### 数学代写|凸优化作业代写Convex Optimization代考|CPD131

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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|Using least-squares

The least-squares problem is the basis for regression analysis, optimal control, and many parameter estimation and data fitting methods. It has a number of statistical interpretations, e.g., as maximum likelihood estimation of a vector $x$, given linear measurements corrupted by Gaussian measurement errors.

Recognizing an optimization problem as a least-squares problem is straightforward; we only need to verify that the objective is a quadratic function (and then test whether the associated quadratic form is positive semidefinite). While the basic least-squares problem has a simple fixed form, several standard techniques are used to increase its flexibility in applications.
In weighted least-squares, the weighted least-squares cost
$$\sum_{i=1}^k w_i\left(a_i^T x-b_i\right)^2,$$
where $w_1, \ldots, w_k$ are positive, is minimized. (This problem is readily cast and solved as a standard least-squares problem.) Here the weights $w_i$ are chosen to reflect differing levels of concern about the sizes of the terms $a_i^T x-b_i$, or simply to influence the solution. In a statistical setting, weighted least-squares arises in estimation of a vector $x$, given linear measurements corrupted by errors with unequal variances.

Another technique in least-squares is regularization, in which extra terms are added to the cost function. In the simplest case, a positive multiple of the sum of squares of the variables is added to the cost function:
$$\sum_{i=1}^k\left(a_i^T x-b_i\right)^2+\rho \sum_{i=1}^n x_i^2,$$
where $\rho>0$. (This problem too can be formulated as a standard least-squares problem.) The extra terms penalize large values of $x$, and result in a sensible solution in cases when minimizing the first sum only does not. The parameter $\rho$ is chosen by the user to give the right trade-off between making the original objective function $\sum_{i=1}^k\left(a_i^T x-b_i\right)^2$ small, while keeping $\sum_{i=1}^n x_i^2$ not too big. Regularization comes up in statistical estimation when the vector $x$ to be estimated is given a prior distribution.

Weighted least-squares and regularization are covered in chapter 6; their statistical interpretations are given in chapter 7.

## 数学代写|凸优化作业代写Convex Optimization代考|Solving linear programs

There is no simple analytical formula for the solution of a linear program (as there is for a least-squares problem), but there are a variety of very effective methods for solving them, including Dantzig’s simplex method, and the more recent interiorpoint methods described later in this book. While we cannot give the exact number of arithmetic operations required to solve a linear program (as we can for leastsquares), we can establish rigorous bounds on the number of operations required to solve a linear program, to a given accuracy, using an interior-point method. The complexity in practice is order $n^2 m$ (assuming $m \geq n$ ) but with a constant that is less well characterized than for least-squares. These algorithms are quite reliable, although perhaps not quite as reliable as methods for least-squares. We can easily solve problems with hundreds of variables and thousands of constraints on a small desktop computer, in a matter of seconds. If the problem is sparse, or has some other exploitable structure, we can often solve problems with tens or hundreds of thousands of variables and constraints.

As with least-squares problems, it is still a challenge to solve extremely large linear programs, or to solve linear programs with exacting real-time computing requirements. But, like least-squares, we can say that solving (most) linear programs is a mature technology. Linear programming solvers can be (and are) embedded in many tools and applications.

Some applications lead directly to linear programs in the form $(1.5)$, or one of several other standard forms. In many other cases the original optimization problem does not have a standard linear program form, but can be transformed to an equivalent linear program (and then, of course, solved) using techniques covered in detail in chapter 4.
As a simple example, consider the Chebyshev approximation problem:
$$\text { minimize } \max _{i=1, \ldots, k}\left|a_i^T x-b_i\right| \text {. }$$
Here $x \in \mathbf{R}^n$ is the variable, and $a_1, \ldots, a_k \in \mathbf{R}^n, b_1, \ldots, b_k \in \mathbf{R}$ are parameters that specify the problem instance. Note the resemblance to the least-squares problem (1.4). For both problems, the objective is a measure of the size of the terms $a_i^T x-b_i$. In least-squares, we use the sum of squares of the terms as objective, whereas in Chebyshev approximation, we use the maximum of the absolute values.

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Using least-squares

$$\sum_{i=1}^k w_i\left(a_i^T x-b_i\right)^2$$

$$\sum_{i=1}^k\left(a_i^T x-b_i\right)^2+\rho \sum_{i=1}^n x_i^2$$

## 数学代写|凸优化作业代写Convex Optimization代考|Solving linear programs

$$\operatorname{minimize} \max _{i=1, \ldots, k}\left|a_i^T x-b_i\right| .$$

## 有限元方法代写

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## MATLAB代写

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