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# 数学代写|优化理论代写Optimization Theory代考|ENGR62 Approximate Solving Problem by the Choice of Informational Operator

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## 数学代写|优化理论代写Optimization Theory代考|Improvement of the Lower Estimate of the Accuracy of the Approximate Solving Problem by the Choice of Informational Operator

Let $\widetilde{I}$ be any class of informational operators [285]. Assume that the class creates the informational operators of one type with different sets of functionals. For example, if a set of values of function is used, then their number or set of nodes can change their number or even the value of function is computed within constant $N$ (or both). Informational operators of different types (the value of the function and its derivatives, the coefficient of the factorize by certain basis, etc.) create different classes. It is possible to introduce the characteristics:
\begin{aligned} &\rho(\Pi, A, \widetilde{I})=\inf {I_N(f)} \underset{I}{\rho}\left(\Pi, A, I_N(P)\right)\left(\rho\left(\Pi, A, I_N(P)\right) \equiv \rho(\Pi, A)\right) \ &\rho(\Pi, \Lambda, \widetilde{I})=\inf {A \in \Lambda} \rho(\Pi, A, \widetilde{I}) \end{aligned}
where $\rho=(\Pi, A, \widetilde{I})$ is a lower boundary of the error of the algorithm $A \in \Lambda$ in the problem class $\Pi$ using information from class $\widetilde{I}$, and $\rho(\Pi, \Lambda, \widetilde{I})$ is a lower bound of the error of algorithms in the computing model $(\Pi, \Lambda, \widetilde{I})$.

Information $I_N^0(P) \in \widetilde{I}$, for which the condition $\rho\left(\Pi, A, I_N^0(P)\right)=\rho(\Pi, A, \widetilde{I})$ is performed, is called an optimal in classes $\Pi, \widetilde{I}$ by using the algorithm $A \in \Lambda$. If $\rho\left(\Pi, A^0, I_N^0(P)\right)=\rho(\Pi, \Lambda, \widetilde{I})$, then the algorithm $A^0 \in \Lambda$ and the information $I_N^0(P) \in \widetilde{I}$ are called optimal in this computational model $(\Pi, \Lambda, \widetilde{I})$.

Likewise, it is possible to introduce the definition of complexity for the problem $P$ and the problem of class $\Pi$ and their characteristics:

• $T(\Pi, A, \widetilde{I}, \varepsilon)=\inf _{I_N(P) \in \widetilde{I}} T\left(\Pi, A, I_N(P), \varepsilon\right)$ is $\varepsilon$-complexity of the algorithm $A \in \Lambda(\varepsilon)$ in the problem of class $\Pi$ within the use of information $\widetilde{I}$.
• $T(\Pi, \Lambda(\varepsilon), \widetilde{I})=\inf _{A \in \Lambda(\varepsilon)} T(\Pi, A, \widetilde{I}, \varepsilon)$ is $\varepsilon$-complexity of the problem in this computation model $(\Pi, \Lambda(\varepsilon), \widetilde{I})$.
• $T(P, A, \tilde{I}, \varepsilon)$ is the $\varepsilon$-complexity of the algorithm $A \in \Lambda(\varepsilon)$ when the problem $P \in \Pi$ is solved using information $\widetilde{I}$.
• $T(P, \Lambda(\varepsilon), \widetilde{I})$ is the $\varepsilon$-complexity of the problem $P$ by using the algorithms $\Lambda(\varepsilon)$ and information $\widetilde{I}$, as well as the definition of complexity optimal algorithm and optimal information.

## 数学代写|优化理论代写Optimization Theory代考|Approximate Information

Approximate Information There is known information (approximated) $I_{N \sigma}(P)$ instead of information (exact) $I_N(P)$ where $\sigma \geq 0$ characterizes the deviation of the approximate information from the exact one. It is possible to consider the characteristics for the approximate information $I_{N \sigma}(P)$ that are similar to those that were given above for $I_N(P)$ assuming that information $I_{N \sigma}(P)$ can be adjusted considering $I_0$-information about the problem of the class $\Pi$. Thus, the central algorithm [270] in this case decreases the effect of error of the information $I_{N \sigma}(P)$ on the approximate solution. Examples of constructing these algorithms are given in $[33,106]$.

## 数学代写|优化理论代写Optimization Theory代考|Improvement of the Lower Estimate of the Accuracy of the Approximate Solving Problem by the Choice of Informational Operator

$$\rho(\Pi, A, \widetilde{I})=\inf I_N(f) \rho_I\left(\Pi, A, I_N(P)\right)\left(\rho\left(\Pi, A, I_N(P)\right) \equiv \rho(\Pi, A)\right) \quad \rho(\Pi, \Lambda, \widetilde{I})=\inf A \in \Lambda \rho(\Pi, A, \widetilde{I})$$

$T(\Pi, A, \widetilde{I}, \varepsilon)=\inf _{I_N(P) \in I} T\left(\Pi, A, I_N(P), \varepsilon\right)$ 是 $\varepsilon$ – 算法的㚆杂性 $A \in \Lambda(\varepsilon)$ 在阶级问题上П在信息使用范围内 $\widetilde{I}$.

$T(\Pi, \Lambda(\varepsilon), \widetilde{I})=\inf _{A \in \Lambda(\varepsilon)} T(\Pi, A, \widetilde{I}, \varepsilon)$ 是 $\varepsilon$-此计算模型中问题的㚆杂性 $(\Pi, \Lambda(\varepsilon), \widetilde{I})$.

$T(P, A, \bar{I}, \varepsilon)$ 是个 $\varepsilon$ – 算法的复杂性 $A \in \Lambda(\varepsilon)$ 当问题 $P \in \Pi$ 使用信自解决 $\widetilde{I}$.

$T(P, \Lambda(\varepsilon), \widetilde{I})$ 是个 $\varepsilon$ – 问题的复杂性 $P$ 通过使用算法 $\Lambda(\varepsilon)$ 和信息 $\widetilde{I}$, 以及复杂度最优算法和最优信息的定义。

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