## 数学代写|凸优化作业代写Convex Optimization代考|МАTH4071

2022年10月10日

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## 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 $\mathrm{m}S=\inf {\mathbf{x c s}} f(\mathbf{x})$ does not exceed an upper bound $\hat{f}^$ for $\mathrm{m}=\min _{\mathbf{x} \in \mathbf{A}} f(\mathbf{x})$. (The minimum among evaluated values of $f(\mathbf{x})$ in $\mathbf{A}$ is a natural upper bound $\hat{f}^$ for $\mathrm{m}$.)

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 $\mathrm{m}=\min _{\mathbf{x} \in \mathbf{A}} f(\mathbf{x})$ 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|Visualization

For the expensive black-box multi-objective optimization problems it seems reasonable to hybridize a computer aided algorithmic search with an interactive human heuristic. Visualization is very important in perception of relevant information by a human expert $[48,122,260,263]$. In this section we investigate possibilities of the visualization of scarce information on the Pareto front using a statistical model of the considered problem.
The following problem of bi-objective optimization is considered:
$$\min _{\mathbf{x} \in \mathbf{A}} \mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_2(\mathbf{x})\right)^T,$$
where the properties of $\mathbf{f}(\mathbf{x})$ and of the feasible region $\mathbf{A} \subseteq \mathbb{R}^d$ are specified later on. We are interested in the approximation and visualization of $\mathbf{P}(\mathbf{f})_O$ using scarce information obtained in the initial/exploration phase of optimization. The necessity of the exploration phase follows from the assumption on the black-box objectives. The human heuristic abilities can be advantageous here in perception of scarce information gained during the exploration. The restriction of information scarcity is implied by the assumption on expensiveness of the objectives. The further search can be rationally planned by the optimizer depending on the results of the exploration. Visualization is expected to aid the perception of the available results.
The exploratory phase assumes that we have values of the objective functions at some number of random points in $\mathbf{A}$ which are independent and uniformly distributed. This exploration method can be seen as an analog of a popular heuristic decision by throwing a coin in the case of a severe uncertain decision situation. Moreover. the uniform distribution of points in the feasible region is the worstcase optimal algorithm for the multi-objective optimization of Lipschitz objectives; see Chapter 6. Although in the latter case the uniformity is understood in the deterministic sense, the random uniform distribution of points is a frequently used simply implementable approximation of the deterministic one. Now we have to extract, nseful for the further search, information from the availahle data, i.e., from a set of $\mathbf{x}_i, \mathbf{y}_i=\mathbf{f}\left(\mathbf{x}_i\right), i=1, \ldots, n$.

In single-objective global optimization, some information on the global minimum of $f(\mathbf{x})$ can be elicited from the sample $z_i=f\left(\mathbf{x}_i\right)$, where $\mathbf{x}_i$ are independent random points, by means of the methods of statistics of extremes; see Section 4.4.

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

(i) 将优化集分支成子集树，
(ii) 对子集的前瞻侏做出决策以进行进一步搜亰，以从及
(iii) 选择被认为具有进一步分支预期的子集。 估”）来表示 $f(\mathbf{x})$ 在子堆上 $\mathbf{A}$ 被构造，并且那些子集 $\mathbf{S} \subset \mathbf{A}$ 鿆拒绝的下限为
$m S=\inf x \operatorname{cs} f(\mathbf{x})$ 不超过上限 帞子 ${\mathrm{f}} \wedge$ 为了 $m=\min {\mathbf{x} \in \mathbf{A}} f(\mathbf{x})$. (评估值中的最小值 $f(\mathbf{x})$ 在 $\mathbf{A}$ 是目然上界 $、$ 帞子 ${f} \wedge \mathrm{s}} \mathrm{s}$ 为.) 分支定界技术是为单目标全局优化开发的最佳确定性技术之一。这些技术自然可以扩展到多 目标情况，如第 5 章所示。在单目标优化的情况下，确定性分支定界技术已在 [238] 和 [237] 中推广到边界是随机而不是确定性的情况，并且是基于关于目标函数。相应的方法称 为分支和概率界方法。在这些方法中，用于检验假设的统计程序 $H_0: M_S \leq \hat{f}^*$ 用于做出 关于集合的前瞻性的决定 $\mathbf{S} \subset \mathbf{A}$ 在阶段(ii)。拒绝假设 $H_0$ 对应于全局最小值的决定 $\mathrm{m}=\min {\mathbf{x} \in \mathbf{A}} f(\mathbf{x})$ 无法到达S. 与确定性决策规则不同，这种拒绝可能是错误的。这可能 导敖全局最大化器夆失。但是，可以控制错误拒绝概率的渐近水平并将其固定。

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

$$\min _{\mathbf{x} \in \mathbf{A}} \mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_2(\mathbf{x})\right)^T$$

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## MATLAB代写

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