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# 数学代写|凸优化代写Convex Optimization代考|Methods with Complete Data

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## 数学代写|凸优化代写Convex Optimization代考|Nonsmooth Models of the Objective Function

In the previous section, we looked at several methods for solving the following problem:
$$\min _{x \in Q} f(x)$$
where $f$ is a Lipschitz continuous convex function and $Q$ is a closed convex set. We have seen that the optimal method for problem (3.3.1) is the Subgradient Method (3.2.14), (3.2.16). Note that this conclusion is valid for the whole class of Lipschitz continuous functions. However, if we are going to minimize a particular function from this class, we can expect that it will not be as bad as in the worst case. We usually can hope that the actual performance of the minimization methods can be much better than the worst-case theoretical bound. Unfortunately, as far as the Subgradient Method is concerned, these expectations are too optimistic. The scheme of the Subgradient Method is very strict and in general it cannot converge faster than in theory. It can also be shown that the Ellipsoid Method (3.2.53) inherits this drawback of subgradient schemes. In practice it works more or less in accordance with its theoretical bound even when it is applied to a very simple function like $|x|^2$.

In this section, we will discuss algorithmic schemes which are more flexible than the Subgradient Method and Ellipsoid Method. These schemes are based on the notion of a nonsmooth model of a convex objective function.
Definition 3.3.1 Let $X=\left{x_k\right}_{k=0}^{\infty}$ be a sequence of points in $Q$. Define
$$\hat{f}k(X ; x)=\max {0 \leq i \leq k}\left[f\left(x_i\right)+\left\langle g\left(x_i\right), x-x_i\right\rangle\right]$$
where $g\left(x_i\right)$ are some subgradients of $f$ at $x_i$. The function $\hat{f}_k(X ; \cdot)$ is called a nonsmooth model of the convex function $f$.

Note that $f_k(X ; \cdot)$ is a piece-wise linear function. In view of inequality (3.1.23), we always have
$$f(x) \geq \hat{f}k(X ; x)$$ for all $x \in \mathbb{R}^n$. However, at all test points $x_i, 0 \leq i \leq k$, we have $$f\left(x_i\right)=\hat{f}_k\left(X ; x_i\right), \quad g\left(x_i\right) \in \partial \hat{f}_k\left(X ; x_i\right) .$$ Moreover, the next model is always better than the previous one: $$\hat{f}{k+1}(X ; x) \geq \hat{f}_k(X ; x)$$
for all $x \in \mathbb{R}^n$.

## 数学代写|凸优化代写Convex Optimization代考|Kelley’s Method

The model $\hat{f}_k(X ; \cdot)$ represents complete information on the function $f$ accumulated after $k$ calls of the oracle. Therefore, it seems natural to develop a minimization scheme, based on this object. Perhaps, the most natural method of this type is as follows.
Kelley’s Method

Choose $x_0 \in Q$.

$k$ th iteration $(k \geq 0)$. Find $x_{k+1} \in \operatorname{Arg} \min _{x \in Q} \hat{f}_k(X ; x)$.

Intuitively, this scheme looks very attractive. Even the presence of a complicated auxiliary problem is not too disturbing, since for polyhedral $Q$ it can be solved by linear optimization methods in finite time. However, it turns out that this method cannot be recommended for practical applications. The main reason for this is its instability. Note that the solution of the auxiliary problem in method (3.3.2) may be not unique. Moreover, the whole set $\operatorname{Arg} \min {x \in Q} \hat{f}_k(X ; x)$ can be unstable with respect to an arbitrary small variation of data $\left{f\left(x_i\right), g\left(x_i\right)\right}$. This feature results in unstable practical behavior of the scheme. At the same time, it can be used to construct an example of a problem for which method (3.3.2) has a very disappointing lower complexity bound. Example 3.3.1 Consider the problem (3.3.1) with $$\begin{gathered} f(y, x)=\max \left{|y|,|x|^2\right}, \quad y \in \mathbb{R}, x \in \mathbb{R}^n, \ Q=\left{z=(y, x): y^2+|x|^2 \leq 1\right} \end{gathered}$$ where the norm is standard Euclidean. Thus, the solution of this problem is $z^=$ $\left(y^, x^\right)=(0,0)$, and the optimal value $f^=0$. Denote by $Z_k^=\operatorname{Arg} \min {z \in Q} \hat{f}_k(Z ; z)$ the optimal set of model $\hat{f}_k(Z ; z)$ and let $\hat{f}_k^=\hat{f}_k\left(Z_k^*\right)$ be the optimal value of the model.

Let us choose $z_0=(1,0)$. Then the initial model of the function $f$ is $\hat{f}_0(Z ; z)=$ $y$. Therefore, the first point, generated by Kelley’s method, is $z_1=(-1,0)$. Hence, the next model of the function $f$ is as follows:
$$\hat{f}_1(Z ; z)=\max {y,-y}=|y|$$

## 数学代写|凸优化代写Convex Optimization代考|Nonsmooth Models of the Objective Function

$$\min _{x \in Q} f(x)$$

$$\hat{f} k(X ; x)=\max 0 \leq i \leq k\left[f\left(x_i\right)+\left\langle g\left(x_i\right), x-x_i\right\rangle\right]$$

$$f(x) \geq \hat{f} k(X ; x)$$

$$f\left(x_i\right)=\hat{f}_k\left(X ; x_i\right), \quad g\left(x_i\right) \in \partial \hat{f}_k\left(X ; x_i\right) .$$

$$\hat{f} k+1(X ; x) \geq \hat{f}_k(X ; x)$$

## 数学代写|凸优化代写Convex Optimization代考|Kelley’s Method

$$\hat{f}_1(Z ; z)=\max y,-y=|y|$$

## MATLAB代写

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