Posted on Categories:Convex optimization, 凸优化, 数学代写

# 数学代写|凸优化代写Convex Optimization代考|Global Efficiency Bounds on Specific Problem Classes

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|凸优化代写Convex Optimization代考|Star-Convex Functions

Let us start from a definition.
Definition 4.1.1 We call the function $f$ star-convex if its set of global minimums $X^$ is not empty and for any $x^ \in X^$ and any $x \in \mathbb{R}^n$ we have $$f\left(\alpha x^+(1-\alpha) x\right) \leq \alpha f\left(x^*\right)+(1-\alpha) f(x) \quad \forall x \in \mathscr{F}, \forall \alpha \in[0,1]$$
A particular example of a star-convex function is a usual convex function. However, in general star-convex function need not to be convex, even in the scalar case. For instance, $f(x)=|x|\left(1-e^{-|x|}\right), x \in \mathbb{R}$, is star-convex, but not convex. Star-convex functions arise quite often in optimization problems related to sum of squares. For example the function $f(x, y)=x^2 y^2+x^2+y^2$ with $(x, y) \in \mathbb{R}^2$ belongs to this class.

Theorem 4.1.4 Assume that the objective function in the problem (4.1.14) is starconvex, and the set $\mathscr{F}$ is bounded: diam $\mathscr{F}=D<\infty$. Let the sequence $\left{x_k\right}$ be generated by method (4.1.16).

1. If $f\left(x_0\right)-f^* \geq \frac{3}{2} L D^3$, then $f\left(x_1\right)-f^* \leq \frac{1}{2} L D^3$.
2. If $f\left(x_0\right)-f^* \leq \frac{3}{2} L D^3$, then the rate of convergence of process (4.1.16) is as follows:
$$f\left(x_k\right)-f\left(x^\right) \leq \frac{3 L D^3}{2\left(1+\frac{1}{3} k\right)^2}, \quad k \geq 0$$ Proof Indeed, in view of inequality (4.1.11) the upper bound on the parameters $M_k$, and definition (4.1.25), for any $k \geq 0$ we have: \begin{aligned} & f\left(x_{k+1}\right)-f\left(x^\right) \leq \min y[ f(y)-f\left(x^\right)+\frac{L}{2}\left|y-x_k\right|^3: \ &\left.y=\alpha x^+(1-\alpha) x_k, \alpha \in[0,1]\right] \ & \leq \min {\alpha \in[0,1]}[ f\left(x_k\right)-f\left(x^\right) \ &\left.-\alpha\left(f\left(x_k\right)-f\left(x^\right)\right)+\frac{L}{2} \alpha^3\left|x^-x_k\right|^3\right] \ & \leq \min _{\alpha \in[0,1]}\left[f\left(x_k\right)-f\left(x^\right)-\alpha\left(f\left(x_k\right)-f\left(x^*\right)\right)+\frac{L}{2} \alpha^3 D^3\right] . \end{aligned}

Let us now look at another interesting class of nonconvex functions.
Definition 4.1.3 A function $f(\cdot)$ is called gradient dominated of degree $p \in[1,2]$ if it attains a global minimum at some point $x^$ and for any $x \in \mathscr{F}$ we have $$f(x)-f\left(x^\right) \leq \tau_f|\nabla f(x)|^p$$
where $\tau_f$ is a positive constant. The parameter $p$ is called the degree of domination.
We do not assume here that the global minimum of function $f$ is unique. Let us give several examples of gradient dominated functions.

Example 4.1.1 (Convex Functions) Let $f$ be convex on $\mathbb{R}^n$. Assume it achieves its minimum at point $x^$. Then, for any $x \in \mathbb{R}^n$ with $\left|x-x^\right|<R$, we have
$$f(x)-f\left(x^\right) \stackrel{(2.1 .2)}{\leq}\left\langle\nabla f(x), x-x^\right\rangle \leq|\nabla f(x)| \cdot R$$
Thus, the function $f$ is a gradient dominated function of degree one on the set $\mathscr{F}=\left{x:\left|x-x^*\right|<R\right}$ with $\tau_f=R$.

Example 4.1.2 (Strongly Convex Functions) Let $f$ be differentiable and strongly convex on $\mathbb{R}^n$. This means that there exists a constant $\mu>0$ such that
$$f(y) \stackrel{(2.1 .20)}{\geq} f(x)+\langle\nabla f(x), y-x\rangle+\frac{1}{2} \mu|y-x|^2$$
for all $x, y \in \mathbb{R}^n$. Then, minimizing both sides of this inequality in $y$, we obtain,
$$f(x)-f\left(x^\right) \leq \frac{1}{2 \mu}|\nabla f(x)|^2 \quad \forall x \in \mathbb{R}^n$$ Thus, $f$ is a gradient dominated function of degree two on the set $\mathscr{F}=\mathbb{R}^n$ with $\tau_f=\frac{1}{2 \mu}$ Example 4.1.3 (Sum of Squares) Consider a system of non-linear equations: $$g(x)=0$$ where $g(x)=\left(g_1(x), \ldots, g_m(x)\right)^T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a differentiable vector function. We assume that $m \leq n$ and that there exists a solution $x^$ to (4.1.33). Let us assume in addition that the Jacobian
$$J^T(x)=\left(\nabla g_1(x), \ldots, \nabla g_m(x)\right)$$
is uniformly non-degenerate on a certain convex set $\mathscr{F}$ containing $x^*$. This means that the value
$$\sigma \equiv \inf {x \in \mathscr{F}} \lambda{\min }\left(J(x) J^T(x)\right)$$
is positive. Consider the function
$$f(x)=\frac{1}{2} \sum_{i=1}^m g_i^2(x)$$

## 数学代写|凸优化代写Convex Optimization代考|Star-Convex Functions

$$f\left(\alpha x^{+}(1-\alpha) x\right) \leq \alpha f\left(x^*\right)+(1-\alpha) f(x) \quad \forall x \in \mathscr{F}, \forall \alpha \in[0,1]$$

$f(x)=|x|\left(1-e^{-|x|}\right), x \in \mathbb{R}$ ，是星凸的，但不是凸的。星凸函数经常出现在与平方和相关的优化问题中。例如函数
$f(x, y)=x^2 y^2+x^2+y^2$ 和 $(x, y) \in \mathbb{R}^2$ 属于这一类。

1. 如果 $f\left(x_0\right)-f^* \geq \frac{3}{2} L D^3$ ，然后 $f\left(x_1\right)-f^* \leq \frac{1}{2} L D^3$.
2. 如果 $f\left(x_0\right)-f^* \leq \frac{3}{2} L D^3$ ，则过程 (4.1.16) 的收玫速度如下:
$f \backslash$ feft $\left(x_{-} k \backslash\right.$ right)-f $\backslash$ left( $(x \wedge \backslash$ right) $\backslash$ leq $\backslash$ frac ${3 L D \wedge 3}{2 \backslash$ left(1+|frac ${1}{3} k \backslash$ right) $\wedge 2}, \mid$ quad $k \backslash$ Igeq 0
证明事实上，鉴于不等式 (4.1.11) 参数的上界 $M_k$, 和定义 (4.1.25), 对于任何 $k \geq 0$ 我们有:

$$f(x)-f \backslash \text { left }(x \wedge \backslash r i g h t) \backslash \text { leq } \backslash \text { tau_f } f \backslash \text { nabla } f(x) \mid \wedge p$$

$$\sigma \equiv \inf x \in \mathscr{F} \lambda \min \left(J(x) J^T(x)\right)$$

$$f(x)=\frac{1}{2} \sum_{i=1}^m g_i^2(x)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。