### 数学代写|凸优化作业代写Convex Optimization代考|ELEN90026

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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|Mathematical optimization

A mathematical optimization problem, or just optimization problem, has the form
$$\begin{array}{ll} \operatorname{minimize} & f_0(x) \ \text { subject to } & f_i(x) \leq b_i, \quad i=1, \ldots, m . \end{array}$$
Here the vector $x=\left(x_1, \ldots, x_n\right)$ is the optimization variable of the problem, the function $f_0: \mathbf{R}^n \rightarrow \mathbf{R}$ is the objective function, the functions $f_i: \mathbf{R}^n \rightarrow \mathbf{R}$, $i=1, \ldots, m$, are the (inequality) constraint functions, and the constants $b_1, \ldots, b_m$ are the limits, or bounds, for the constraints. A vector $x^{\star}$ is called optimal, or a solution of the problem (1.1), if it has the smallest objective value among all vectors that satisfy the constraints: for any $z$ with $f_1(z) \leq b_1, \ldots, f_m(z) \leq b_m$, we have $f_0(z) \geq f_0\left(x^{\star}\right)$.

We generally consider families or classes of optimization problems, characterized by particular forms of the objective and constraint functions. As an important example, the optimization problem (1.1) is called a linear program if the objective and constraint functions $f_0, \ldots, f_m$ are linear, i.e., satisfy
$$f_i(\alpha x+\beta y)=\alpha f_i(x)+\beta f_i(y)$$
for all $x, y \in \mathbf{R}^n$ and all $\alpha, \beta \in \mathbf{R}$. If the optimization problem is not linear, it is called a nonlinear program.

This book is about a class of optimization problems called convex optimization problems. A convex optimization problem is one in which the objective and constraint functions are convex, which means they satisfy the inequality
$$f_i(\alpha x+\beta y) \leq \alpha f_i(x)+\beta f_i(y)$$ for all $x, y \in \mathbf{R}^n$ and all $\alpha, \beta \in \mathbf{R}$ with $\alpha+\beta=1, \alpha \geq 0, \beta \geq 0$. Comparing (1.3) and (1.2), we see that convexity is more general than linearity: inequality replaces the more restrictive equality, and the inequality must hold only for certain values of $\alpha$ and $\beta$. Since any linear program is therefore a convex optimization problem, we can consider convex optimization to be a generalization of linear programming.

## 数学代写|凸优化作业代写Convex Optimization代考|Applications

The optimization problem (1.1) is an abstraction of the problem of making the best possible choice of a vector in $\mathbf{R}^n$ from a set of candidate choices. The variable $x$ represents the choice made; the constraints $f_i(x) \leq b_i$ represent firm requirements or specifications that limit the possible choices, and the objective value $f_0(x)$ represents the cost of choosing $x$. (We can also think of $-f_0(x)$ as representing the value, or utility, of choosing $x$.) A solution of the optimization problem (1.1) corresponds to a choice that has minimum cost (or maximum utility), among all choices that meet the firm requirements.

In portfolio optimization, for example, we seek the best way to invest some capital in a set of $n$ assets. The variable $x_i$ represents the investment in the $i$ th asset, so the vector $x \in \mathbf{R}^n$ describes the overall portfolio allocation across the set of assets. The constraints might represent a limit on the budget (i.e., a limit on the total amount to be invested), the requirement that investments are nonnegative (assuming short positions are not allowed), and a minimum acceptable value of expected return for the whole portfolio. The objective or cost function might be a measure of the overall risk or variance of the portfolio return. In this case, the optimization problem (1.1) corresponds to choosing a portfolio allocation that minimizes risk, among all possible allocations that meet the firm requirements.
Another example is device sizing in electronic design, which is the task of choosing the width and length of each device in an electronic circuit. Here the variables represent the widths and lengths of the devices. The constraints represent a variety of engineering requirements, such as limits on the device sizes imposed by the manufacturing process, timing requirements that ensure that the circuit can operate reliably at a specified speed, and a limit on the total area of the circuit. A common objective in a device sizing problem is the total power consumed by the circuit. The optimization problem (1.1) is to find the device sizes that satisfy the design requirements (on manufacturability, timing, and area) and are most power efficient.

In data fitting, the task is to find a model, from a family of potential models, that best fits some observed data and prior information. Here the variables are the parameters in the model, and the constraints can represent prior information or required limits on the parameters (such as nonnegativity). The objective function might be a measure of misfit or prediction error between the observed data and the values predicted by the model, or a statistical measure of the unlikeliness or implausibility of the parameter values.

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Mathematical optimization

$$\text { minimize } f_0(x) \text { subject to } f_i(x) \leq b_i, \quad i=1, \ldots, m .$$

$$f_i(\alpha x+\beta y)=\alpha f_i(x)+\beta f_i(y)$$

$$f_i(\alpha x+\beta y) \leq \alpha f_i(x)+\beta f_i(y)$$

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## MATLAB代写

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