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# 数据科学代写|数据分析代写Data Analysis代考|Characterizing Minima of Smooth Functions

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## 数据科学代写|数据分析代写Data Analysis代考|Characterizing Minima of Smooth Functions

The results of Section 2.2 give us the tools needed to characterize solutions of the unconstrained optimization problem
$$\min _{x \in \mathbb{R}^n} f(x)$$
where $f$ is a smooth function.
We start with necessary conditions, which give properties of the derivatives of $f$ that are satisfied when $x^*$ is a local solution. We have the following result.

Theorem 2.4 (Necessary Conditions for Smooth Unconstrained Optimization)
(a) Suppose that $f$ is continuously differentiable. If $x^$ is a local minimizer of (2.11), then $\nabla f\left(x^\right)=0$.
(b) Suppose that $f$ is twice continuously differentiable. If $x^$ is a local minimizer of (2.11), then $\nabla f\left(x^\right)=0$ and $\nabla^2 f\left(x^\right)$ is positive semidefinite. Proof We start by proving (a). Suppose for contradiction that $\nabla f\left(x^\right) \neq 0$, and consider a step $\alpha \nabla f\left(x^\right)$ away from $x^$, where $\alpha$ is a small positive number. By setting $p=\alpha \nabla f\left(x^\right)$ in formula (2.3) from Theorem 2.1, we have $$f\left(x^ \quad \alpha \nabla f\left(x^\right)\right)=f\left(x^\right) \quad \alpha \nabla f\left(x^* \quad \gamma \alpha \nabla f\left(x^\right)\right)^T \nabla f\left(x^\right),$$
for some $\gamma \in(0,1)$. Since $\nabla f$ is continuous, we have that
$$\nabla f\left(x^* \quad \gamma \alpha \nabla f\left(x^\right)\right)^T \nabla f\left(x^\right) \geq \frac{1}{2}\left|\nabla f\left(x^\right)\right|^2,$$ for all $\alpha$ sufficiently small, and any $\gamma \in(0,1)$. Thus, by substituting into (2.12), we have that $$f\left(x^ \quad \alpha \nabla f\left(x^\right)\right)=f\left(x^\right) \quad \frac{1}{2} \alpha\left|\nabla f\left(x^\right)\right|^2\right),$$
for all positive and sufficiently small $\alpha$. No matter how we choose the neighborhood $\mathcal{N}$ in the definition of local minimizer, it will contain points of the form $x^-\alpha \nabla f\left(x^\right)$ for sufficiently small $\alpha$. Thus, it is impossible to choose a neighborhood $\mathcal{N}$ of $x^$ such that $f(x) \geq f\left(x^\right)$ for all $x \in \mathcal{N}$, so $x^*$ is not a local minimizer.

## 数据科学代写|数据分析代写Data Analysis代考|Convex Sets and Functions

Convex functions take a central role in optimization precisely because these are the instances for which it is easy to verify optimality and for which such optima are guaranteed to be discoverable within a reasonable amount of computation.

A convex set $\Omega \subset \mathbb{R}^n$ has the property that
$$x, y \in \Omega \Rightarrow(1-\alpha) x+\alpha y \in \Omega \text { for all } \alpha \in[0,1] .$$
For all pairs of points $(x, y)$ contained in $\Omega$, the line segment between $x$ and $y$ is also contained in $\Omega$. The convex sets that we consider in this book are usually closed.
The defining property of a convex function is the following inequality: $f((1-\alpha) x+\alpha y) \leq(1-\alpha) f(x)+\alpha f(y), \quad$ for all $x, y \in \mathbb{R}^n$, all $\alpha \in[0,1]$.
$(2.15)$
The line segment connecting $(x, f(x))$ and $(y, f(y))$ lies entirely above the graph of the function $f$. In other words, the epigraph of $f$, defined as
$$\text { epi } f:=\left{(x, t) \in \mathbb{R}^n \times \mathbb{R} \mid t \geq f(x)\right} \text {, }$$
is a convex set. We sometimes call a function satisfying (2.15) as weakly convex function, to distinguish it from the special class called strongly convex functions, defined in Section 2.5 .

The concepts of “minimizer” and “solution” for the case of convex objective function and constraint set become more elementary in the convex case than in the general case of Section 2.1. In particular, the distinction between “local” and “global” solutions goes away.

## 数据科学代写|数据分析代写Data Analysis代考|Characterizing Minima of Smooth Functions

2.2 节的结果为我们提供了表征无约束优化问题解所需的工具
$$\min _{x \in \mathbb{R}^n} f(x)$$

## 数据科学代写|数据分析代写Data Analysis代考|Convex Sets and Functions

$$x, y \in \Omega \Rightarrow(1-\alpha) x+\alpha y \in \Omega \text { for all } \alpha \in[0,1] .$$

$(2.15)$

\left 缺少或无法识别的分隔符

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