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# 数学代写|凸优化代写Convex Optimization代考|Lower complexity bounds for $\mathcal{F}_L^{\infty, 1}\left(R^n\right)$

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## 数学代写|凸优化代写Convex Optimization代考|Lower complexity bounds for $\mathcal{F}_L^{\infty, 1}\left(R^n\right)$

Before we go forward with optimization methods, let us check our possibilities in minimizing smooth convex functions. In this section we obtain the lower complexity bounds for optimization problems with objective functions from $\mathcal{F}_L^{\infty, 1}\left(R^n\right)$ (and, consequently, $\mathcal{F}_L^{1,1}\left(R^n\right)$ ).
Recall that our problem class is as follows.

In order to make our considerations simpler, let us introduce the following assumption on iterative processes.

Assumption 2.1.4 An iterative method $\mathcal{M}$ generates a sequence of test points $\left{x_k\right}$ such that
$$x_k \in x_0+\operatorname{Lin}\left{f^{\prime}\left(x_0\right), \ldots, f^{\prime}\left(x_{k-1}\right)\right}, \quad k \geq 1 .$$
This assumption is not absolutely necessary and it can be avoided by a more sophisticated reasoning. However, it holds for the majority of practical methods.

## 数学代写|凸优化代写Convex Optimization代考|Lower complexity bounds for $\mathcal{S}_{\mu, L}^{\infty, 1}\left(R^n\right)$

Let us get the lower complexity bounds for unconstrained minimization of functions from the class $\mathcal{S}{\mu, L}^{\infty, 1}\left(R^n\right) \subset \mathcal{S}{\mu, L}^{1,1}\left(R^n\right)$. Consider the following problem class.

As in the previous section, we consider the methods satisfying Assumption 2.1.4. We are going to find the lower complexity bounds for our problem in terms of condition number $Q_f=\frac{L}{\mu}$.

Note that in the description of our problem class we do not say anything about the dimension of the space of variables. Therefore formally, this class includes also the infinite-dimensional problems.

We are going to give an example of some bad function defined in the infinite-dimensional space. We could do that also in a finite dimension, but the corresponding reasoning is more complicated.

Consider $R^{\infty} \equiv l_2$, the space of all sequences $x=\left{x^{(i)}\right}_{i=1}^{\infty}$ with finite norm
$$|x|^2=\sum_{i=1}^{\infty}\left(x^{(i)}\right)^2<\infty$$

## 数学代写|凸优化代写Convex Optimization代考|Lower complexity bounds for $\mathcal{F}_L^{\infty, 1}\left(R^n\right)$

$$x_k \in x_0+\operatorname{Lin}\left{f^{\prime}\left(x_0\right), \ldots, f^{\prime}\left(x_{k-1}\right)\right}, \quad k \geq 1 .$$

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