# 数学代写|最优化理论作业代写optimization theory代考|Optimality Criteria on Simple Regions

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## 数学代写|最优化理论作业代写optimization theory代考|Optimality Criteria on Simple Regions

Definition 1.1.1 (Local minimum, maximum) Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a real valued function and $M \subset \mathbb{R}^n$. A point $\bar{x} \in M$ is called a local minimum (respectively, a strict local minimum) for $f_{\mid M}$ if there exists an open subset $U \subset \mathbb{R}^n, \bar{x} \in U$, such that $f(x) \geq f(\bar{x})$ (respectively, $f(x)>f(\bar{x})$ ) for all $x \in(M \cap U) \backslash{\bar{x}}$.

If the set $U$ above can be chosen to be $\mathbb{R}^n$, then $\bar{x}$ is called a global minimum (respectively, strict global minimum) for $f_{\mid M}$.

A point $\bar{x}$ is called a (strict) local (respectively, global) maximum for $f_{\mid M}$ if $\bar{x}$ is a (strict) local (respectively, global) minimum for $(-f)_{\mid M}$.

Remark 1.1.2 In literature the word optimum is used to indicate minimum and maximum, respectively.
The nonnegative orthant of $\mathbb{R}^n$ will be denoted by $\mathbb{H}^n$ :
$$\mathbb{H}^n=\left{x=\left(x_1, x_2, \ldots, x_n\right) \in \mathbb{R}^n \mid x_i \geq 0,1 \leq i \leq n\right}$$
Example 1.1.3 Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $f(x)=c^{\top} x$, where $c=$ $\left(c_1, c_2\right)^{\top}$. Consider the origin for $f_{\mid \mathbb{H}^2}$ for various values of $c_1, c_2$ (see Figure 1.1). If both $c_1>0$ and $c_2>0$, then $0 \in \mathbb{H}^2$ is a strict local (and global) minimum for $f_{\mid \mathbb{H}^2}$. If both $c_1 \geq 0$ and $c_2 \geq 0$, then $0 \in \mathbb{H}^2$ is a local (and global) minimum for $f_{\mid \mathbb{H}^2}$. However, if $c_1<0$ or $c_2<0$, then $0 \in \mathbb{H}^2$ is not a local minimum for $f_{\mid \mathbb{H}^2}$.

## 数学代写|最优化理论作业代写optimization theory代考|Optimality Criteria of First and Second Order

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be (Fréchet-) differentiable at $\bar{x}$. We denote by $D f(\bar{x})$ the row vector $\left(\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_n}\right)_{\mid \bar{x}}$ of partial derivatives evaluated at $\bar{x}$. With $C^k\left(\mathbb{R}^n, \mathbb{R}\right)$ we denote the space of $k$-times continuously differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$. In this notation, $C^0\left(\mathbb{R}^n, \mathbb{R}\right)$, respectively, $C\left(\mathbb{R}^n, \mathbb{R}\right)$, stands for the space of continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$. For $U \subset \mathbb{R}^n, V \subset \mathbb{R}^n, U$ open, the space $C^k(U, V)$ is defined in an analogous way. We will simply write $f \in C^k$ if $U$ and $V$ are known.

Theorem 1.2.1 Let $\bar{x} \in \mathbb{R}^n$ be a local minimum for $f: \mathbb{R}^n \rightarrow \mathbb{R}$ If $f$ is differentiable at $\bar{x}$, then $D f(\bar{x})=0$.

Proof. Suppose that $D f(\bar{x}) \neq 0$. Then, there is a vector $\xi \in \mathbb{R}^n$ with $D f(\bar{x}) \xi=\alpha<0$ (for example, $\xi=-D^{\top} f(\bar{x})$ ). Define $\varphi(t)=f(\bar{x}+t \xi)$; see also Figure 1.6. The chain rule yields $\varphi^{\prime}(t)=D f(\bar{x}+t \xi) \cdot \xi$. The Taylor expansion of $\varphi$ around $t=0$ gives: $$\varphi(t)=\varphi(0)+t \varphi^{\prime}(0)+o(|t|)=\underbrace{\varphi(0)}{f(\bar{x})}+t[\underbrace{\varphi^{\prime}(0)}{\alpha<0}+\frac{o(|t|)}{t}],$$ where $o(\cdot)$ is a function $h(\cdot)$ forwhich $\frac{h(|t|)}{|t|} \rightarrow 0$ as $t \rightarrow 0$. Hence, there exists a $\bar{t}>0$ such that $\left|\frac{o(|t|)}{t}\right| \leq\left|\frac{\alpha}{2}\right|$ for $t \in(0, \bar{t})$. Consequently, for $t \in(0, \bar{t})$ we have $\varphi(t) \leq \varphi(0)+\frac{1}{2} \alpha t$; in particular, $\varphi(t)<\varphi(0)$, i.e. $f(\bar{x}+t \xi)<f(\bar{x})$. But then $\bar{x}$ cannot be a local minimum for $f$.

# 最优化代写

## 数学代写|最优化理论作业代写optimization theory代考|Optimality Criteria on Simple Regions

$\mathbb{R}^n$的非负正交用$\mathbb{H}^n$表示:
$$\mathbb{H}^n=\left{x=\left(x_1, x_2, \ldots, x_n\right) \in \mathbb{R}^n \mid x_i \geq 0,1 \leq i \leq n\right}$$

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