## 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|凸优化作业代写凸优化代考| Pareto Front的上下估计

$$L_{\mathbf{A}}(\mathbf{w})=\min _{\mathbf{x} \in \mathbf{A}} g(\mathbf{x}, \mathbf{w})$$

(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。

## 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

.

(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。

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