The Internet provides the user with more quick way search for information in comparison with traditional ones. The search for information in ISERN1 can be performed using several methods, which differ significantly both in the efficiency and quality of the search, and in the type of information retrieved. Depending on the goals and objectives method seeker information retrieval in IRERN1 are used individually or in combination with each other.

1. Direct appeal to 1LH. The simplest method search, which implies the presence of an address and is reduced to the client's request to a server of a certain type, that is, sending a request using a certain protocol.

Typically, this process begins after entering the address in the appropriate line of the browser program or selecting the address description in the browser window.

When directly addressing the address, you can use an abbreviated notation of the standard 1ЖЬ - omit elements by default. For example, omit the name of the protocol (the protocol is selected by the lower-level domain or the default service is taken); omit the default file name (depending on the server configuration) and the last "/" character; omit the server name and use relative addressing of directory names.

Note that this method is the basis for the operation of more complex technologies, since as a result of complex processes everything comes down to a direct call to the address 1LH.

2. Using a set of links. Most servers that represent general hypertext materials offer links to other servers (they contain 1JB addresses of other resources). This way of finding information is called linkset search. Since all sites in the VWV space are effectively linked, information retrieval can be done by sequentially browsing the linked pages using a browser.

It should be noted that network administrators do not set themselves the goal of placing a complete set of links on the main topics of their server and constantly monitoring their correctness, therefore, this search method does not provide completeness and does not guarantee the reliability of obtaining information. Although this one is completely manual method search looks like a complete anachronism in a network containing more than 60 million nodes, "manual" viewing of Yeb pages is often the only possible one at the final stages of information retrieval, when mechanical "digging" gives way to deeper analysis. The use of catalogs, classified and thematic lists and all kinds of small reference books also applies to this type of search.

3. Use of specialized search engines: search engines, resource directories, metasearch, search for people, teleconference addresses, search in file archives, etc.

The main idea of ​​search engines (servers) is to create a database of words found in Mehrnew documents, in which, for each word, a list of documents containing this word will be stored. The search is carried out in the content of documents. Documents entering SheteG are registered in search engines using special programs and do not require human intervention. Based on this, we receive complete, but by no means reliable information.

Despite the abundance of words and word forms in natural languages, most of them are used infrequently, which was noticed by the linguist Zipf back in the late 40s. XX century In addition, the most common words are conjunctions, prepositions and articles, that is, words that are completely useless when searching for information. As a result, the dictionary of the largest search engine, 11th: epe1 AYaY ^ a, is only a few GB in size. Since all morphological units in the dictionary are ordered, the search for the desired word can be performed without sequential scanning. The presence of lists of documents in which the search word is found allows the search engine to perform operations with these lists: merge, intersect or subtract them.

Search engine queries can be of two types: simple and complex.

At simple request indicates a word or a set of words not separated by any signs. For a complex query, words can be separated from each other logical operators and their combinations. These operators take precedence.

The correctness and the number of documents issued by the search engine depends on how the query is formulated, whether it is simple or complex.

Many search engines use or coexist with thematic directories for searching. Therefore, it can be quite difficult to classify search engines. Most of them can be attributed equally to both search engines and classification catalogs.

The most famous search engines include the following: American(AltaVista, Hot Bot, Lycos, Open Text, Mckinley, Excite, Cuiwww); russians(Yandex, Search, Aport, Tela, Rambler).

In resource directories, a hierarchical (tree-like) and / or network model of the database is used, since any resource with a URL, description and other information is subject to a certain classification - it is called a classifier. The sections of the classifier are called headings. The library analogue of the catalog is a systematic catalog.

The classifier is developed and improved by a team of authors. Then it is used by another group of specialists, called systematizers. The taxonomists, knowing the classifier, read the documents and assign them classification indices indicating which sections of the classifier these documents correspond to.

There are techniques that make it easier to find information using directories. These techniques are called link and link, and both are used by the creators of directories on the Internet. The above techniques are used in a situation where a document can be classified into one of several sections of the classifier, and the person conducting the search may not know to which section.

A reference is used when the creators of the classifier and organizers are able to make a clear decision about assigning a document to one of the sections of the classifier, and the user, in search of this document, can turn to another section. Then in this other section a reference is placed (Cm.) to that section of the classifier, which actually contains information about documents of this type.

For example, information about maps of countries can be placed in the sections "Science-Geography-Country", "Economy-Geography-Country", "References-Map-Country". It is decided that country maps are placed in the second section "Economy-Geography-Country", and references to it are placed in the other two sections. This technique is actively used at Yahoo !.

Link (See also) it is used in a less unambiguous situation when even the creators of the classifier and systematizers are not able to make a clear decision about assigning documents to a certain section of the classifier. It is especially useful in directories that use the network database model.

The following classification catalogs are distributed: European(Yellow Web, Euroseek); American(Yahoo !, Magellan, Infoseek, etc.); russians(WWW, Stars, Weblist, Rocit, Au).

The advantage of metasearch over search engines and directories is that it provides a single interface or point of access to Internet indexes.

There are two types of multiple access tools:

• 1) multiple access services from their " home pages»Provide a menu with a choice of search tools. The popularity of these services is due to the fact that so many search engines are presented in the form of a menu. They allow for easy navigation from one search engine to another without having to remember the URLs or type them into the viewer. Most Popular Multiple Access Services All-in-One(http://www.allonesearch.com); C / Net(http://www.search, com); Internet sleep(http://isleuth.com);
• 2) meta-indexes, often called multi- or integrated search services, provide a single search form in which the user types search query, sent to several search engines at the same time, and the individual results are presented in a single list. This type of service is valuable when you need a maximum sample of documents on a particular subject and when the document is unique.

Another advantage of the meta-index is that the search results of each search engine are quite unique, that is, the meta-index does not produce duplicate links.

The main disadvantage of this search engine is that it does not allow the use of the individual properties of different search engines.

Most Popular Meta-Indexes Beaucoup(http: //www.bea coup.com); Pathfinder(http://www.medialingua.ru/www/wwwsearc.htm).

It should be noted that the division between these two services is rather vague. Some of the larger sections offer access to individual search engines as well as meta-index searches.

So far, the search has been mainly focused on hypertext content. However, you can just as well search for other Internet resources. For this, there are both specialized search engines (searching only for resources of the same type), and "ordinary" search engines that offer additional features search for non-hypertext documents.

Search for people. There is no single list or directory of addresses Email, just as there is no single printed telephone directory for the whole world. There are several commercial and non-commercial referral services, but most involve a particular region or discipline. They are compiled different methods and can be collected by special computer programs from an Internet newsgroup post, or started by individuals who do not necessarily own the addresses. These directories are often referred to as "white pages" and include directories of email and postal addresses, and phone numbers... One of the most reliable ways to find information about personal contacts, if you know the organization to which the person belongs, is to contact home page organizations. Another way is to use personal directories.

As a result of use, the search engine should return the e-mail URL of the correct person.

Main personal directories: Who where(http: // www. whowhere.com); Yahu people(http://yahoo.com/search/people); Four 11(http: //www.four 1 l.com).

There are not so many specialized search engines that search for conference URLs, in particular, these are DejaNews(http://www.dejanews.com is the most sophisticated search system in newsgroups (Usenet). It is characterized by an abundance of advanced search capabilities, useful filters for "cleaning" the result, formal-logical syntax of queries and the ability to search for files.

Many search engines provide the ability to search for conferences as additional service(Yahoo !, Alta Vista, Anzwers, Galaxy, Info Seek, etc.). You can enter the conference search mode using the Usenet button.

Search in file archives. The Internet contains a huge amount of resources. A large part of them are file archives on FTP servers. To find them, specialized search engines are used. The files are registered using special programs, and the file names are indexed.

Some non-specialized search engines also provide the ability to search file archives. For example, by entering search.ftp into AltaVista, we will get links to servers that specialize in finding files on FTP archives. As a result of use, the search engine should return the U RL address of the file.

The main search engines in file archives: Archie(http://archie.de); Filez(http://www.filez.com); FFP-Search(http: // ftpsearch.city.ru).

1. Purpose and classification of search engine optimization methods

Due to the complexity of the design objects, the quality criteria and limitations of the parametric optimization problem (1.5), as a rule, are too complicated for the application of classical methods of extremum search. Therefore, in practice, preference is given to methods of search engine optimization. Let's consider the main stages of any search method.

The initial data in the search methods are the required accuracy of the method  and the starting point of the search X 0.

Then the size of the search step h is selected, and according to a certain rule, new points X k +1 are obtained from the previous point X k, for k = 0,1,2, ... New points are obtained until the condition for stopping the search is satisfied ... The last search point is considered to be the solution to the optimization problem. All search points constitute the search path.

The search methods can differ from each other in the procedure for choosing the step size h (the step can be the same at all iterations of the method or calculated at each iteration), the algorithm for obtaining a new point, and the condition for stopping the search.

For methods using a constant step size, h should be chosen significantly less than the accuracy h »Öe). If for the chosen step size h it is not possible to obtain a solution with the required accuracy, then it is necessary to decrease the step size and continue the search from the last point of the available trajectory.

It is customary to use the following as the search termination conditions:

all adjacent search points are worse than the previous one;

çФ (X k +1) - Ф (X k) ç £ e, that is, the values ​​of the objective function Ф (Х) at neighboring points (new and previous) differ from each other by no more than the required accuracy e;

that is, all partial derivatives at the new search point are practically equal to 0 or differ from 0 by an amount not exceeding the specified precision e.

The algorithm for obtaining a new search point X k +1 from the previous point X k is different for each of the search methods, but any new search point must be no worse than the previous one: if the optimization problem is the problem of finding a minimum, then Ф (X k +1) £ Ф (X k).

Search engine optimization methods are usually classified according to the order of the derivative of the objective function used to obtain new points. So, in the methods of searching for the zero order, it is not required to calculate the derivatives, but the function Ф (Х) itself is sufficient. First order search methods use first partial derivatives, and second order methods use a second derivative matrix (Hessian matrix).

The higher the order of the derivatives, the more reasonable is the choice of a new search point and the fewer the number of method iterations. But at the same time, the complexity of each iteration increases due to the need for the numerical calculation of the derivatives.

The efficiency of the search method is determined by the number of iterations and by the number of calculations of the objective function Ф (Х) at each iteration of the method (N). Let's consider the most common search methods in order of decreasing number of iterations.

For zero-order search methods, the following is true: in the random search method, it is impossible to predict in advance the number of calculations Ф (Х) at one iteration N, and in the coordinate descent method N £ 2 × n, where n is the number of controlled parameters X = (x1, x2. ,…, Xn).

For the first-order search methods, the following estimates are valid: in the gradient method with a constant step N = 2 × n; in the gradient method with step splitting N = 2 × n + n 1, where n 1 is the number of calculations Ф (Х) necessary to check the step splitting condition; in the steepest descent method N = 2 × n + n 2, where n 2 is the number of calculations Ф (Х) required to calculate the optimal step size; and in the Davidon - Fletcher - Powell (DFT) method N = 2 × n + n 3, where n 3 is the number of calculations Ф (Х) required to calculate the matrix approximating the Hessian matrix (for the values ​​n 1, n 2, n 3 the relation n 1< n 2 << n 3).

And finally, in the second-order method - Newton's method N = 3 × n 2. When obtaining these estimates, an approximate calculation of the derivatives is assumed by the formulas of finite differences / 6 /:

that is, to calculate the first-order derivative, you need to know two values ​​of the objective function Ф (Х) at adjacent points, and for the second derivative - the values ​​of the function at three points.

In practice, the steepest descent method and the DFT method have found wide application, as methods with optimal ratio the number of iterations and their complexity.

2. Zero order search methods

2.1. Random search method

In the method of random search, the initial data are the required accuracy of the method e, the starting point of the search X 0 = (x1 0, x2. 0, ..., xn 0) and the size of the search step h. The search for new points is carried out in a random direction, on which the given step h is postponed (Fig. 2.1), thus obtaining a test point X ^ and checking whether the test point is better than the previous search point. For the problem of finding a minimum, this means that

Ф (X ^) £ Ф (X k), k = 0,1,2 ... (2.4)

If condition (2.4) is satisfied, then the test point is included in the search trajectory X k +1 = X ^. Otherwise, the test point is excluded from consideration and a new random direction is selected from the point X k, k = 0,1,2 ,.

Despite the simplicity this method, its main drawback is the fact that it is not known in advance how many random directions will be required to obtain a new point of the search trajectory X k +1, which makes the cost of carrying out one iteration too large. In addition, since the information about the objective function Ф (Х) is not used when choosing the search direction, the number of iterations in the random search method is very large.

In this regard, the random search method is used to study little-studied design objects and to leave the zone of attraction of the local minimum when searching for the global extremum of the objective function / 6 /.

2.2. Coordinate descent method

Unlike the random search method, in the coordinate descent method, directions parallel to the coordinate axes are selected as possible search directions, and movement is possible both in the direction of increasing and decreasing the coordinate value.

The initial data in the coordinate descent method are the step size h and the starting point of the search X 0 = (x1 0, x2. 0,…, xn 0). We start the movement from the point X 0 along the x1 axis in the direction of increasing coordinates. We get a test point X ^ with coordinates (x1 0 + h, x2 0,…, xn 0), for k = 0.

Let us compare the value of the function Φ (X ^) with the value of the function at the previous search point X k. If Ф (X ^) £ Ф (X k) (we assume that it is required to solve the problem of minimizing the objective function Ф (X)), then the test point is included in the search trajectory (X k +1 = X ^).

Otherwise, we exclude the sample point from consideration and get a new sample point, moving along the x1 axis in the direction of decreasing the coordinate. We get a test point X ^ = (x1 k -h, x2. K,…, xn k). We check if Ф (X ^)> Ф (X k), then we continue to move along the x 2 axis in the direction of increasing coordinates. We get a test point X ^ = (x1 k, x2. K + h,…, xn k), etc. When constructing a search trajectory, repeated movement along the points included in the search trajectory is prohibited. Obtaining new points in the coordinate descent method continues until a point X k is obtained, for which all neighboring 2 × n sample points (in all directions x1, x2., ..., xn in the direction of increasing and decreasing the value of each coordinate) will be worse, that is, Ф (X ^)> Ф (X k). Then the search stops and the last point of the search trajectory X * = X k is selected as the minimum point.

3. First order search methods

3.1. Structure of the gradient search method

In the first-order search methods, the vector gradient of the objective function grad (Ф (X k)) is selected as the direction of the search for the maximum of the objective function Φ (X), for the search for the minimum - the vector antigradient -grad (Φ (X k)). In this case, the property of the gradient vector is used to indicate the direction of the fastest change in the function:

To study the methods of first-order search, the following property is also important: the vector gradient grad (Ф (Х k)) is directed along the normal to the level line of the function Ф (Х) at the point Х k (see Fig. 2.4). Level lines are curves on which the function takes on a constant value (Ф (Х) = сnst).

In this chapter, we will look at 5 modifications of the gradient method:

constant step gradient method,

gradient method with step splitting,

steepest descent method,

Davidon-Fletcher-Powell method,

two-level adaptive method.

3.2. Constant step gradient method

In the gradient method with a constant step, the initial data are the required accuracy e, the starting point of the search X 0 and the search step h.

New points are obtained using the formula.

Search engine optimization Is a set of measures to increase the position of sites or their individual web pages in the search results search engines.

The main search engine optimization tools are:

programming,

marketing,

special methods of working with content.

More often than not, a higher position of a site in the search results brings more interested users to the site. When analyzing the effectiveness of search engine optimization, the cost of the target visitor is determined, taking into account the time taken to bring the site to the indicated positions, and the number of users who will stay on the site and take any action is also taken into account.

The essence of search engine optimization is to create pages, the content of which is convenient both for the user to read and for indexing by search robots. The search engine enters the optimized pages into its database in such a way that when the user queries for keywords, the site is placed at the top of the search results. the likelihood of a user visiting the site increases. Consequently, on the contrary, if the optimization was not carried out, then the site's ranking in the search result will be low (far from the first page), and the probability that the user will visit such a site is minimal.

It is not uncommon for search engine robots to be unable to read a web page. Such a site does not appear in the results at all search results, and the likelihood that visitors will find it generally tends to zero.

The main goal of a site's search engine optimization is to improve the site's position in search engine results. To do this, you should analyze existing methods optimization and identify the most effective among them.

Search engine optimization techniques developed taking into account the basic principles of information retrieval systems. Therefore, first of all, it is necessary to evaluate the parameters of the site, by which the search engines calculate its relevance, namely:

keyword density (modern search engine algorithms analyze the text and filter out pages where keywords occur too often),

site citation index (by the way, the network offers many tools to increase the citation of the site, i.e. you can simply buy units), which depends on the authority and the number of web resources that link to the site,

organization of links from sites, the subject of which is identical to the subject of the optimized site.

Thus, all the factors that affect the position of the site in the search results page of the system can be divided into internal and external. Accordingly, optimization requires working with both external and internal factors: bringing the text on the pages in line with key queries; improving the quantity and quality of content on the site; stylistic design of the text, etc.

Search engine optimization methods. Most specialists use search engine optimization without the use of unfair and prohibited methods, which implies a set of measures to increase website traffic, which is based on the analysis of the behavior of targeted visitors.

The research carried out in the work made it possible to highlight the most effective methods of search engine optimization:

increasing the visibility of the site by search engine robots;

improving the usability of the site for visitors;

improving the content on the site;

analysis of requests related to the promoted site and its categories;

search for sites of related topics to create affiliate programs and exchange links.

The analysis of the most common methods of internal search engine optimization, such as:

selection and placement in the site code of meta tags containing short description site content; this method allows you to highlight keywords and phrases for which the optimized site should be found by search engines,

the use of "friendly URLs", which makes the site convenient not only for users, but also for search engines, which will take into account the topic of the page,

optimization of texts on the site, that is, ensuring the correspondence of texts to meta tags. The text should contain words designated in the meta tags as keywords. At the same time, do not forget that an overabundance of keywords in the text can be harmful. First of all, the text may simply become unreadable. In addition, search engines may regard this as spam. It is also possible to increase the "weight" of a word in the text due to the use of formatting elements.

Due to the complexity and low level of knowledge of design objects, both quality criteria and limitations of the parametric optimization problem are, as a rule, too complex for the application of classical methods of extremum search. Therefore, in practice, preference is given to methods of search engine optimization. Consider the main stages of any search method.

The initial data in the search methods are the required accuracy of the method e and the starting point of the search NS 0 .

Then the value of the search step is selected h, and according to some rule, new points are obtained NS k +1 by previous point NS k at k= 0, 1, 2, ... Receiving new points continues until the search termination condition is met. The last search point is considered to be the solution to the optimization problem. All search points constitute the search path.

Search methods differ from each other in the procedure for selecting the step size h(the step can be the same at all iterations of the method or calculated at each iteration), the algorithm for obtaining a new point and the condition for stopping the search.

For methods using a constant step size, h much less precision should be chosen e... If at the selected step size h If it is not possible to obtain a solution with the required accuracy, then it is necessary to reduce the step size and continue the search from the last point of the existing trajectory.

It is customary to use the following as the search termination conditions:

1) all neighboring search points are worse than the previous one;

2) ç F (X k +1 ) –Ф (X k ) ç £ e, that is, the values ​​of the objective function F (X) at neighboring points (new and previous) differ from each other by no more than the required accuracy e;

3) ,i = 1, …, n, that is, all partial derivatives at the new search point are practically equal to 0, that is, they differ from 0 by an amount that does not exceed the accuracy of e.

Algorithm for obtaining a new search point NS k+1 to the previous point NS k its own for each of the search methods, but any new search point must be no worse than the previous one: if the optimization problem is the problem of finding the minimum, then F (X k +1 ) £ F (X k ).

Search engine optimization methods are usually classified according to the order of the derivative of the objective function used to obtain new points. So, in the methods of searching for the zero order, it is not required to calculate the derivatives, but the function itself is sufficient F (X). First order search methods use first partial derivatives, and second order methods use a second derivative matrix (Hessian matrix).

The higher the order of the derivatives, the more reasonable is the choice of a new search point and the fewer the number of method iterations. But at the same time, the laboriousness of each iteration is due to the need for the numerical calculation of the derivatives.

The efficiency of the search method is determined by the number of iterations and by the number of calculations of the objective function F (X) at each iteration of the method.

Consider the most common search methods by arranging them in decreasing order of the number of iterations.

For zero order search methods the following is true: in the random search method, the number of calculations cannot be predicted in advance F (X) at one iteration N, and in the coordinate descent method N£ 2 × n, where n- the number of controlled parameters X = (x 1 , x 2 .,…, x n ).

For first order search methods the following estimates are valid: in the gradient method with a constant step N = 2 × n; in gradient method with step splitting N=2 × n + n 1 , where n 1 - number of calculations F (X), necessary to check the conditions for crushing the step; in the steepest descent method N = 2 × n + n 2 , where n 2 - number of calculations F (X), necessary to calculate the optimal step size; and in the Davidon-Fletcher-Powell (DFP) method N = 2 × n + n 3 , where n 3 - number of calculations F (X), necessary for calculating the matrix approximating the Hessian matrix (for the quantities n 1 , n 2 , n 3 the relation is true n 1 < n 2 < n 3 ).

And finally in the second order method- Newton's method N = 3 × n 2 .

When obtaining these estimates, it is assumed that the derivatives are approximately calculated by the formulas of finite differences, that is, to calculate the first-order derivative, two values ​​of the objective function are required F (X), and for the second derivative - the values ​​of the function at three points.

In practice, the steepest descent method and the DFT method have found wide application, as methods with an optimal ratio of the number of iterations and their complexity.

Let's start looking at zero order search methods. In the method of random search, the initial data are the required accuracy of the method e, the starting point of the search NS 0 = (x 1 0 , x 2 0 , …, x n 0 ) and the size of the search step h.

The search for new points is carried out in a random direction, on which the given step is postponed h thus get a trial point and checking if the probe point is better than the previous search point. For the problem of finding a minimum, this means that:

(6.19)

If given condition is satisfied, then the test point is included in the search trajectory (
). Otherwise, the test point is excluded from consideration and a new random direction is selected from the point NS k , k= 0, 1, 2, ... (Fig. 6.3).

NS k +1

F (X)

Despite the simplicity of this method, its main drawback is the fact that it is not known in advance how many random directions will be required to obtain a new point of the search trajectory NS k +1 , which makes the cost of performing one iteration too large.

Rice. 6.3. To a random search method

In addition, since the choice of the search direction does not use information about the objective function F (X), the number of iterations in the random search method is very large.

In this regard, the random search method is used to study little-studied design objects and to leave the zone of attraction of the local minimum when searching for the global extremum of the objective function.

Unlike the random search method, in the coordinate descent method, directions parallel to the coordinate axes are selected as possible search directions, and movement is possible both in the direction of increasing and decreasing the coordinate value.

The initial data in the coordinate descent method are the step size h and the starting point of the search NS 0 = (x 1 0 , x 2 . 0 ,…, x n 0 ) ... We start the movement from a point NS 0 along the x-axis 1 in the direction of increasing coordinates. Get a test point
(x 1 k + h, x 2 k ,…, x n k), k= 0. Let us compare the value of the function F (X) with the value of the function at the previous search point X k.

If
(we assume that it is required to solve the problem of minimizing F (X), then the test point is included in the search trajectory (
) .

Otherwise, we exclude the test point from consideration and get a new test point, moving along the axis x 1 in the direction of decreasing coordinates. Get a test point
(x 1 k – h, x 2 k ,…, x n k). Check if
, then we continue to move along the x 2 axis in the direction of increasing coordinates. Get a test point
(x 1 k + h, x 2 k ,…, x n k), etc.

When constructing a search trajectory, repeated movement along the points included in the search trajectory is prohibited.

Obtaining new points in the coordinate descent method continues until a point X k is obtained, for which all neighboring 2 × n sample points (in all directions x 1 , x 2 , …, x n in the direction of increasing and decreasing the coordinate value) will be worse, that is,
... Then the search stops and the last point of the search trajectory is selected as the minimum point X * = X k .

Consider the work of the coordinate descent method using an example (Fig. 2.21): n = 2, X = (x 1 , x 2 ), Ф (x 1 , x 2 )  min, F (x 1 , x 2 ) = (x 1 – 1) 2 + (x 2 – 2) 2 , h= 1, X 0 = (0, 1) .

We start moving along the axis x 1 upward

coordinates. Get the first trial point

(x 1 0 + h, x 2 0 ) = (1, 1), F() = (1-1) 2 + (1-2) 2 = 1,

F (X 0 ) = (0-1) 2 + (1-2) 2 = 2,

F ( ) < Ф(Х 0 )  NS 1 = (1, 1).

x 1 from point NS 1

=(x 1 1 + h, x 2 1 ) = (2, 1), F ( ) = (2-1) 2 + (1-2) 2 = 2,

F (X 1 ) = (1-1) 2 + (1-2) 2 = 1,

that is F ( )> Ф (Х 1 ) - the trial point with coordinates (2, 1) is excluded from consideration, and the search for the minimum continues from the point NS 1 .

We continue to move along the axis x 2 from point NS 1 in the direction of increasing coordinates. Get a test point

= (x 1 1 , x 2 1 + h) = (1, 2), F ( ) = (1-1) 2 + (2-2) 2 = 0,

F (X 1 ) = (1-1) 2 + (1-2) 2 = 1,

F ( ) < Ф(Х 1 )  NS 2 = (1, 2).

We continue to move along the axis x 2 from point NS 2 in the direction of increasing coordinates. Get a test point

= (x 1 2 , x 2 2 + h) = (1, 3), F ( ) = (1-1) 2 + (3-2) 2 = 1,

F (X 2 ) = (1-1) 2 + (2-2) 2 = 0,

that is F ( )> Ф (Х 2 ) - the trial point with coordinates (1, 3) is excluded from consideration, and the search for the minimum continues from the point NS 2 .

5. We continue to move along the axis x 1 from point NS 2 in the direction of increasing coordinates. Get a test point

= (x 1 2 + h, x 2 2 ) = (2, 2), F ( ) = (2-1) 2 + (2-2) 2 =1,

F (X 2 ) = (1-1) 2 + (2 - 2) 2 = 0,

that is F (X ^ )> Ф (Х 2 ) - the trial point with coordinates (2, 2) is excluded from consideration, and the search for the minimum continues from the point NS 2 .

6. We continue to move along the axis x 1 from point NS 2 in the direction of decreasing coordinates. Get a test point

= (x 1 2 - h, x 2 2 ) = (0, 2), F ( ) = (0-1) 2 +(2-2) 2 = 1,

F (X 2 ) = (1-1) 2 + (2 - 2) 2 = 0,

that is F ( )> Ф (Х 2 ) - the trial point with coordinates (0, 2) is excluded from consideration, and the search for the minimum is over, since for the point NS 2 the search termination condition is satisfied. The minimum point of the function F (x 1 , x 2 ) = (x 1 – 1) 2 + (x 2 - 2) 2 is NS * = X 2 .

In first-order search methods, as the direction of searching for the maximum of the objective function F (X) the vector is the gradient of the objective function grad(F (X k )) , for finding the minimum - the vector antigradient - grad(F (X k )) ... In this case, the property of the gradient vector is used to indicate the direction of the fastest change in the function:

.

For the study of first-order search methods, the following property is also important: vector gradient grad(F (X k )) directed along the normal to the level line of the function F (X) at the point NS k .

Level lines Are the curves on which the function takes on a constant value ( F (X) = const).

V this section five modifications of the gradient method are considered:

- gradient method with constant step,

- gradient method with step splitting,

- the steepest descent method,

- Davidon-Fletcher-Powell (DFP) method,

- two-level adaptive method.

In the gradient method with a constant step, the initial data are the required accuracy e, starting point of search NS 0 and search step h.

NS k + 1 = NS k - h× gradF(NS k ), k = 0,1,2, ... (6.20)

Formula (2.58) is applied if for the function F (X) you need to find the minimum. If the problem of parametric optimization is posed as the problem of finding the maximum, then to obtain new points in the gradient method with a constant step, the following formula is used:

NS k + 1 = NS k + h× gradF(NS k ), k = 0, 1, 2, ... (6.21)

Each of formulas (6.20), (6.21) is a vector relation including n equations. For example, given NS k +1 = (x 1 k +1 , x 2 k +1 ,…, x n k +1 ), NS k =(x 1 k , x 2 k ,…, x n k ) :

(6.22)

or, in scalar form,

(6.23)

In general, (2.61) can be written:

(6.24)

As a condition for terminating the search, in all gradient methods, as a rule, a combination of two conditions is used: ç F (X k +1 ) - Ф (X k ) ç £ e or
for all i =1, …, n.

Consider an example of finding a minimum using the gradient method with a constant step for the same function as in the coordinate descent method:

n = 2, X = (x 1 , x 2 ),  =0.1,

F (x 1 , x 2 ) = (x 1 – 1) 2 + (x 2 – 2) 2  min, h = 0,3, NS 0 = (0, 1).

Get the point NS 1 according to the formula (2.45):

F (X 1 ) = (0.6–1) 2 + (1.6–2) 2 = 0.32, Ф (X 0 ) = (0 –1) 2 + (1–2) 2 = 2.

 F (X 1 ) - Ф (X 0 ) =1,68 >  = 0,1  continue searching.

Get the point NS 2 according to the formula (2.45):

F (X 2 ) = (0.84–1) 2 + (1.84–2) 2 = 0.05,

F (X 1 ) = (0,6 –1) 2 + (1,6–2) 2 = 0,32.

 F (X 1 ) - Ф (X 0 ) =0,27 >  = 0,1  continue searching.

Similarly, we get X 3:

F (X 3 ) = (0.94–1) 2 + (1.94–2) 2 = 0.007,

F (X 3 ) = (0,84 –1) 2 + (1,84–2) 2 = 0,05.

Since the search termination condition is satisfied, it found NS * = X 3 = (0.94, 1.94) with accuracy  = 0.1.

The search path for this example is shown in Fig. 6.5.

The undoubted advantage of gradient methods is the absence of unnecessary costs for obtaining sample points, which reduces the cost of carrying out one iteration. In addition, due to the use of an effective search direction (gradient vector), the number of iterations is also noticeably reduced in comparison with the coordinate descent method.

In the gradient method, you can slightly reduce the number of iterations if you learn to avoid situations where several search steps are performed in the same direction.

In the gradient method with step splitting, the procedure for selecting the step size at each iteration is implemented as follows.

e, starting point of search NS 0 h(usually h= 1). New points are obtained using the formula:

NS k + 1 = NS k - h k × gradF(NS k ), k = 0,1,2, ..., (6.25)

where h k- step size by k-th iteration of the search, at h k the condition must be met:

F (X k – h k × gradF (X k )) £ F (X k ) - e × h k ×½ gradF (X k ) ½ 2 . (6.26)

If the value h k is such that inequality (2.64) is not satisfied, then the step is split until this condition is satisfied.

Splitting a step is performed according to the formula h k = h k × a, where 0< a < 1.Такой подход позволяет сократить число итераций, но затраты на проведение одной итерации при этом несколько возрастают.

This makes it easy to replace and supplement procedures, data and knowledge.

In the steepest descent method, at each iteration of the gradient method, the optimal step in the direction of the gradient is selected.

The initial data is the required accuracy e, the starting point of the search is X 0.

New points are obtained using the formula:

NS k + 1 = NS k - h k × gradF(NS k ), k = 0,1,2, ..., (6.27)

where h k = arg minF (X k – h k × gradF (X k )) , that is, the choice of the step is made according to the results of one-dimensional optimization with respect to the parameter h (at 0< h < ¥).

The main idea of ​​the steepest descent method is that at each iteration of the method, the maximum possible step size is selected in the direction of the steepest decrease in the objective function, that is, in the direction of the antigradient vector of the function F (X) at the point NS k... (fig. 2.23).

When choosing the optimal step size, it is necessary from the set NS M = (X½ X = X k – h× gradF (X k ), h Î / h = 22 (2 h-1)2=8(2h-1)=0.

Hence, h 1 = 1/2 is the optimal step at the first iteration of the steepest descent method. Then

NS 1 = NS 0 – 1/2gradF (X 0 ),

x 1 1 =0 -1/2 = 1, x 2 1 = 1-1/2 = 2  NS 1 = (1, 2).

Let's check the fulfillment of the search termination conditions at the search point NS 1 = (1, 2). The first condition is not met

F (X 1 ) -F (X 0 ) = 0-2 =2 >  = 0.1, but it is true

that is, all partial derivatives with precision  can be considered equal to zero, the minimum point is found: X * = X 1 = (1, 2). The search trajectory is shown in Fig. 6.7.

Thus, the steepest descent method found the minimum point of the objective function in one iteration (due to the fact that the function level lines F (x 1 , x 2 ) = (x 1 – 1) 2 + (x 2 – 2) 2 . ((x 1 – 1) 2 + (x 2 –2) 2 = const- the equation of the circle, and the antigradient vector from any point is exactly directed to the minimum point - the center of the circle).

In practice, the objective functions are much more complex, the lines also have a complex configuration, but in any case, the following is true: of all gradient methods, the steepest descent method has the smallest number of iterations, but the search for the optimal step by numerical methods poses a certain problem, since in real problems arising when designing radio electronic devices, the use of classical methods for finding an extremum is practically impossible.

For optimization problems under uncertainty (optimization of stochastic objects), in which one or more controlled parameters are random variables, a two-level adaptive search optimization method is used, which is a modification of the gradient method.

NS 0 and the initial value of the search step h(usually
). New points are obtained using the formula:

NS k + 1 = NS k - h k + 1 × gradФ (X k), k= 0,1,2,…, (6.28)

where is the step h k +1 can be calculated using one of two formulas: h k +1 = h k + l k +1 × a k, or h k +1 = h k × exp(l k +1 × a k ) ... As a reduction factor, one usually chooses l k =1/ k, where k Is the iteration number of the search method.

The meaning of the coefficient l k lies in the fact that at each iteration, some adjustment of the step size is made, while more number iterations of the search method, the closer the next search point is to the extremum point and the more accurate (less) the step must be adjusted in order to prevent moving away from the extremum point.

The magnitude a k determines the sign of such a correction (for a k> 0 the step increases, and for a k <0 уменьшается):

a k = sign ((gradF(NS k ), gradF(NS))} ,

that is a k Is the sign of the dot product of the vectors of the objective function gradients at the points NS k and , where =NS k – h k × gradF (X k ) trial point, and h k Is the step that was used to get the point NS k at the previous iteration of the method.

The sign of the dot product of two vectors allows us to estimate the value of the angle between these vectors (we denote this angle  ). If   9, then the dot product must be positive, otherwise negative. In view of the above, it is easy to understand the principle of adjusting the step size in the two-level adaptive method. If the angle between anti-gradients    (acute angle), then search direction from point NS k is selected correctly, and the step size can be increased (Fig. 6.8).

Rice. 6.8. Selecting the search direction when   

If the angle between the antigradients    (obtuse angle), then search direction from point NS k removes us from the minimum point NS*, and the step must be reduced (fig. 6.9).

Rice. 6.9. Selecting the search direction when  > 

The method is called two-level, since at each iteration of the search, not one, but two points are analyzed and two antigradient vectors are constructed.

This, of course, increases the cost of carrying out one iteration, but allows adaptation (tuning) of the step size h k +1 on the behavior of random factors.

Despite the simplicity of its implementation, the steepest descent method is not recommended as a “serious” optimization procedure for solving the problem of unconstrained optimization of a function of many variables, since it works too slowly for practical use.

The reason for this is the fact that the steepest descent property is a local property, so frequent changes in the search direction are necessary, which can lead to an inefficient computational procedure.

A more accurate and efficient method for solving the parametric optimization problem can be obtained using the second derivatives of the objective function (second order methods). They are based on the approximation (i.e., approximate replacement) of the function F (X) function j(NS),

j(X) = F (X 0 ) + (X - X 0 ) T × gradF (X 0 ) + ½ G(X 0 ) × (X - X 0 ) , (6.29)

where G(X 0 ) - Hessian matrix (Hessian, matrix of second derivatives), calculated at the point NS 0 :

¶ 2 F (X) ¶ 2 F (X) . . . ¶ 2 F (X)

¶ x 1 2 ¶ x 1 ¶ x 2 ¶ x 1 ¶ x n

G(X) = ¶ 2 F (X) ¶ 2 F (X) . . . ¶ 2 F (X)

¶ x 2 ¶ x 1 ¶ x 2 2 ¶ x 2 ¶ x n

¶ 2 F (X) ¶ 2 F (X) . . . ¶ 2 F (X)

¶ x n ¶ x 1 ¶ x n ¶ x 2 ¶ x n 2 .

Formula (2.67) represents the first three terms of the expansion of the function F (X) in the Taylor series in the vicinity of the point NS 0 , therefore, when approximating the function F (X) function j(NS) an error of no more than ½½ occurs X-X 0 ½½ 3.

Taking into account (2.67) in Newton's method, the initial data are the required accuracy e, starting point of search NS 0 and obtaining new points is made according to the formula:

NS k +1 = X k – G -1 (NS k ) × gradФ (X k), k=0,1,2,…, (6.30)

where G -1 (NS k ) - the matrix inverse to the Hessian matrix, calculated at the search point NS k (G(NS k ) × G -1 (NS k ) = I,

I = 0 1… 0 is the identity matrix.

Consider an example of finding the minimum for the same function as in the gradient method with a constant step and in the coordinate descent method:

n = 2, X = (x 1 , x 2 ),  = 0.1,

F (x 1 , x 2 ) = (x 1 – 1) 2 + (x 2 – 2) 2  min, NS 0 =(0, 1).

Get the point NS 1 :

X 1 = X 0 - G –1 (X 0) ∙ grad Ф (X 0),

where

grad Ф (X 0) = (2 ∙ (x 1 0 –1)), 2 ∙ (x 1 0 –1) = (–2, –2), that is

or

x 1 1 = 0 – (1/2∙(–2) + 0∙(–2)) = 1,

x 2 1 = 1 – (0∙(–2) + 1/2∙(–2)) = 2,

X 1 = (1, 2).

Let's check the fulfillment of the search termination conditions: the first condition is not fulfilled

F (X 1 ) -F (X 0 ) = 0 - 2  = 2 >  = 0.1,

but fair

that is, all partial derivatives with accuracy  can be considered equal to zero, the minimum point is found: X * = X 1 = (12). The search trajectory coincides with the trajectory of the steepest descent method (Fig. 2.24).

The main disadvantage of Newton's method is the cost of calculating the inverse Hessian G -1 (NS k ) at each iteration of the method.

The DFT method overcomes the shortcomings of both the steepest descent method and Newton's method.

The advantage of this method is that it does not require the calculation of the inverse Hessian, and as the search direction in the DFT method, the direction is chosen - N k × gradF(X k), where N k- a positive definite symmetric matrix that is recalculated at each iteration (step of the search method) and approximates the inverse Hessian G -1 (NS k ) (N k ® G -1 (NS k ) with increasing k).

In addition, the DFT method, when applied to find the extremum of a function of n variables, converges (that is, gives a solution) in no more than n iterations.

The computational procedure of the DFT method includes the following steps.

The initial data are the required precision e, the starting point of the search NS 0 and the initial matrix N 0 (usually the identity matrix, N 0 = I).

On k-th iteration of the method, the search point X k and the matrix N k (k = 0,1,…).

Let's designate the search direction

d k = -N k × gradФ (X k).

Find the optimal step size l k in the direction d k using the methods of one-dimensional optimization (in the same way as in the steepest descent method, the quantity in the direction of the antigradient vector was chosen)

H. Denote v k = l k × d k and get a new search point NS k +1 = X k + v k .

4. Check the fulfillment of the search termination condition.

If ½ v k ½£ e or ½ gradF (X k +1 ) ½£ e, then the solution is found NS * = X k +1 ... Otherwise, we continue the calculations.

5. Denote u k = gradФ (X k +1) - gradФ (Х k) and the matrix N k +1 we will calculate by the formula:

H k +1 = H k + A k + B k , (6.31)

where A k = v k ... v k T / (v k T × u k ) , B k = - H k × u k ... u k T ... H k / (u k T × H k × u k ) .

A k and V k Are auxiliary size matrices n NS n (v k T matches a string vector, v k means a column vector, the result of multiplication n-dimensional line on n-dimensional column is a scalar (number) and multiplying column by row gives a matrix of size n x n).

6. Increase the iteration number by one and go to step 2 of this algorithm.

The DFT method is a powerful optimization procedure that is effective in optimizing most functions. For one-dimensional optimization of the step size in the DFT method, interpolation methods are used.

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