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

#### Doug I. Jones

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## 数学代写|凸优化作业代写Convex Optimization代考|Probabilistic Bounds in Multi-Objective

Randomization is one of the most important ideas used in the construction of heuristic methods for multi-objective optimization. Mathematically substantiated stochastic methods for non-convex multi-objective optimization attracted interest of researchers quite recently. A natural idea is to generalize single-objective optimization methods to the multi-objective case, and a prospective candidate is the well-developed method based on the statistical methods of extremal values. A brief review of this approach, known as a branch and probability bound (BPB) method is presented in Section 4.4. Some statistical procedures which are well known in single-objective optimization can be extended to multi-objective problems using scalarization. In this chapter, a version of multi-objective BPB method is described and discussed; this method is an extension of the BPB method developed for the case of a single-objective function in [237], see also [239, Section 2.6.1]. The considered extension is based on the Tchebycheff scalarization method briefly discussed in Chapter 2.

Let us recall that by means of the Tchebycheff scalarization a multi-objective optimization problem
$$\min {\mathbf{x}} \in \mathbf{A f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_2(x), \ldots, f_m(\mathbf{x})\right)^T, \mathbf{A} \subset \mathbb{R}^d,$$ is reduced to the single-objective optimization problem where the objective function is defined using the following formula: \begin{aligned} &g(\mathbf{x}, \mathbf{w})=\max {i=1, \ldots, m} w_i\left(f_i(\mathbf{x})-u_i\right), \ &w_t \geq 0 \text { for } i=1, \ldots, m \text { and } \sum_{i=1}^m w_i=1, \end{aligned} and $\mathbf{u}$ is a so-called utopian vector
$$u_i<\min f_i(\mathbf{x}) \text { for } i=1, \ldots, m .$$
The minimizer of (8.2) is a weakly Pareto optimal decision of the respective multi-objective optimization problem. Similarly, by the modified versions of the Tchebycheff scalarization, defined by (2.5) and (2.6), exclusively the Pareto optimal decisions can be found.

More precisely this can be formulated as follows [135]. Let $\mathbf{y} \in \mathbf{P}(\mathbf{f})O$ be any Pareto optimal objective vector, then it corresponds to a minimum of an objective function (2.5) defined by the modified Tchebycheff scalarization. The parameters of the objective function (2.5) should satisfy the following requirements: $\rho \geq 0$ should be sufficiently small, and weights should be defined by the following formulas: $w_i=$ $\beta /\left(y_i-u_i\right), i=1, \ldots, m$, where $\mathbf{u}$ is a utopian vector. On the other hand, the set $\left{\mathbf{f}\left(\mathbf{x}{\mathbf{w}}\right): w_i \geq 0, \sum_{i=1}^m w_i=1\right}$ is coincident with $\mathbf{P}(\mathbf{f})O$, where $\mathbf{x}{\mathbf{w}}$ is an optimal decision of the scalarized problem corresponding to the vector of weights w. Let us recall that when the original Tchebycheff scalarization (8.2) is used, the weakly Pareto optimal solutions can be obtained besides of Pareto optimal solutions.

## 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} .
$$This representation is valid for some positive constants c and \alpha; more generally, c=c(t) is a slowly varying function for t \simeq \mathrm{m} but the results cited below are also valid for this slightly more general case. The value of c is irrelevant but the value of \alpha, which is called “tail index,” is important. We shall assume that the value of \alpha is known. As discussed below this can always be considered true in the algorithms we consider. Several good estimates of m are known for given \alpha, see [239, Section 2.4]. We shall use one of them, the optimal linear estimator based on the use of k order statistics. This estimator has the form$$
\widehat{\mathrm{m}}{n, k}=c \sum{i=1}^k\left[u_i / \Gamma(i+2 / \alpha)\right] y_{i, n},
$$where \Gamma(\cdot) is the Gamma-function,$$
u_i= \begin{cases}(\alpha+1), & \text { for } i=1, \ (\alpha-1) \Gamma(i), & \text { for } i=1, \ldots, k-1, \ (\alpha-\alpha k-1) \Gamma(k), & \text { for } i=k,\end{cases}

1 / c= \begin{cases}\sum_{i=1}^k 1 / i, & \text { for } \alpha=2 \ \frac{1}{\alpha-2}(\alpha \Gamma(k+1) / \Gamma(k+2 / \alpha)-2 / \Gamma(1+2 / \alpha)), & \text { for } \alpha \neq 2\end{cases}
$$## 凸优化代写 ## 数学代写|凸优化作业代写Convex Optimization代考|多目标的概率边界 随机化是构建用于多目标优化的启发式方法的最重要的思想之一。用于非凸多目标优化的数 学证实的随机方法最近引起了研究人员的兴趣。一个自然的想法是将单目标优化方法推广到 多目标情况，而前瞻性候选方法是基于极值统计方法的成孰方法。 4.4 节介绍了这种方法的 简要回顾，称为分支和概率界 (BPB) 方法。一些在单目标优化中广为人知的统计过程可以使 用标量化扩展到多目标问题。在本章中，描述和讨论了多目标 BPB 方法的一个版本；该方法 是在 [237] 中针对单目标函数的情况开发的 BPB 方法的扩展，另请参见 [239，第 2.6 .1 节]。所考虑的扩展基于第 2 章中简要讨论的 Tchebycheff 标量化方法。 让我们回顾一下，通过 Tchebycheff 标量化一个多目标优化问题$$
\min \mathbf{x} \in \mathbf{A f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_2(x), \ldots, f_m(\mathbf{x})\right)^T, \mathbf{A} \subset \mathbb{R}^d,
$$简化为单目标优化问题，其中目标函数使用以下公式定义:$$
g(\mathbf{x}, \mathbf{w})=\max i=1, \ldots, m w_i\left(f_i(\mathbf{x})-u_i\right), \quad w_t \geq 0 \text { for } i=1, \ldots, m \text { and }
$$和 \mathbf{u} 是一个所佣的乌托邦向量$$
u_i<\min f_i(\mathbf{x}) \text { for } i=1, \ldots, m .
$$(8.2) 的最小化器是各个多目标优化问题的弱帕累托最优决策。类似地，通过由 (2.5) 和 (2.6) 定义的 Tchebycheff 标量的修改版本，可以仅找到帕男托最优决策。 更准确地说，这可以表述如下[135]。让 \mathbf{y} \in \mathbf{P}(\mathbf{f}) O 是任何帕累托最优目标向量，则它对应 于由改进的 Tchebycheff 标量化定义的目标函数 (2.5) 的最小值。目标函数 (2.5) 的参数应 满足以下要求: \rho \geq 0 应该足够小，并且权重应该由以下公式定义: w_i= \beta /\left(y_i-u_i\right), i=1, \ldots, m ，在哪里u是一个乌托邦向量。另一方面，集 \mathbf{P}(\mathbf{f}) O ，在哪里 \mathbf{x w} 是对应于权重向量 w 的标量化问题的最优决策。让戔们回想一下，当 使用原始 Tchebycheff 标量化（8.2) 时，除了帕累托最优解之外，还可以获得弱帕累托最 优解。 ## 数学代写|凸优化作业代写Convex Optimization代考|关于函数最小值的统计推断 让 f(\mathbf{x}) 是在可行域上给定的函数 \mathbf{A} 然后让 \mathbf{x} 1, \ldots, \mathbf{x}_n ，是 \mathrm{A} 中相同分布的点。请注意，在 本节中，考虑了对标量值函数的最小值的估计。此外，我们提供估计 m=\min \mathbf{x} \in \mathrm{Af}(\mathbf{x}) ， 并展示如何构建置信区间 \mathrm{m}. \mathbf{x}_j(j=1, \ldots, n). 样本 \mathbf{Y} 是独立的，其其础累积分布函数由 \ \ \mathrm{G}(\mathrm{t})= loperatorname {\mathrm{Pr}} {\backslash \operatorname{mathbf}{\mathrm{x}} \backslash in \mathbf {\mathrm{A}} 给出: \mathrm{f}(\backslash mathbf {\mathrm{X}}){\mathrm{f}(\backslash) mathbf {\mathrm{X}}) \mathrm{\backslash} in \mathbf {\mathrm{A}} • Denoteby和arandomvariablewhichhascumulativedistribution function \mathrm{G}(\mathrm{t}), \operatorname{andby}{1, \mathrm{n}} \backslash \mathrm{eq} \backslash / dots \backslash leq \mathrm{y}{-}{\mathrm{n}, \mathrm{n}} theorderstatisticscorrespondingtothesample\mathbf {\mathrm{Y}}. Theparameter isatthesametimethelowerendpointo ftherandomvariable 和, i. e., \backslash \mathrm{mathrm}{\mathrm{m}}=e s \sin f 和. Thatis, 数学 {\mathrm{m}} issuchthat \mathrm{G}(\backslash \mathrm{mathrm}{\sim \mathrm{m}})=0 and \mathrm{G}(\backslash mathrm{\sim \mathrm{m}}+ Ivarepsilon )>0 foran y \backslash 伐普西隆 >0 \$$ 。 适用于非常广泛的功能 $f(\mathbf{x})$ 和分布 $P$ ，累积分布函数 $G(\cdot)$ 可以显示为具有以下表示 $t \simeq \mathrm{m}$ : $$G(t)=c(t-\mathrm{m})^\alpha+\mathrm{o}\left((t-\mathrm{m})^\alpha\right), t \downarrow \mathrm{m} .$$ 这种表示对一些正常数有效 $c$ 和 $\alpha$; 电普遍， $c=c(t)$ 是一个缓慢变化的函数 $t \simeq \mathrm{m}$ 但下面引 用的结果也适用于这种稍微更一般的情况。的价值 $c$ 无关㘯要，但价值 $\alpha$ ，也就是所佣的“尾 指数”, 很重要。我们将假设 $\alpha$ 是已知的。正如下面所讨论的，这在我们考虑的算法中总是被 认为是正确的。 几个不错的估计 $m$ 众所周知 $\alpha$ ，见 [239，第 $2.4$ 节]。我们将使用其中之一，基于使用的最优 线性估计器 $k$ 订单统计。这个估计量的形式 $$\widehat{\mathrm{m}} n, k=c \sum i=1^k\left[u_i / \Gamma(i+2 / \alpha)\right] y{i, n},$$
在哪里 $\Gamma(\cdot)$ 是 Gamma 函数，
$u_i={(\alpha+1), \quad$ for $i=1,(\alpha-1) \Gamma(i), \quad$ for $i=1, \ldots, k-1,(\alpha-\alpha k-1) \mathrm{I}$
$1 / c=\left{\sum_{i=1}^k 1 / i, \quad\right.$ for $\alpha=2 \frac{1}{\alpha-2}(\alpha \Gamma(k+1) / \Gamma(k+2 / \alpha)-2 / \Gamma(1+2 / \alpha))$

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