### 数学代写|随机过程统计代写Stochastic process statistics代考|МТН 3016

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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|Generalized Reduced Gradient Method

When solving an equality-constrained optimization problem, we are looking for a solution $\bar{x}$ that provided us the minimum (or the maximum) value for the objective function, but strictly complying with the constraints. In some cases, the objective function can be directly modified to include all the constraints. In the generalized reduced gradient method, the constraints are manipulated to put some of the variables of the problem as function of other variables, then replacing those variables in the objective function for those generated functions. Thus, an unconstrained problem, or at least an equalityconstrained problem with a reduced number of constraints, can be obtained. The modified objective function could be a single-variable function; it should then be solved by the basic principles of calculus. Otherwise, it can be a multivariable function with no constraints. Thus, a gradient-based method may be useful to solve the reduced problem. In other cases, it is not possible to obtain explicit functionalities for all the variables, and some equality constraints could not be included in the objective function. Nevertheless, the modified problem will have a reduced number of equality constraints, and it could be solved by using a method for equality-constrained optimization problems, such as the method of Lagrange multipliers.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Equality- and Inequality-Constrained Optimization

A more general optimization problem involves a feasible region bounded by equality and inequality constraints. A general way to represent an equalityand inequality-constrained optimization problem is as follows:
optimize $Z=z(\bar{x})$
s.t.
$h(\bar{x})=0$
$g(\bar{x}) \leq 0$
(2.44)
For these type of problems, an optimal solution for $z(\bar{x})$ that complies with both the equality and the inequality constraints must be obtained. The equality constraints must always be satisfied as $h(\bar{x})=0$. Nevertheless, the inequality constraints can be complied in the form $g(\bar{x})=0$, for which it is mentioned that the constraint is active; or it can be satisfied in the form $g(\bar{x})<0$, and the constraint is inactive. The solution of the optimization problem will depend on the number of active and inactive constraints.

To find a solution to an equality- and inequality-constrained optimization problem, an approach similar to that for the method of Lagrange multipliers can be implemented. A new objective function, which includes all the equality and inequality constraints, can be stated as follows:
$$\text { optimize } L=z(\bar{x})+\sum_{i=1}^{m} \lambda_{i} h_{i}(\bar{x})+\sum_{j=1}^{n} \mu_{j} g_{j}(\bar{x})$$
This function is known as the augmented Lagrangian function. As before, $\lambda_{i}$ are the Lagrange multipliers. The new variables $\mu_{j}$ are known as the Karush-Kuhn-Tucker multipliers because of the contributions of William Karush, Harold W. Kuhn, and Albert W. Tucker to the solution method of such problems. The necessary conditions for the augmented Lagrangian function can be stated as follows:
$$\begin{gathered} \frac{\partial L}{\partial \bar{x}}=\nabla z(\bar{x})+\sum_{i=1}^{\mathrm{m}} \lambda_{i} \nabla h_{i}(\bar{x})+\sum_{j=1}^{n} \mu_{j} \nabla g_{j}(\bar{x}) \ \frac{\partial L}{\partial \lambda_{i}}=h_{i}(\bar{x})=0 \ \frac{\partial L}{\partial \mu_{j}}=g_{j}(\bar{x})=0 \end{gathered}$$

## 数学代写|随机过程统计代写Stochastic process statistics代考|Active Set Strategy

As aforementioned, when solving the necessary conditions for the augmented Lagrangian function, it is necessary to know which of the inequality constraints are active, because only those constraints must be included in the function $L$. To detect the active inequalities, the active set strategy can be used. The steps of the method are explained in this section.

1. Set all the Karush-Kuhn-Tucker multipliers to zero. This implies that all the inequality constraints are inactive.
2. Solve the system of equations given by the necessary conditions. This provides the intermediate solution $\bar{x}=\bar{x}_{\mathrm{INT}}$.
3. If for any $j$, all the inequality constraints are satisfied at the solution found in step 2, i.e., $g_{j}\left(\bar{x}{\mathrm{INT}}^{}\right) \leq 0$, and all the Karush-Kuhn-Tucker multipliers are positive, then an optimal solution has been found, and $\bar{x}^{}=\bar{x}{\text {INT }}^{*}$.
4. If one or more of the inequality constraints are not satisfied at the solution $\bar{x}{\mathrm{LT} \text {, }}^{*}$ or one or more $\mu{j}$ are negative, the solution found in step 2 is outside the feasible region. Thus, the constraint with the largest violation, $g_{k}(\bar{x})$, is turned into active and added to the augmented Lagrangian function.

The steps for the active set strategy are represented as a flowchart in Figure 2.7. An example is presented to show the application of this methodology.
Example 2.7: Solve the problem presented in Example
2.5. Nevertheless, to avoid unfeasible solutions where
the number of stages is equal or less than $N_{\min }$, the
following inequality constraint should be added:
$$N \geq N_{\min }+1$$
To solve this problem, the inequality constraint is first modified into the standard form $g(\bar{x}) \leq 0$ as follows:
$$6-N \leq 0$$

## 数学代写|随机过程统计代写Stochastic process statistics代考|Equality- and Inequality-Constrained Optimization

H(X¯)=0
G(X¯)≤0
(2.44)

优化 大号=和(X¯)+∑一世=1米λ一世H一世(X¯)+∑j=1nμjGj(X¯)

∂大号∂X¯=∇和(X¯)+∑一世=1米λ一世∇H一世(X¯)+∑j=1nμj∇Gj(X¯) ∂大号∂λ一世=H一世(X¯)=0 ∂大号∂μj=Gj(X¯)=0

## 数学代写|随机过程统计代写Stochastic process statistics代考|Active Set Strategy

1. 将所有 Karush-Kuhn-Tucker 乘数设置为零。这意味着所有的不等式约束都是无效的。
2. 求解由必要条件给出的方程组。这提供了中间解决方案X¯=X¯我ñ吨.
3. 如果对于任何j，在步骤 2 中找到的解满足所有不等式约束，即Gj(X¯我ñ吨)≤0，并且所有的 Karush-Kuhn-Tucker 乘数都是正的，那么已经找到了一个最优解，并且X¯=X¯INT ∗.
4. 如果在解决方案中不满足一个或多个不等式约束X¯大号吨, ∗或一个或多个μj是否定的，则在步骤 2 中找到的解在可行域之外。因此，违反最大的约束，Gķ(X¯), 变为活动状态并添加到增广拉格朗日函数中。

2.5 中提出的问题。

ñ≥ñ分钟+1

6−ñ≤0

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## MATLAB代写

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