### 数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

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• Foundations of Data Science 数据科学基础

## 数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

Despite the achievements in the application software of modern computers, today there are many problems for which it is impossible to obtain a solution with given accuracy at limited computing resources. This is all about the problems of mathematical modeling, crystallography, radio astronomy, control of fleeting processes, cryptanalysis, and problems of high dimension.

As a rule, the solution of the applied problems is reduced to the solving typical classes of problems of computational and applied mathematics. Thus, it is important to create methods for building high-speed efficient algorithms for calculating $\varepsilon$-solutions of problems that use minimal computer memory for software. This will improve applied mathematical software and provide an opportunity to solve problems with less computing resources and reduce losses from the uncertainty of conclusions based on approximate solutions.

The main attention in the chapter is given to the creation of the elements of the complexity theory. With the use of it, this would be possible to construct effective complexity algorithms for computation of $\varepsilon$-solutions problems of numerical mathematics with limited computing resources.

Important results in the theory of computing optimization on the computing machinery were obtained by M. S. Bakhvalov, P. S. Bondarenko, V. V. Voievodin, H. Vozhniakovsky, V. V. Ivanov, M. P. Komeichuk, I. M. Molchanov, S. M. Nikolski, A. Sard, I. V. Sergienko, S. L. Sobolev, J. Traub, and others. These results allow estimating $\varepsilon$.

Computational complexity is less investigated than other characteristics. The complexity of the problem in time essentially depends on the computing model (computer architecture). A question of problem classes narrowing, the ways of input data presentation, and the complete use of a priori information on the problem are relevant for computational complexity minimizing of algorithm complexity of $\varepsilon$ solution constructing.

Today, many works are devoted to the study of the possibility of increasing the high speed of computing algorithms by paralleling the computations using traditional (with the focus on sequential computation) numerical methods. The general disadvantage of most of these studies lies in their obtainment of ideal computational models that lead to incomplete use of a priori information about the problem.

This chapter is devoted to the presentation of the general provisions of the complexity theory, statement of problems, algebraic and analytic complexity, and complexity of real computational processes. Key attention is given to the asymptotic qualities of “fast” algorithms, computer architecture, and the complexity and specificity of the characteristic estimate use. There are examples of the elements use of the complexity theory to the $\varepsilon$-solution construction of some practical important problems of computational and applied mathematics [285].

## 数学代写|优化算法代写optimization algorithms代考|General Provisions. Statement of the Problem

Let $F\left(I_{0}\right), A(X)$, and $C(Y)$ be the classes of problems of computational (or applied) mathematics, algorithms, and models of computing tools (computers), and $I_{0}$, $X, Y$ are a multitude of parameters that are dependent on the essentially suitable classes.

It is assumed that for the $\varepsilon$-solution constructing of the problem $f \in F$ (approximate solution, any error that does not exceed $\varepsilon \geq 0$ ), we use the algorithm $a \in A$ that was implemented on the computer $c \in C$ that is oriented on the use of information $I_{0}$ on class $F$ and information $I_{n}(f)$ on the specific class problem. The information (information operator) $I_{n}(f)$ can be given, for example, as a set of functionals $I_{n}(f)=\left(i_{1}(f), i_{2}(f), \ldots, i_{n}(f)\right)^{T}$ from the elements of the problem $f$.

Therefore, computation model is used for $\varepsilon$-solution construction that is described using $I_{0}, I_{n}(f), X, Y$.

The quality of the computational process (CP) of input data reduction, the result of which is $\varepsilon$-solution that is characterized by the computational complexity-the amount of a random computational resource that is necessary to the $\varepsilon$-solution constructing that is also called cost or expenses. The most widely used computing complexity characteristics is a processing time $T=T\left(I_{n}(f), X, Y, \varepsilon\right)$ and computer memory $M=M\left(I_{n}(f), X, Y, \varepsilon\right)$ that are required for $\varepsilon$-solution computing. Dependence of characteristics $T, M$ from $I_{0}$ is not specified since this information does not change.

They say that the problem has a restricted (algebraic) complexity (in this computational model) if there is an algorithm $a \in A$, by which it can be accurately solved $(\varepsilon=0)$ with limited computational complexity.

The problem has unrestricted (analytic) complexity if it cannot be solved precisely $(\varepsilon=0)$ in this computational model with restricted computational complexity.
A specific problem can have an algebraic or analytic complexity depending on input data and set of the computing model operations.

## 数学代写|优化算法代写optimization algorithms代考|solving problem computation of a system of linear algebraic

For example, solving problem computation of a system of linear algebraic equations by Gaussian elimination has an algebraic complexity providing that input data is given accurately and arithmetic operations are performed accurately either. If this condition is not performed, then the problem has an analytic complexity.

In real sets of operations, the great majority of problems of computational and applied mathematics are the problems of unlimited computational complexity; in other words, they are solved approximately ( $\varepsilon>0$ ). The exception is combinatorial and some algebraic problems [3].

The theory of analytic computational complexity is engaged in the optimization of the processes of approximate solving problems. The problems of algebraic complexity are used as an auxiliary in the theory of analytic complexity. On the other hand, the problems of algebraic complexity can have very high complexity and can be solved approximately [10].

The general situation of an approximate $\varepsilon$-solution of a problem constructing with constrained computing resources can be described by the following conditions $[14,106,114,237]$ :
$$\begin{gathered} E(I, X, Y) \leq \varepsilon, \ T(I, X, Y, \varepsilon) \leq T_{0}(\varepsilon), \ M(I, X, Y, \varepsilon) \leq M_{0}(\varepsilon), \end{gathered}$$
where $\varepsilon, T_{0}, M_{0}$ are the given numbers.
The quality of the approximate solution is characterized in the general case by the global error $\left(E\left(I_{n}(f), X, Y\right)\right)$, i.e., the sum of the three components: $E_{H}\left(I_{0}, I_{n}(f), Y\right)$ are the errors that are caused by inaccurate input information; $E_{\mu}\left(I_{0}, I_{n}(f), X\right)$ are the errors of the method; and $E_{z}\left(I_{n}(f), X, Y\right)$ are the errors through rounding $[106,114]$. Computations are often considered in the absence of some or all components of global error. All these can be some real computing situations or the results of idealization of computing conditions to simplify the research [106].
Thus, in the general case, it is needed to compute an approximate solving problem $f \in F$ using the model $I_{0}, I_{n}(f), X, Y$ under constraints (2.1), (2.2), and (2.3).

Further on, we will assume (if nothing other is not expected) that memory $M$ can be increased to the necessary volume; in other words, the constrain (2.3) can be removed but, apparently, by increasing the characteristic of $T$ (process time). This can be done, for example, by increasing a share of “slow” (disk) memory in the general structure of computer memory. Considering that within $\varepsilon \rightarrow 0, M_{0}(\varepsilon) \rightarrow \infty$ (for example, when it comes to rounding errors or errors in the method in stepwise algorithms), we will assume that $\varepsilon \geq \varepsilon_{0}>0$, where $\varepsilon_{0}$ is a given number.
Consider the problem of -solution finding (2.1), (2.2), and (2.3) [285].
Let $A(\varepsilon, X)(A=A(\varepsilon, X) \subseteq A(X))$ be a multitude of CA for which the condition (2.1) is used; in other words the algorithms for $\varepsilon$-solution computation for the given conditions. CA $A\left(\varepsilon, T_{0}\right)$ for which the conditions (2.1), (2.2) are used will be called $T$-effective, and $\left(A\left(\varepsilon, T_{0}\right) \subseteq A(\varepsilon, X)\right)$ is a multitude of $T$-effective CA.

## 数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

MS Bakhvalov, PS Bondarenko, VV Voievodin, H. Vozhniakovsky, VV Ivanov, MP Komeichuk, IM Molchanov, SM Nikolski, A. Sard, IV Sergienko, SL Sobolev, J. 特劳布等人。这些结果允许估计e.

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