## 管理科学代写|决策论代写Management Science Models for Decision Making代考|OPMT3197

2022年10月6日

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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
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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## 管理科学代写|决策论代写Management Science Models for Decision Making代考|Convex and Concave Functions

A real-valued function $g(y)$ defined over some convex subset $\Gamma \subset R^n$ ( $\Gamma$ may be $R^n$ itself) is said to be a convex function if
$$g\left(\alpha y^1+(1-\alpha) y^2\right) \leq \alpha g\left(y^1\right)+(1-\alpha) g\left(y^2\right)$$
for all $y^1, y^2 \in \Gamma$, and $0 \leq \alpha \leq 1$. This inequality defining a convex function is called Jensen’s inequality after the Danish mathematician who introduced it.

To interpret Jensen’s inequality geometrically, introduce an $(n+1)$ th axis for plotting the function value. So points in this space $R^{n+1}$ are $\left(y, y_{n+1}\right)^T$, where on the $y_{n+1}$ th axis we plot the function value $g(y)$ to get a geometric representation of the function.

The set of all points $\left{(y, g(y))^T: y \in \Gamma\right}$ in this space $R^{n+1}$ is a surface, which is the surface or graph of the function $g(y)$.

The line segment $\left{\left(\alpha y^1+(1-\alpha) y^2, \alpha g\left(y^1\right)+(1-\alpha) g\left(y^2\right)\right)^T: 0 \leq \alpha \leq 1\right}$ joining the two points $\left(y^1, g\left(y^1\right)\right)^T,\left(y^2, g\left(y^2\right)\right)^T$ on the graph of the function is called the chord of the function between the points $y^1, y^2$ or on the one-dimensional line interval joining $y^1$ and $y^2$. If we plot the function curve and the chord on the line segment $\left{\alpha y^1+(1-\alpha) y^2: 0 \leq \alpha \leq 1\right}$, then Jensen’s inequality requires that the function curve lie beneath the chord. See Fig. $2.1$ where the function curve and a chord are shown for a function $\theta(\lambda)$ of one variable $\lambda$.

The real-valued function $h(y)$ defined on a convex subset $\Gamma \subset R^n$ is said to be a concave function if $-h(y)$ is a convex function, that is, if
$$h\left(\alpha y^1+(1-\alpha) y^2\right) \geq \alpha h\left(y^1\right)+(1-\alpha) h\left(y^2\right)$$
for all $y^1, y^2 \in \Gamma$ and $0 \leq \alpha \leq 1$; see Fig. 2.2. For a concave function $h(y)$, the function curve always lies above every chord.

## 管理科学代写|决策论代写Management Science Models for Decision Making代考|Piecewise Linear (PL) Functions

Definition: Piecewise Linear (PL) Functions: Considering real-valued continuous functions $f(x)$ defined over $R^n$, these are nonlinear functions that may not satisfy the linearity assumptions over the whole space $R^n$, but there is a partition of $R^n$ into convex polyhedral regions, say $R^n=K_1 \cup K_2 \cup \ldots \cup K_r$ such that $f(x)$ is an affine function within each of these regions individually, that is, for each $1 \leq t \leq r$
there exist constants $c_0^t, c^t=\left(c_1^t, \ldots, c_n^t\right)$ such that $f(x)=f_t(x)=c_0^t+c^t x$ for all $x \in K_t$, and for every $S \subset{1, \ldots, r}$, and at every point $x \in \cap_{t \in S} K_t$, the different functions $f_t(x)$ for all $t \in S$ have the same value.

Now we give some examples of continuous PL functions defined over $R^1$. Denote the variable by $\lambda$.

Each convex polyhedral subset of $R^1$ is an interval; so a partition of $R^1$ into convex polyhedral subsets expresses it as a union of intervals: $\left[-\infty, \lambda_1\right]={\lambda: \lambda \leq$ $\left.\lambda_1\right},\left[\lambda_1, \lambda_2\right]=\left{\lambda: \lambda_1 \leq \lambda \leq \lambda_2\right}, \ldots,\left[\lambda_{r-1}, \lambda_r\right],\left[\lambda_r, \infty\right]$, where $\lambda_1, \ldots, \lambda_r$ are the boundary points of the various intervals, usually called the breakpoints in this partition.

The function $\theta(\lambda)$ is a PL function if there exists a partition of $R^1$ like this such that inside each interval of this partition the slope of $\theta(\lambda)$ is a constant, and its value at each breakpoint agrees with the limits of $\theta(\lambda)$ as $\lambda$ approaches this breakpoint from the left, or right; that is, it should be of the form tabulated below:

Notice that the PI function $\theta(\lambda)$ defined in the table above is continuous, and at ēāch of thẻ breảkpooints $\bar{\lambda} \in\left{\lambda_1 \ldots . \lambda_r\right}$ wẻ vërify thả
$$\lim {\epsilon \rightarrow 0^{-}} \theta(\bar{\lambda}+\epsilon)=\lim {\epsilon \rightarrow 0^{+}} \theta(\bar{\lambda}+\epsilon)=\theta(\bar{\lambda}) .$$
Here are numerical examples of continuous PL functions:
Example 2.1.

# 决策论代写

## 管理科学代写|决策论代写管理科学决策模型代考|凸函数和凹函数

$$g\left(\alpha y^1+(1-\alpha) y^2\right) \leq \alpha g\left(y^1\right)+(1-\alpha) g\left(y^2\right)$$
，则称其为凸函数。这个定义凸函数的不等式被称为延森不等式，以引入它的丹麦数学家的名字命名

$$h\left(\alpha y^1+(1-\alpha) y^2\right) \geq \alpha h\left(y^1\right)+(1-\alpha) h\left(y^2\right)$$
，则称其为凹函数;见图2.2。对于凹函数$h(y)$，函数曲线总是位于每个弦的上方

## 管理科学代写|决策论代写管理科学决策模型代考|分段线性(PL)函数

$$\lim {\epsilon \rightarrow 0^{-}} \theta(\bar{\lambda}+\epsilon)=\lim {\epsilon \rightarrow 0^{+}} \theta(\bar{\lambda}+\epsilon)=\theta(\bar{\lambda}) .$$

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## MATLAB代写

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