### 数学代写|随机过程统计代写Stochastic process statistics代考|STAT3921

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程统计代写Stochastic process statistics代考|Optimization with Constraints: Feasible Region

The objective function is the heart of an optimization problem, since it is the function that should be minimized or maximized. When the optimization situation involves only the analysis of the objective function, as shown in Figure 1.3, it is said that the problem is unconstrained, and the search for the solution occurs in the entire set of $R^{n}$. Nevertheless, for most of the practical problems, the variables are not unbounded; thus, the search space is reduced by the introduction of bounds for the variables. Furthermore, most of the variables appearing on the objective function are related to other variables through a mathematical model, which implies that the search space for the objective function is even more reduced. The region in the $n$-dimensional space limited by those bounds for the variables and the equations relating the set of variables is known as the feasible region. In Figure $1.4 a$, a feasible region in a bidimensional space can be observed, which is enclosed by the constraints $h_{1}(x), h_{2}(x)$, and $g_{1}(x)$. On the other hand, Figure $1.4 \mathrm{~b}$ shows a tridimensional feasible region, where the constraint $g(\bar{x})$ is a plane, and the objective function is a surface with various minimums. Until now, two types of constraints have been mentioned, $h(\bar{x})$ and $g(\bar{x})$. The difference between the two types of constraints is explained. The equations $h(\bar{x})$ are equality constraints, i.e., they have the structure $h(\bar{x})=0$. Such constraints should be strictly complied. Thus, they reduce even more the number of feasible solutions. On the other hand, the equations $g(\bar{x})$ are inequality constraints, i.e.,$g(\bar{x}) \leq 0$. Such constraints can be complied as an equality, $g(\bar{x})=0$, for which it is said that the constraint is active; or they can be complied as an inequality, $g(\bar{x})<0$, for which it said that the constraint is inactive. The inequality constraints, thus, are more easily complied, because they allow the feasible region to include higher number of solutions.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Multiobjective Optimization

At this point, the objective function has been mentioned as a measurement of the goodness of a given solution. It has been stated that an optimization problem, where there is only the necessity of maximizing/minimizing a given objective function, is known as an unconstrained problem. On the other hand, the optimization problem may involve constraints, which limits

the solution space for the objective function. Nevertheless, there are cases on which it is desired to simultaneously optimize two or more objective functions subject to the same set of constraints and variables. This is called multiobjective optimization. This type of problems can be represented in the following general formulation:
optimize $\bar{Z}=\left(Z_{1}, Z_{2}, \ldots, Z_{k}\right)$
s.t.
$h(\bar{x})=0$
$g(\bar{x}) \leq 0$
where $k$ is the total number of objectives, $\bar{Z}$ is the vector of objective functions, and $Z_{1}=z_{1}(\bar{x}), Z_{2}=z_{2}(\bar{x})$, and so on are the individual objective functions. As it is observed, the feasible regions, i.e., the constraints of the optimization problem, are the same for all the objectives. Thus, the only difference with the optimization problem given in Equation $1.8$ is that there are more than one objective functions, and the solution complies with all of them and also with the constraints.

Let us imagine about a problem with two objectives. Both objective functions may have their minimum (or maximum) on the same (or almost on the same) point $\bar{x}$, as shown in Figure $1.5 \mathrm{a}$. Nevertheless, such a problem could even be stated as a problem with a single objective. A more interesting situation occurs when the minimum of the objective functions occur for a different set of $\bar{x}$ values, which implies that when one of the objectives is being reduced, the other one increases (Figure $1.5 \mathrm{~b}$ ). When two objective functions follow such performance, it is mentioned that they are in competence. This implies that, when the minimum for one of the objective functions occurs, the other objective has a value far from its own minimum. Thus, assuming, in general, a situation with $k$ objective functions, the solution to the optimization problem should be a compromise

among all the objectives, since the better solution for one of the objectives could be a considerably inadequate solution for the others. An important concept in multiobjective optimization is the nondominated solution. In mathematical terms, it can be mentioned that $\bar{x}^{}$ is a nondominated solution for a minimization problem if, for any other feasible solution $\bar{x}$, the following relationship is true: $$Z_{p}(\bar{x}) \leq Z_{p}\left(\bar{x}^{}\right)$$

## 数学代写|随机过程统计代写Stochastic process statistics代考|Weighted Sum Method

In this method, the optimization problem shown in Equation $1.9$ is modified as follows:
\begin{aligned} &\text { optimize } Z_{\text {mult }}=\sum_{i=1}^{k} \omega_{i} Z_{i} \ &\text { s.t. } \ &h(\bar{x})=0 \ &g(\bar{x}) \leq 0 \end{aligned}
Thus, the vector of objectives is substituted by a linear combination of the individual objectives. The parameters $\omega_{i}$ are known as weights. In the weighing method, the function $Z_{\text {mult }}$ is used for generating the Pareto front. In the first approach, $k$ individual optimization problems are solved for each objective function. In other words, the problem is first solved by setting one of the $\omega_{i}^{\prime}$ s as 1 and the other weights as zero, and so on, until the optimal for each individual objective function has been found. Those points represent the extremes at the Pareto front. Then, the intermediate solutions are obtained by testing different combinations of $\omega_{i}$, which provide more or less importance to each $Z_{i}$. Certainly, particular care should be provided on the scale of each objective functions for a proper selection of the values of the weights, in order to have equality on the contributions of each individual objective to $Z_{\text {mult- }}$

H(X¯)=0
G(X¯)≤0

## 数学代写|随机过程统计代写Stochastic process statistics代考|Weighted Sum Method

优化 从很多 =∑一世=1ķω一世从一世  英石  H(X¯)=0 G(X¯)≤0

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## MATLAB代写

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