### 数学代写|凸优化作业代写Convex Optimization代考|Summary and discussion

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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|凸优化作业代写Convex Optimization代考|Summary and discussion

In this chapter, we have revisited some mathematical basics of sets, functions, matrices, and vector spaces that will be very useful to understand the remaining chapters and we also introduced the notations that will be used throughout this book. The mathematical preliminaries reviewed in this chapter are by no means complete. For further details, the readers can refer to [Apo07] and [WZ97] for Section 1.1, and [H.J85] and [MS00] for Section 1.2, and other related textbooks.
Suppose that we are given an optimization problem in the following form:
\begin{aligned} \text { minimize } & f(\boldsymbol{x}) \ \text { subject to } & \boldsymbol{x} \in \mathcal{C} \end{aligned}
where $f(\boldsymbol{x})$ is the objective function to be minimized and $\mathcal{C}$ is the feasible set from which we try to find an optimal solution. Convex optimization itself is a powerful mathematical tool for optimally solving a well-defined convex optimization problem (i.e., $f(\boldsymbol{x})$ is a convex function and $\mathcal{C}$ is a convex set in problem $(1.127)$ ), or for handling a nonconvex optimization problem (that can be approximated as a convex one). However, the problem (1.127) under investigation may often appear to be a nonconvex optimization problem (with various camouflages) or a nonconvex and nondeterministic polynomial-time hard (NP-hard) problem that forces us to find an approximate solution with some performance or computational efficiency merits and characteristics instead. Furthermore, reformulation of the considered optimization problem into a convex optimization problem can be quite challenging. Fortunately, there are many problem reformulation approaches (e.g., function transformation, change of variables, and equivalent representations) to conversion of a nonconvex problem into a convex problem (i.e., unveiling of all the camouflages of the original problem).

The bridge between the pure mathematical convex optimization theory and how to use it in practical applications is the key for a successful researcher or professional who can efficiently exert his (her) efforts on solving a challenging scientific and engineering problem to which he (she) is dedicated. For a given opti-

mization problem, we aim to design an algorithm (e.g., transmit beamforming algorithm and resource allocation algorithm in communications and networking, nonnegative blind source separation algorithm for the analysis of biomedical and hyperspectral images) to efficiently and reliably yield a desired solution (that may just be an approximate solution rather than an optimal solution), as shown in Figure 1.6, where the block “Problem Reformulation,” the block “Algorithm Design,” and the block “Performance Evaluation and Analysis” are essential design steps before an algorithm that meets our goal is obtained. These design steps rely on smart use of advisable optimization theory and tools that remain in the cloud, like a military commander who needs not only ammunition and weapons but also an intelligent fighting strategy. It is quite helpful to build a bridge so that one can readily use any suitable mathematical theory (e.g., convex sets and functions, optimality conditions, duality, KKT conditions, Schur complement, S-procedure, etc.) and convex solvers (e.g., CVX and SeDuMi) to accomplish these design steps.

The ensuing chapters will introduce fundamental elements of the convex optimization theory in the cloud on one hand and illustrate how these elements were collectively applied in some successful cutting edge researches in communications and signal processing through the design procedure shown in Figure $1.6$ on the other hand, provided that the solid bridges between the cloud and all the design blocks have been constructed.

## 数学代写|凸优化作业代写Convex Optimization代考|Lines and line segments

In this chapter we introduce convex sets and their representations, properties, illustrative examples, convexity preserving operations, and geometry of convex sets which have proven very useful in signal processing applications such as hyperspectral and biomedical image analysis. Then we introduce proper cones (convex cones), dual norms and dual cones, generalized inequalities, and separating and supporting hyperplanes. All the materials on convex sets introduced in this chapter are essential to convex functions, convex problems, and duality to be introduced in the ensuing chapters. From this chapter on, for simplicity, we may use $\mathbf{x}$ to denote a vector in $\mathbb{R}^{n}$ and $x_{1}, \ldots, x_{n}$ for its components without explicitly mentioning $\mathbf{x} \in \mathbb{R}^{n}$.

Mathematically, a line $\mathcal{L}\left(\mathbf{x}{1}, \mathbf{x}{2}\right)$ passing through two points $\mathbf{x}{1}$ and $\mathbf{x}{2}$ in $\mathbb{R}^{n}$ is the set defined as
$$\mathcal{L}\left(\mathbf{x}{1}, \mathbf{x}{2}\right)=\left{\theta \mathbf{x}{1}+(1-\theta) \mathbf{x}{2}, \theta \in \mathbb{R}\right}, \mathbf{x}{1}, \mathbf{x}{2} \in \mathbb{R}^{n}$$
If $0 \leq \theta \leq 1$, then it is a line segment connecting $\mathbf{x}{1}$ and $\mathbf{x}{2}$. Note that the linear combination $\theta \mathbf{x}{1}+(1-\theta) \mathbf{x}{2}$ of two points $\mathbf{x}{1}$ and $\mathbf{x}{2}$ with the coefficient sum equal to unity as in (2.1) plays an essential role in defining affine sets and convex sets, and hence the one with $\theta \in \mathbb{R}$ is referred to as the affine combination and the one with $\theta \in[0,1]$ is referred to as the convex combination. Affine combination and convex combination can be extended to the case of more than two points in the same fashion.
Affine sets and affine hulls
A set $C$ is said to be an affine set if for any $\mathbf{x}{1}, \mathbf{x}{2} \in C$ and for any $\theta_{1}, \theta_{2} \in \mathbb{R}$ such that $\theta_{1}+\theta_{2}=1$, the point $\theta_{1} \mathbf{x}{1}+\theta{2} \mathbf{x}_{2}$ also belongs to the set $C$. For instance, the line defined in $(2.1)$ is an affine set. This concept can be extended to more than two points, as illustrated in the following example.

## 数学代写|凸优化作业代写Convex Optimization代考|Relative interior and relative boundary

Affine hull defined in (2.13) and affine dimension of a set defined in (2.14) play an essential role in convex geometric analysis, and have been applied to dimension reduction in many signal processing applications such as blind separation (or unmixing) of biomedical and hyperspectral image signals (to be introduced in Chapter 6). To further illustrate their characteristics, it would be useful to address the interior and the boundary of a set w.r.t. its affine hull, which are, respectively, termed as relative interior and relative boundary, and are defined below.

The relative interior of $C \subseteq \mathbb{R}^{n}$ is defined as
\text { relint } \begin{aligned} C &={\mathbf{x} \in C \mid B(\mathbf{x}, r) \cap \text { aff } C \subseteq C, \text { for some } r>0} \ &=\text { int } C \text { if aff } C=\mathbb{R}^{n} \quad(c f .(1.20)), \end{aligned}
where $B(\mathbf{x}, r)$ is a 2 -norm ball with center at $\mathbf{x}$ and radius $r$. It can be inferred from (2.16) that
$$\text { int } C= \begin{cases}\text { relint } C, & \text { if affdim } C=n \ 0, & \text { otherwise. }\end{cases}$$
The relative boundary of a set $C$ is defined as
\begin{aligned} \text { relbd } C &=\mathbf{c l} C \backslash \text { relint } C \ &=\mathbf{b d} C \text {, if int } C \neq \emptyset \text { (by }(2.17)) \end{aligned}
For instance, for $C=\left{\mathbf{x} \in \mathbb{R}^{n} \mid|\mathbf{x}|_{\infty} \leq 1\right}$ (an infinity-norm ball), its interior and relative interior are identical, so are its boundary and relative boundary; for $C=\left{\mathbf{x}_{0}\right} \subset \mathbb{R}^{n}$ (a singleton set), int $C=\emptyset$ and bd $C=C$, but relbd $C=\emptyset$. Note that affdim $(C)=n$ for the former but affdim $(C)=0 \neq n$ for the latter, thereby providing the information of differentiating the interior (boundary) and the relative interior (relative boundary) of a set. Some more examples about the relative interior (relative boundary) of $C$ and the interior (boundary) of $C$, are illustrated in the following examples.

Example 2.2 Let $C=\left{\mathrm{x} \in \mathbb{R}^{3} \mid x_{1}^{2}+x_{3}^{2} \leq 1, x_{2}=0\right}=\mathrm{cl} C$. Then relint $C=$ $\left{\mathbf{x} \in \mathbb{R}^{3} \mid x_{1}^{2}+x_{3}^{2}<1, x_{2}=0\right}$ and relbd $C=\left{\mathbf{x} \in \mathbb{R}^{3} \mid x_{1}^{2}+x_{3}^{2}=1, x_{2}=0\right}$ as shown in Figure $2.3$. Note that int $C=\emptyset$ since affdim $(C)=2<3$, while bd $C=$ cl $C \backslash$ int $C=C$.

Example $2.3$ Let $C_{1}=\left{\mathbf{x} \in \mathbb{R}^{3} \mid|\mathbf{x}|_{2} \leq 1\right}$ and $C_{2}=\left{\mathbf{x} \in \mathbb{R}^{3} \mid|\mathbf{x}|_{2}=1\right}$. Then int $C_{1}=\left{\mathbf{x} \in \mathbb{R}^{3} \mid|\mathbf{x}|_{2}<1\right}=$ relint $C_{1}$ and int $C_{2}=$ relint $C_{2}=0$ due to $\operatorname{affdim}\left(C_{1}\right)=\operatorname{affdim}\left(C_{2}\right)=3$.

From now on, for the conceptual conciseness and clarity in the following introduction to convex sets, sometimes we address the pair (int $C$, bd $C$ ) in the context without explicitly mentioning that a convex set $C$ has nonempty interior. However, when int $C=\emptyset$ in the context, one can interpret the pair (int $C$, bd $C$ ) as the pair (relint $C$, relbd $C$ ).

## 数学代写|凸优化作业代写Convex Optimization代考|Summary and discussion

最小化 F(X)  受制于 X∈C

## 数学代写|凸优化作业代写Convex Optimization代考|Lines and line segments

\mathcal{L}\left(\mathbf{x}{1}, \mathbf{x}{2}\right)=\left{\theta \mathbf{x}{1}+(1-\theta) \ mathbf{x}{2}, \theta \in \mathbb{R}\right}, \mathbf{x}{1}, \mathbf{x}{2} \in \mathbb{R}^{n}\mathcal{L}\left(\mathbf{x}{1}, \mathbf{x}{2}\right)=\left{\theta \mathbf{x}{1}+(1-\theta) \ mathbf{x}{2}, \theta \in \mathbb{R}\right}, \mathbf{x}{1}, \mathbf{x}{2} \in \mathbb{R}^{n}

A 集C被称为仿射集，如果对于任何X1,X2∈C并且对于任何θ1,θ2∈R这样θ1+θ2=1, 点θ1X1+θ2X2也属于集合C. 例如，定义在(2.1)是一个仿射集。这个概念可以扩展到两点以上，如下例所示。

## 数学代写|凸优化作业代写Convex Optimization代考|Relative interior and relative boundary

（2.13）中定义的仿射壳和（2.14）中定义的集合的仿射维数在凸几何分析中发挥着重要作用，并已应用于许多信号处理应用中的降维，例如生物医学和高光谱图像信号（将在第 6 章中介绍）。为了进一步说明它们的特性，有必要通过其仿射壳来解决集合的内部和边界，它们分别称为相对内部和相对边界，并在下面定义。

重新安装 C=X∈C∣乙(X,r)∩ 亲 C⊆C, 对于一些 r>0 = 整数 C 如果 C=Rn(CF.(1.20)),

整数 C={ 重新安装 C, 如果 affdim C=n 0, 除此以外。

重磅 C=ClC∖ 重新安装 C =bdC, 如果 int C≠∅ （经过 (2.17))

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## MATLAB代写

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