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# 数学代写|凸优化代写Convex Optimization代考|Complexity bounds for global optimization

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## 数学代写|凸优化代写Convex Optimization代考|Complexity bounds for global optimization

Let us try to apply the formal language, introduced in the previous section, to a particular problem class. Consider, for example, the following problem:
$$\min _{x \in B_n} f(x)$$
In our terminology, this is a constrained minimization problem without functional constraints. The basic feasible set of this problem is $B_n$, an $n$-dimensional box in $R^n$ :
$$B_n=\left{x \in R^n \mid 0 \leq x^{(i)} \leq 1, i=1 \ldots n\right} .$$
Let us measure distances in $R^n$ using $l_{\infty}$-norm:
$$|x|_{\infty}=\max {1 \leq i \leq n}\left|x^{(i)}\right| .$$ Assume that, with respect to this norm, the objective function $f(x)$ is Lipschitz continuous on $B_n$ : $$|f(x)-f(y)| \leq L|x-y|{\infty} \quad \forall x, y \in B_n,$$
with some constant $L$ (Lipschitz constant).

## 数学代写|凸优化代写Convex Optimization代考|Identity cards of the fields

After the pessimistic results of the previous section, first of all we should understand what could be our goal in theoretical analysis of optimization problems. It seems, everything is clear for general global optimization. But maybe the goals of this field are too ambitious? Maybe in some practical problems we would be satisfied by much less “optimal” solution? Or, maybe there are some interesting problem classes, which are not so dangerous as the class of general continuous functions?
In fact, each of these questions can be answered in a different way. And this way defines the style of research (or rules of the game) in the different fields of nonlinear optimization. If we try to classify these fields, we can easily see that they differ one from another in the following aspects:
Goals of the methods.
Classes of functional components.

Description of the oracle.
These aspects define in a natural way the list of desired properties of the optimization methods. Let us present the “identity cards” of the fields,

• which we are going to consider in the book.
• Name: General global optimization. (Section 1.1)
• Goals: Find a global minimum.
• Functional class: Continuous functions.
• Oracle: $0-1-2$ order black box.
• Desired properties: Convergence to a global minimum.
• Features: From theoretical point of view, this game is too short. We always lose it.
• Problem sizes: There are examples of solving problems with thousands of variables. However, no guarantee for success even for very small problems.
• History: Starts from 1955. Several local peaks of interest related to new heuristic ideas simulated annealing, neural networks, genetic algorithms.

## 数学代写|凸优化代写Convex Optimization代考|Complexity bounds for global optimization

$$\min {x \in B_n} f(x)$$ 用我们的术语来说，这是一个没有功能约束的约束最小化问题。本问题的基本可行集为$B_n$，是$R^n$中的一个$n$维方框: $$B_n=\left{x \in R^n \mid 0 \leq x^{(i)} \leq 1, i=1 \ldots n\right} .$$ 让我们使用$l{\infty}$ -norm来测量$R^n$中的距离:
$$|x|_{\infty}=\max {1 \leq i \leq n}\left|x^{(i)}\right| .$$假设，对于这个范数，目标函数$f(x)$在$B_n$: $$|f(x)-f(y)| \leq L|x-y|{\infty} \quad \forall x, y \in B_n,$$上是Lipschitz连续的

## 数学代写|凸优化代写Convex Optimization代考|Identity cards of the fields

oracle的描述。

Oracle: $0-1-2$订单黑盒子。

## MATLAB代写

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