观 $f(\mathbf{x})$ 在某些时候 $\mathbf{A}$ 的极值是否使用确定性的下界(通常称为“低估”) $f(\mathbf{x})$ 的子集上 $\mathbf{A}$ 是构造出来的,那些子集呢 $\mathbf{S} \subset \mathbf{A}$ 被拒绝的下界是为了什么 $\mathrm{m}S=\inf {\mathbf{x} \in \mathbf{S}} f(\mathbf{x})$ 是否超过上限 $\hat{f}^$ 为 $\mathbf{m}=\min {\mathbf{x} \in \mathbf{A}} f(\mathbf{x})$。的评估值中的最小值 $f(\mathbf{x})$ 在 $\mathbf{A}$ 这是一个自然的上限吗 $\hat{f}^$ 为 $\mathrm{m}\iota$ )


分支和定界技术是为单目标全局优化开发的最好的确定性技术之一。这些技术可以很自然地扩展到第5章所示的多目标情况。在单目标优化的情况下,确定性分支和定界技术在[238]和[237]中被推广到边界是随机的而不是确定性的情况下,并且是基于关于目标函数最小值的统计推断构建的。相应的方法称为分支和概率定界方法。在这些方法中,应用检验假设$H_0: M_S \leq \hat{f}^*$的统计程序,对阶段(ii)中集合$\mathbf{S} \subset \mathbf{A}$的前瞻性作出决定。拒绝假设$H_0$对应于$\mathbf{S}$中不能达到全局最小值$\mathrm{m}=\min _{\mathbf{x} \in \mathrm{A}} f(\mathbf{x})$的决定。与确定性决策规则不同,这种拒绝可能是错误的。这可能导致全局最大化器丢失。然而,错误拒绝概率的渐近水平是可以控制的,并且是固定的

数学代写|凸优化作业代写Convex Optimization代考

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