due to the totality of options, you can induce a histogram, evaluate, the scales are often increased garni options, i, nareshti, you can praise the solution - continue the search, or settle down with the knowledge of the solutions.

Irrespective of the versatility and simplicity of the procedure of vipadical probing, it cannot be separated by a significant calculation of labor. To that greater breadth of methods straight poshuk solution.

4.5.3. Methods of insane optimization

Necessary to reach the extremum in the most common forms, to develop the system of non-linear equations to the development of the system of non-linear equations - the task is even more foldable and laborious (to develop the solution of non-linear equations to the simple problem of optimization). Therefore, it is practical to zastosovuyt іnshі pіdhodі to optіzіzatsії funktsіy, vzglyady yakikh chnemо z about direct methods. Nadali here speaks about the minimization, to that the extremum is the minimum.

Ninі razrobleno impersonal numerical methods for zavdan as insane, and mental optimization. The accuracy of the numerical method is characterized by a variety of factors: speed of speed, time spent on one iteration, total memory of the EOM, the method required for the implementation, the class of tasks, etc., etc. One and the same method, effective for the implementation a leader of one type may appear absolutely unacceptable for a leader of another type.

Below, look at the main methods of solving problems of non-linear programming. Keep in mind that the range of such methods is already wide and is overwhelmed. In addition, for a number of methods, which are being considered, there are various modifications. More detailed information can be taken from

For example, in

Let's look at the direct methods of insane optimization, if the exchange is daily.

The sense of direct methods in insane optimization fields in the sequence of points X, X, …, X, such

f(X)>f(X)>… …>f(X). Since the point X can be taken as a sufficient point, you can choose to choose closer to the minimum point. The transition (iteration) from point X to point X, k = 0,1,2, ... consists of two stages:

– vibrate straight ruhu from the dots X;

– vyznachennya krok vzdovzh tsy directly.

Methods for prompting such sequences are often called descent methods, but there is a transition from large values ​​of a function to smaller ones.

Mathematically, methods and descent are described by spіvvіdshennyam

X =X + a k p , k =0,1,2,...,

de p - single vector, which means straight downhill;

a k - Dovzhina kroku.

Different methods and descent are introduced one way in one way to choose p and ak. Really zastosovuyutsya less methods, yakі mayut zbіzhnistyu. The stench allows for the end of the kіlkіst krokіv to take away the minimum point, or go to it close. The quality of similar iterative methods is evaluated for the speed of profitability.

Theoretically, in the methods of descent, the task is broken for an infinite number of iterations. In practice, the calculation is pinned, with victorious criteria (minds), the grains of the iterative process. For example, can you buti mind a little increase-

	argument		X[ k] − X[ k − 1 ]

f(X[k]) - f(X[k-1])< γ . Здесь k – номер итерации; ε , γ – задан-

nі the value of the accuracy of rozvyazannya tasks.

Methods for searching for the minimum point are called determinants, so that the offending parameters of the transition from X to X (directly, the value of the croc) are chosen unambiguously for the information available at the point X. As soon as the transition is victorious, there is a turnaround mechanism, then the algorithm is called a turnaround at least.

Deterministic algorithms of insane minimization are divided into classes depending on the type of information that is victorious. If dermal iterations stop less than the value of minimizing functions, then the method is called the zero-order method. Also, moreover, it is necessary to calculate the first lower functions, which are minimized, then there is a place for methods of the first order,

for the need for additional calculation of other similar ones - methods of a different order.

It should be noted that with the implementation of the insane minimization of the methods of the first and other orders, as a rule, there is a greater high degree of profitability, lower methods of the zero order. However, in practice, the calculation of the first and other similar functions of a large number of sizable labor workers. In a number of behaviors, stench cannot be removed from visual analytic functions. Pokhіdnі different numerical methods sign pardons, yakі can zastosuvannya such methods. In addition, the criterion of optimality can be tasks not in an obvious way, but in a system of equals. And here it is analytically and numerically to know the pokhіdnі staє more smoothly, if not impossible. For this reason, the methods of the zero order are considered here.

Methods of a one-dimensional search. Translation of the method of one-dimensional search - numerical search for the extremum of a function of one argument f(x ) - to achieve a wide and good appearance in literature. Therefore, here we look at only one method, which, with the permission of the authors, is one of the most effective - the method of "golden feathering".

The idea of ​​the method is applied in the next short interval of non-significance - the interval of the value of the argument x, to avenge the minimum point, - to the longest, which does not exceed

admissible error to the result ε. How the last interval can be considered is set by the minds of the zavdannya permissible area of ​​\u200b\u200bthe value of the argument either, at times, if the rest is not left (or) right between, the area is in the middle of the admissible, on the extent to which the shukan minimum is indicated by the forward analysis.

In the middle of the interval, there are two points x = y 0 і x = z 0, which mark the “golden cut” - breaking up on two non-linear parts so that the extension of the greater part to the last of the entire interval is reduced to the greater part of the smaller part. Obviously, the qi points of the spread are symmetrical to the center of the interval (Fig. 26). The coordinates of the points of the "golden cut" can be found from the following proportions:

b − y0		y0 − a	= δ ,	z0 − a		b − z0			= δ,
b - a		b-y		b - a				− a

The stars are easy to take away δ = (1–δ )/δ і diti equal: δ 2 + δ –1=0. As a result, we take away the visible parts that define the "golden cut" interval: δ = 0.618, 1-δ = 0.382. The “golden span” is important for power: the point y 0 is one point of the “golden cut” of the interval, the point z 0 is one of the points of the “golden cut” of the interval. Whom has changed.

checks a simple rosary: ​​0.382/0.618 = 0.618 and (0.618–0.382)/0.618 = 0.382.

Algorithm for searching for a minimum, promptings based on the “golden cut” method, transferring to the skin iteration as one of the short intervals of the left or the right point of the “golden cut” in such a rank, so that the minimum is to shuffle, take the middle:

1. Set k =0, default interval of non-significance, acceptable error to the result ε.

2. Calculate the coordinates of the point of the "golden cut":

y k \u003d a k +0.382 (b k -a k), z k \u003d a k +0.618 (b k -a k).

3. Calculate the values ​​of the target function at the known points

f (y k ) and f (z k ).

4. Yakscho f (yk) ≤ f (zk) (Fig. 26, but), put ak + 1 = ak, bk + 1 = zk, zk + 1 = yk, yk + 1 = ak + zk –yk, k =k+1. In the other direction (Fig. 26, b) a k + 1 = y k, b k + 1 = b k, y k + 1 = z k, z k + 1 = y k + b k -z k, k = k +1.

5. Verify the mind of the completed joke

b k + 1 − a k + 1 ≤ ε. At the same time, the solution is to choose the point x = (y k + 1 + z k + 1 ) 2 . In the next step, go to cut 2.

The efficiency of the “golden cut” method is calculated by the fact that the skin iteration requires only a one-time calculation of the value of the target function.

Direct search method (Hooke-Jeeves method). Deco-

tori cob point X. By changing the components of the vector X, one circles around the given point, as a result of which one finds a point (new basis), which shows directly, in which the change of the function f (X) is changed, which is minimized. At the chosen one directly, the descent is changed, reconsidering that the value of the function changes. The procedure is repeated cyclically until it is possible to know straight down the descent for the improvement of the received teeth.

Algorithm for the Direct Poshuk Method in Yourself infamous looking can be formulated like this:

1. Set by the coordinates x i , i = 1,2, ... n of the cod point (k = 0), by the vector of cod increment of coordinates

∆ X \u003d (∆ x 1, ∆ x 2, ..., ∆ x n ) in the process of cooling the surroundings, the smallest allowable values ​​of the ε component of ∆ X, which will be accelerated by the multiplier λ ≥ 1, which means the speed of the descent, the scale factor d > 1.

2. Take X for the "old basis": X b \u003d X. Calculate

value f (X b).

3. Change the skin coordinate x b i, i = 1,2, ... n,

point X b by the value ∆ x i , then take x i = x b i + ∆ x i then

x i = x b i - ∆ x i. Calculate the values ​​of f (X ) at the test points that are maintained, and equal them to the values ​​of f (X b ). Yakscho f(X)< < f (X б ), то соответствующая координата х i приобретает новое значение, вычисленное по одному из приведенных выражений. В противном случае значение этой координаты остается неизменным. Если после изменения последней n -й координаты f (X )

4. Create a descent straight from the “old” to the “new” basis through the rest, so that the coordinates of the new point are calculated

X: x i \u003d x i + λ (x i -x bi), i \u003d 1.2, ... n. Calculate the value of f(X). How to win the mind f (X )

The “new” basis is taken “old” (X b = X, f (X b) = f (X)) and go to step 5. In the other case, take x i = x i, i = 1.2, .. .n.

5. Yak i p. 3, using the help of change the skin coordinate of the point X

functions f (X ) with the values ​​of f (X b ), subtract from item 4. How f (X )

6. How for all i ∆ x i<ε , вычисления прекращаются. В противном случае уменьшают значения ∆ х i в d раз и переходят к п. 3.

The work of the algorithm is illustrated in fig. 27. Show lines

equal to the minimized function f (x 1 , x 2 ), then the lines that control minds f (x 1 , x 2 ) = f 1 = const, f (x 1, x 2) = f 2 = const and so on. Here f 1 > f 2 > f 3 . Sutsilni lini - the results of a one-time vikonanny pp. 3...5 (directly looking for a change in function and descent), dotted line - advancing descent.

The advantage of the direct search method is the simplicity of computer programming. Vіn vіmagає znanny tsіl'ovoї ї ї ї ї ї ї ї y explicit vyglyadі, і navit easily vrakhovuє zahovuє deyakі zminnі, і navіt folded іn the area of ​​zakuku.

The shortcoming of the direct search method is that in times of strongly twisted, bent, or volodymy kutas, the linear equal function of the veins can appear late without delay, to the point of minimum through the border of the circle directly, which is being analysed.

Deformed bagatohedron method (Nelder-Mead method) the one with the minimization of functions n change f(X) for the n-world open space will be a bagatohedron, what to revenge n +1 top. It is obvious that the skin vertex corresponds to the actual vector Xi . Calculate the value of the target function f(Xi ), i=1,2,…, n +1, at the skin of the vertices of the bagatohedron, the maximum value of these vertices and the most visible vertex xh . Through this vertex and the center of gravity of the other vertices, draw a projecting line, where the point is found xq with smaller values ​​of the target function, lower at the top Xh (Fig. 28, a ). Let's turn off the top xh . 3 vertices that are missing, і points xq a new bagatohedron will be created, from which the procedure is described. In the process of victoring such operations, the bagatohedron changes the dimensions, which made the name of the method mind-boggling.

We introduce the following values: X – coordinate vector of the i-ї vertex of the bagatohedron at the k-th step of the search, i= 1,2,…n +1, k= 1,2,…; h - number of the vertex, which has a value

tires, cream X . Coordinates of the center of gravity of virahu-

	xj [n+2, k] =		n+ 1
follow the formula			∑ xj[i, k] − xj[h, k]		J = 1,2, ... n.
			j=1

Applied algorithm to the bagatohedron method, which is deformed, bends in the offensive:

1. Set by the coefficients of fermentationα , stretching γ >1, embossing β<1 , допустимой погрешностью определения координат

points to the minimum ε. Select the coordinates of the vertices of the outward bagatohedron X, i = 1,2, ... n +1, k = 1.

2. Calculate the values ​​of the goal function at all vertices f (X), i = 1,2, ... n +1, i find the points X, X (in Fig. 28, b, the points are similar X 2 and X 1), and also X.

3. Design a point X through the center of

women: X \u003d X + α (X - X).

4. If f (X ) ≤ X , mark the stretching operation

nya: X \u003d X + γ (X -X). In the next step, go to step 6.

5. There will be a new bagatohedron: yakscho f(X)

by replacing X by X, otherwise by replacing X by X. Continue the calculation of item 2 with k =k +1.

6. What X >f (X )>X for all i , not equal to h ,

pinpoint the squeezing operation: X = X + β (X - X). There will be a new bagatohedron by replacing X with X and continuing the calculation of item 2 with k =k +1.

7. If f (X )>X, then, taking the vertex X, there will be a new bagatohedron, similar to the flow one, changing the lengths of all edges twice: X \u003d X + 0.5 (X -X ) and continue calculating z p. 2 for k=k+1.

At pp. 6, 7 before proceeding to paragraph 2, it is necessary to recheck the vikonnannya mind the completion of the search for a minimum, for example, according to the condition

view max n ∑ + 1 (x j [ i ,k ] − x j [ n + 2,k ] ) 2< ε 2 .

i j = 1

W After an additional operation, expansion and compression of expansion, the shape of the deformed bagatohedron is adapted to the topography of the target function. As a result, the bagatohedron rises up on the old frail surface, changing straight at the curved depressions, clinging to the outskirts of the minimum, which indicates the effectiveness of the examined method.

α =1, 2≤ γ ≤3, 0.4≤β ≤0.6.

The method of overt coordinates (Rosenbrock's method). This is the essence of the field in the last rotations of the coordinate system, it is possible to directly change the most significant change in the goal function (Fig. 29). 3 cob points X zdіysnyuyut descent at the speck X along lines parallel to the coordinate axes. On the next iteration, one of the axes may pass straight ahead x'1 = X-X, reshta - at straight lines perpendicular to x'1 . The descent of the bridging axes should be brought to the point X what gives you the opportunity to inspire a new vector x''1 = X-X and on yoga base a new system directly ask

points to the minimum X.

IN On the view of other methods of the zero order, the Rosenbrock method of orientations to the detection of the optimal point in the skin direction, and not just fixing the sound in all directions. The size of the crack in the process of searching continuously changes fallow in the relief of the surface level. Using the wrapping of coordinates from the regulation to make Rosenbrock's method effective for solving folding optimization tasks.

IN zokrema, a given method for proving the richness of other effectives while minimizing about the “outer” functions (with strongly twisted surfaces of the level), the resultant shards of the resultant directly for the sake of pragma roztashuvatisya vzdovzh osі “yaru”.

Parallel dot method (Powell method). The essence of the field lies in the subsequent conduct of a single search for a minimum goal function n + 1 straight be-yakim with the help of single-mode methods. On the first iteration like the first n coordinates are directly selected, like(n+1)th directly vikoristovuєtsya first їх (Fig. 30). On the skin offensive iteration, the sound starts from the other directly in front of the iteration, apparently the number directly changes to one;(n+1)th directly offensive iteration is given by the vector X-X[n+1] - From the minimum point, found on the first shortest of the forward iteration, through the minimum point, found on the remaining third of the iteration.

The most acceptable variant of the solution, as it is accepted on the managerial level, no matter what food, it is accepted to be optimal, and the process of yoga is supposed to be optimized.

Vzaєmozalezhnіst i skladnіst organіzatsіynih, sotsіalno-ekonomіchnih, tehnіchnih that іnshih aspektіv upravlіnnya virobnitstvom in danii hour zvoditsya up of acceptance upravlіnskogo rіshennya, yak zachіpaє Velika Quantity rіznomanіtnih faktorіv scho tіsno pereplіtayutsya one of a through scho staє nemozhlivim zrobiti analіz cutaneous okremo of vikoristannyam traditsіynih analіtichnih . methods.

Most of the factors act as the primary ones in the process of making a decision, and the stench (for its own sake) does not give any sizable characteristics. It is also true that it is practically unmistakable. At the link with the cim of vinyl, the need for the creation of special methods, building security of choice of important management decisions at the borders of folding organizational, economic, technical tasks (expert assessment, follow-up of operations and methods of optimization and etc.).

Methods, directed at the follow-up of operations, to seek the best solutions for such management areas, such as organization of manufacturing processes and transportation, planning of large-scale manufacturing, material and technical delivery.

Methods for optimizing the solution depend on the successive equal numerical estimates of low factors, the analysis of which traditional methods zdіysniti is not possible. Optimal solution - best medium possible options to an economic system, and the most acceptable of all the taken elements of the system is suboptimal.

The essence of the methods of follow-up operations

As it was guessed earlier, the stench is formed by methods and optimization of management decisions. Their basis is mathematical (deterministic), imaginative models that represent the follow-up process, the type of activity or the system. Such models represent a general characteristic of a particular problem. The stench is the basis for the adoption of an important managerial decision, the process of searching for the optimally accepted option.

Perelik nutrition, yakі play a role for non-intermediate kerіvnikіv The choice of such methods is allowed under the hour of choice of analysis methods:

• steps of priming of selected options;
• more stench than alternatives;
• steps to the appearance of the initial factors;
• some criterion of optimality of the decision.

These are ways of optimizing decisions (managerial) orientation on the search for rational solutions for a large number of firms, businesses, or their productions. The stench is based on the main achievements of statistical, mathematical and economic disciplines (theory of Igor, mass service, graphs, optimal programming, mathematical statistics).

Methods of expert assessments

These ways of optimizing managerial decisions can be complicated, if the task is often not enough to formalize, and also the solution cannot be found with the help of mathematical methods.

Expertise - tse doslіdzhennya foldable special meals at the stage of development of the first managerial decision by senior persons, yakі mayut special luggage knowledge and significant information, for otrimannya vysnovkіv, recommendations, thoughts, assessments. In the process of expert research, there are new achievements in science and technology within the framework of the expert's specialization.

Examined methods and optimization of low management decisions (expert assessments) are effective in lower management tasks in the field of manufacturing:

1. The development of folding processes, phenomena, situations, systems that are characterized by non-formalized, clear characteristics.
2. The ranking of that appointment is determined by the given criteria of the actual factors that act as the initial ways of functioning of the development of the virus system.
3. Examined methods and optimization are especially effective in the field of predicting trends in the development of the system of production, and inspire mutual modality from the outer environment.
4. The promotion of the superiority of the expert assessment is more important than the quality of the functions, which may have a sizable and succinct character, for the additional averaged thoughts of the qualifications of specialists.

The first goal is to use methods and optimization of low management decisions (expert assessment).

Classification of analysis methods

Methods for solving optimization problems, depending on the number of parameters, can be subdivided into:

• One-dimensional optimization methods.
• Methods of optimization of richness.

They are also called "numerical optimization methods". To be precise, these algorithms are just a joke.

As part of the zastosuvannya, similar methods are used:

• direct optimization methods (zero order);
• gradient methods (1st order);
• methods of the 2nd order and in.

Most of the methods of rich optimization are brought closer to the task of another group of methods (one-dimensional optimization).

One-Way Optimization Methods

Whether numerical methods of optimization are based on the approximate calculation of such її characteristics, such as the value of the target function and the functions that set the allowable multiplier, they are better. So, for a skin care task, the choice of characteristics for calculation can be found in the presence of important powers of the analyzed function, the obvious possibilities and possibilities for collecting and processing information.

Use the following methods to solve optimization problems (one-dimensional):

• Fibonacci method;
• dichotomy;
• golden cut;
• subwar crocodile.

Fibonacci method

For the cob, it is necessary to insert the coordinates of point x on the gap as a number, which is more costly in relation to the margin (x - a) to the margin (b - a). Also, a can be seen to move the coordinate 0, and b - 1, the midpoint -?

Assuming that F0 and F1 are equal and equal to each other, F2 is equal to 2, F3 - 3, ..., then Fn = Fn-1 + Fn-2. Also, Fn are the Fibonacci numbers, and the Fibonacci search is the optimal strategy of the so-called successive search to the maximum through those that can be reached closely related to them.

Within the framework of the optimal strategy, it is accepted to choose xn - 1 = Fn-2: Fn, xn = Fn-1: Fn. Whenever there are two intervals (abo), skins of which can act as sounds of an interval of insignificance, the point (decreased) to a new interval matime or coordinates, or. Then, like xn - 2, a point is accepted, so that one of the representations of coordinates can be moved again. In order to win F(xn - 2), the value of the function, decreased in the forward space, it becomes possible to shorten the interval of non-significance and the transfer of one value of the function to the decays.

At the finish stage, you can go to such an interval of insignificance, like , moreover, the middle point is reduced in front of the front crochet. As x1 the point is set, as the coordinate ½+ε can be entered, and the residual non-insignificance interval will be either [½, 1] by reference to .

At the 1st stage, the duration of the th interval shortened to Fn-1: Fn (singles). At the final stages, the shortness of the days of the regular intervals is represented by the numbers Fn-2: Fn-1, Fn-3: Fn-2, …, F2: F3, F1: F2 (1 + 2ε). Again, the length of such an interval, as a residual variant of the future value (1 + 2ε): Fn.

To change ε, then asymptotically 1: Fn is more rn, when n→∞, and r = (√5 - 1) : 2, which is approximately 0.6180.

Varto means that asymptotically for significant n skins the offensive crochet according to Fibonacci is exactly the interval with the indicated coefficient. This result needs to be equal to 0.5 (coefficient of sounding of the interval of non-significance in the framework of the bisection method for searching for the zero of a function).

Dichotomy method

In order to reveal a single goal function, it is necessary to know the її extremum for the interval (a; b). For which the whole abscissa is divided into two or more equivalent parts, then it is necessary to determine the value of this function at 5 points. Dalі vybiraєtsya minimum among that number. The extremum of the function is to blame for lying in the middle of the gap (a"; b"), which lies to the minimum point. The cordonies sound like two girls. Like a minimum of rotting at t. a chi b, vin sounds at chotiri razi. The new interval is also subdivided into equal parts. In connection with this, that the value of the function at three points was assigned at the front stage, it was necessary to calculate the value of the function at two points.

Golden recut method

For the present, the value of n is the coordinate of such points, as xn and xn-1 are close to 1 - r, which is 0.3820, and r ≈ 0.6180. The given value is even close to the optimal strategy.

Let's assume that F(0.3820) > F(0.6180), also called an interval. However, looking at those that are 0.6180 * 0.6180 ≈ 0.3820 ≈ xn-1, then at this point F is already known. Also, the skin stage, starting from the 2nd, needs only one calculation of the target function, in its skin the shortest period of the analyzed interval with a coefficient of 0.6180.

On the vіdminu vіd Fibonacci, in this method it is not necessary to fix the number n even to the cob of the cob.

The “golden cut” of the cutter (a; b) is a cut, if it is extended to the larger part (a; c), it is identical to the larger part of r to the smaller one, then (a; c) to (c; b). It doesn't matter to guess what is meant by the formula that you have looked at. Otzhe, with suttєvih n, the Fibonacci method is transformed into dani.

Warning method

The essence is the search for a direct change in the function of the function, the ruh of which is directly at the time of the success of the search for a progressively growing crop.

We assign the cob coordinate M0 of the F(M) function, the minimum value to h0, just for fun. Let's change the function t. M0. Dali robimo krok and know the significance of this function at this point.

If the function is less important, if it was on the front crochet, then the attacking crucible will grow straight ahead, increasing it in front by 2 times. When the value is greater than the front one, it is necessary to change the direction directly, and then start to collapse at the chosen one directly with the h0 croc. This algorithm can be modified.

Methods of rich optimization

I don’t take the zero-order guesswork method to the point of respecting the pokhіdnі minimizovanoї funktsії, through which it can be effective in times of vindication of any difficulties with the calculation of the pokhіdnih.

The group of methods in the 1st order is called gradient, so that for the installation directly, the gradient of the function is a vector, the storage of which is the private simplistic minimization of the function for the specific optimization parameters.

In the group of methods of the 2nd order, there are 2 similarities (they have to overcome the obezzhenie through the presence of difficulties in their calculation).

Perelik methods of insane optimization

When vikoristanny rich search without zastosuvannya similar methods of insane optimization:

• Hook and Jeeves (zdіysnennya 2 vidіv poshuku - for a reason that doslіdzhuychiy);
• minimization for the correct simplex (search for the point of minimum of the function for additional alignment on the skin edge of the iteration and the value at the vertices of the simplex);
• cyclic coordinate descent (as an orientation as a guide to the search for coordinate vectors);
• Rosenbrock (foundations on the stosuvanni one-way minimization);
• minimization after the deforming simplex (modification of the method of minimization after the correct simplex: adding the procedure of embossing, stretching).

In a situation similar to that of the process of a rich search, they see the method of the most obvious descent (the most fundamental procedure for minimizing a differentiated function with a large number of changes).

Also see more such methods, yak vikoristovuyut pov'yazanі straight (Devidon-Fletcher-Powell method). Yogo is the essence - appointed directly to the joke like Dj * grad (f (y)).

Classification of mathematical methods in optimization

Mentally, out of the roaming of functions (tsile), the stench is bubbling:

• z 1 min;
• rich fruits.

Fallow as a function (linear or non-linear) there is a large number of mathematical methods, directing to the extremum for the achievement of the set task.

According to the criterion of stosuvannya, similar mathematical methods and optimization are divided into:

• methods of calculation of 1 random goal function;
• bagatovimirnі (1-a good-vector value-gradient).

Vykhodyachi s efficiency calculation, іsnuyut:

• methods of Swedish calculation of extremum;
• simplified calculation.

Tse mental classification of analysis methods.

Optimization of business processes

Methods here can win over different, fallow in the face of problems that are virulent. It is accepted to see the following methods of optimizing business processes:

• vinjatki (changes in the cost of the main process, elimination of the reasons for the oversight and entry control, the speed of transport routes);
• simplification (easier clearance, reduced complexity of the product structure, rozpodil robit);
• standardization (definition of special programs, methods, technologies, etc.);
• accelerated (parallel engineering, stimulation, operational design of the last steps, automation);
• changes (changes in the sphere of siroviny, technologies, methods of work, personnel development, work systems, obligatory commitment, order of processing);
• security of mutual relations (for organizational units, personnel, working system);
• vision and inclusion (what necessary processes, components).

Podatkova optimization: methods

Russian legislation gives taxes to the payer due to the richness of the possibility of quickly expanding the taxes, through which it is customary to see such ways, directing them to minimization, like zagalni (classic) and special.

Global methods and tax optimization:

• implementation of the company's corporate policy with the maximum possible taxation of data Russian legislation povnovazhen (the order of write-off of MSHP, vibіr method of rozrahunka viruchki vіd realіzacії product ta іn);
• optimization for an additional contract (laying down the land, reading and writing the right wording);
• zastosuvannya raznomanіtnyh pіlg, podtkovyh vyzvolen.

Another group of methods can also be beaten by all firms, but against the stink, you can still finish the area of ​​congestion. Special methods for optimizing taxes are as follows:

• change money (an operation that transfers a tight tax return, replaces it with another one that allows you to reach a similar goal, but at the same time you can win a simple tax order).
• podіlu vіdnosin (replacing only part of the state operation);
• extension of the tax payment (postponing the moment of the appearance of the tax return to a later calendar period);
• direct shortening of the object of taxable taxation (allowance for the richness of taxable operations chi maina without the negative impact of the main government activity of business).

Classic methods of insane optimization

Entry

As you can see, the classical task of insane optimization may look like:

Іsnuyut analytical and numerical methods of solving these problems.

We are looking forward to solving analytical methods and solving the problem of insane optimization.

Insane optimization methods occupy a significant place in the MO course. The reason is mindful of the uninterrupted zastosuvannya їх pіd hоvіshennya їх pіd hіrіshennya іn low optimization zavdan, і navіt pіd іnіt hаrіl_zatsiії vіrіshennya vyshennya znachny ї zavdan umovnoї opіmіzatsії (zavdan MT).

1. Necessary mind for the local minimum (maximum) point

Let me deliver the minimum value of the function. Apparently, the function of this point is non-negative, that is.

We know that the vikorist function is laid out on the outskirts of the t. Taylor series.

de, - the sum of the members is low in the order of such an increment (two) and ind.

Z (4) is obviously clear that

Let's say something

Z urahuvannyam (6) may: . (7)

Let's say that it's positive, tobto. . We choose from our own, todi tvir, scho superechit (1).

Therefore, it is quite obvious.

Razmirkovuyuchi similarly to other variables, it is necessary to take the necessary mind for the points of the local minimum of the function of rich variables

It is easy to show that the points of the local maximum need to be the same as the points of the local minimum, tobto. minds (8).

Apparently, the result of the confirmation will be an uneven image: - the mind of a non-positive increase in the function to the point of a local maximum.

Take away the necessary mind not to give advice on power supply: whether a stationary point is a minimum point or a maximum point.

You can take away food, having drunk enough mind. Use your mind to transfer the next matrix of other similar goal functions.

2. Sufficient mind for the local minimum (maximum) point

Let's expand the functions in the vicinity of a point in a Taylor series up to quadratic warehouses.

The layout (1) can be given in a shorter, more understandable way: "scalar tvir vector" and "vector-matrix tvir".

The matrix of two similar types of function is similar to the change.

The increment of the function on the base (1") can be recorded in the following way:

Vrakhovuyuchi necessary mind:

Imagine (3) in the view:

The quadratic form is called the differential quadratic form (DKF).

Since the DKF is positively assigned, then the stationary point is the point of the local minimum.

Although the DCF and the matrix that it represents are negatively assigned, then the stationary point is the point of the local maximum.

Otzhe, that sufficient intelligence is necessary for the point of the local minimum to be able to look

(when you need to think, you can write it like this:

Enough mind.

Obviously, it is necessary that sufficient intelligence of the local maximum can be seen:

Let's think of a criterion that allows us to determine: what is the quadratic form of the matrix that it represents, positively assigned, and negatively assigned.

3. Criterion of Sylvester

Allows for input on nutrition: chi є quadratic form that matrix, which її represents, is positively assigned, chi is negatively assigned.

It is called the Hessian matrix.

The head of the Hesse matrix

that DKF, as you might imagine, will be positively assigned, as all the main signs of the Hessian matrix () are positive (then the sign scheme may come:

However, there may be another sign scheme for the main signifiers in the Hessian matrix, for example, then the matrix and DKF are negatively assigned.

4. Euler's method - the classic method of solving the problem of insane optimization

The whole method of foundations is based on the necessary and sufficient minds, which have been developed in 1.1 - 1.3; it is necessary to know local extremums only without interruptions of differentiation functions.

Algorithm for what to do is simple:

1) vikoristovuyuchi nebhіdnі razum formєmo system in the form of non-linear rivnyan. Significantly, it is impossible to analytically break down the system in a snarky fashion; sled zastosuvat numeric method and rozvyazannya systems of non-linear lines (NU) (div. ES). For reasons Euler's method will be an analytic-numerical method. Virishuyuchi indicated the system of alignments, we know the coordinates of the stationary point .;

2) doslіzhuєmo DKF and the Hessian matrix, as you can imagine. For Sylvester's additional criterion, it is determined whether a point is a stationary point as a minimum point or as a maximum point;

3) calculable value of the target function at the extreme point

Using the Euler method, solve the following problem of insane optimization: to know 4 stationary points of a function of the form:

Z'yasuvati character tsikh points, chi є stink points minimum, chi sіdlovimi (div.). encourage graphic display tsієї functions in the open space and on the flat (behind the auxiliary line).

5. Classical task of mental optimization and virishennia: Exclusion method and Lagrange multiplier method (MML)

As you can see, the classical task of mental optimization may look like:

A graph that explains the formulation of the problem (1), (2) in space.

Linear line

Also, the ODR of the examined tasks is a kind of curve, presented to equals (2 ").

Yak is obviously a baby, a point is a point of an insane global maximum; point - the point of the mental (visible) local minimum; point - the point of the mental (visible) local maximum.

Orders (1"), (2") can be changed using the switch-on (substitution) method, changing the alignment (2") while changing, and substituting the solution (1").

The original problem (1"), (2") is thus transformed into the problem of insane function optimization, which can be easily solved by Euler's method.

The method of inclusion (substitution).

Let the function of laying waste in the form of changes:

they are called fallow churns (or churning camps); apparently, you can enter a vector

Reshta of change are called independent change decisions.

Obviously, you can talk about the vector-stovpchik:

that vector.

For the classic manager of mental optimization:

The system (2) is applicable to the switch-off (substitution) method, but it can be allowed only if the fallow mills (stations) are used, then. guilty buti otrimani takі virazi for fallow crops:

Chi zavzhdi the system of equals (2) can be rozvyazat schodo fallow zminnyh - do not zavzhdy, it is possible less at the drop, if the vyznachnik, the ranks of the Jacobian, the elements of which you can look:

not closer to zero (div. recursive theorem in MA course)

As you can see, the functions that are due to be uninterrupted functions, which are differentiating, in a different way, are the elements of the primary function that are to be calculated in the stationary point of the function.

Substitute s (3) into the function (1), maybe:

Dosledzhuvana function to the extremum can be worked out by the Euler method - the method of crazy optimization of a continuously differentiated function.

Also, the exclusion (substitution) method allows you to convert the task of classical smart optimization to the task of insane optimization of a function - the function of minds (4), which allows you to take away the virus system (3).

A shortcoming of the exclusion method: difficulties, which are sometimes impossible to remove the system of expressions (3). Vіlniy vіd tоgo nedolіku, аlе vimagає vikonannya wash (4) ММЛ.

5.2. Method of Lagrange multipliers. Necessary brainwashing with the classic manager of mental optimization. Lagrange function

MML allows for classic mental optimization:

Convert to the crazy optimization of a specially designed function - the Lagrange function:

de - Lagrange multipliers;

As you can see, in the sum, which is formed from the vihіdnoї tsіl'ovoї funktії and "vivified" sum of functions, - the functions that represent їх obmezhennia (2) vihіdnogo zavdannya.

Let the point be the point of the insane extremum of the function, either way, or (the last differential of the function at the point).

Vikoristovuyuchi concept of fallow and independent change - fallow change; - independent change, just imagined (5) in a roaring look:

Z (2) obviously the system is equal to the mind:

The result of calculating the total differential for skin function

Imagine (6) for the "burnt" looking, vicarist concept of fallow and independent change:

Respectfully, scho (6") on vіdminu vіd (5") is a system that is folded from equals.

Let's multiply the skin-e-equation of the system (6") by the e-equal-th Lagrange multiplier.

Let's arrange the Lagrange multipliers in such a rank, so that the square arms under the sign of the first sum (in other words, the coefficients for the differentials of the independent changes) reach zero.

The term "we order" by Lagrange multipliers with a designated rank means that it is necessary to destroy the system for equal time.

The structure of such a system of equalities can be easily taken away by equating the square arc under the sign of the first sum to zero:

Rewrite (8) at the sight

The system (8") is a system of linear alignments, which includes: .

Shards at the key expression (7) the first sum is equal to zero, it is easy to understand, like the other sum is equal to zero, then. May the system of equals come:

The system of equals (8) is composed of equals, and the system of equals (10) is composed of equals; all equal in two systems, but not in

Daily equalization is given by the equalization system (2):

Otzhe, є system іz rivnyan for znakhodzhennya nevіdomih:

Subtracting the result - the system of equals (11) becomes the main difference of the MML.

It is easy to understand that the equalization system (11) can be taken even more simply by introducing the specially constructed Lagrange function (3).

Deisno

Also, the system of equalities (11) can be represented in the view (vicory (12), (13)):

The system of equalizations (14) is a necessary intellectual optimization in the classic mental optimization.

Found in the result of the rozv'yazannya tsієї system of the value of the vector is called mentally-stationary point.

In order to understand the nature of the mentally-stationary point, it is necessary to speed up with sufficient minds.

5.3 Enough brain for the classic head of brain optimization. MML algorithm

Qi mind allow you to understand, chi є mentally-stationary point is a point of a local mental minimum, or a point of a local mental maximum.

It's simple, just like before, as if you had taken enough brains in tasks for an insane extreme. You can take enough brains and set the classic mental optimization.

The result of this follow-up:

de is the point of the local mental minimum.

de - point of the local mental maximum, - Hessian matrix with elements

The Hessian matrix is ​​possible.

The versatility of the Hesse matrix can be changed, vicorist to the mind's unevenness to zero of the Jacobian: . For whom you can think, you can change the fallows through the independent changes, either the Hessian matrix is ​​the same, or the other. it is necessary to talk about a matrix with elements

then enough to think motherly looking:

Krapka local mental minimum.

Krapka local intellectual maximum.

Proof: MML algorithm:

1) we add the Lagrange function: ;

2) vikoristovuyuchi nebhіdnі mind, form the system equal:

3) from the top of the line of the system, a point is known;

4) vikoristovuyuchi sufficient mind, it is important that the point is the point of the local mind minimum or maximum, then we know

1.5.4. Graph-analytical method of solving the classical task of mental optimization in space and yogo modification with the most simple tasks of IP and AP

This method is a vicorist's geometric interpretation of the classical task of mental optimization and grounding on the basis of the most important facts, the most important task.

B - zagalna dotichna for the function that function, which represents the GDR.

As you can see from the little point - the point of the insane minimum, the point is the point of the mental local minimum, the point is the point of the mental local maximum.

It is known that at the points of mental local extremums, the curve and the double lines are equal

From the course of the MA, it is clear that at the point of torsion the mind is victorious

de - kutovy koefіtsієnt dotichny, carried out ї vіdpovіdnoy іnієyu equal; - kutovy coefficient of dotichny, carried out to the function

Vidomy viraz (MA) for these coefficients:

Let us know that the coefficients are equal.

to that about tse "talking" is necessary to understand

Pointing more allows you to formulate the DFA algorithm by the method of solving the classical problem of mental optimization:

1) there will be a family of linear equal goals functions:

2) future ODR

3) due to the method of introducing a corrected increase in the function, it is known that the nature of the extreme points is clear;

4) doslіdzhuєmo vzjaєmodіyu іnііy іnііnіі іnіmіnії і znahodіyachi іn somy іz system іііnіnіnі coordinates mentally stationary points - local mental minima i local mental maxima.

5) countable

It should be especially noted that the main stages of the DFA method of decomposing the classical problem of mental optimization are different from the main stages of the DFA method of decoupling the problems of NP and LP, the inclusion is only in the ODT, as well as the well-known expansion of the extremal points in the ODRx qi points obov'yazkovo are located at the vertices swollen bagatokutnik, which represents the GDR).

5.5. About practical sense MML

Let's imagine the classical task of mental optimization for the look:

de - Changing values, which represent applied technical and economic tasks of changing resources.

At the expanse of zavdannya (1), (2) look out:

de - changing value. (2")

Come on - the point of mental extremum:

When changing, change

It is necessary to change the value of the target function:

Let's calculate the cost:

3 (3), (4), (5). (6)

Let's put (5") in (3) and take:

Z (6) that the Lagrange multiplier characterizes the "reaction" of the value (orthogonal to the value of the goal function) on the change of the parameter.

At the zagalny vpadku (6) nabuvaє vzglyadu:

Z (6), (7), which is a multiplier, characterizes the change when changing a given resource by 1.

Yakshcho - the maximum surplus or the minimum varity, then it characterizes the change in value when changing by 1.

5.6. The classical problem of mental optimization is like a task about the value of the saddle point of the Lagrange function:

A pair is called a saddle point, which means unevenness.

It is obvious that (1). (2)

Z (2), sho. (3)

As you can see, the system (3) is to revenge equals, similar to quiet equals, as to represent the necessary intelligence in the classical task of mental optimization:

de is the Lagrange function.

In connection with the analogy of systems equal (3) and (4), classically, the task of intellectual optimization can be considered as the problem of finding the saddle point of the Lagrange function.

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Basic understanding of modeling. Zagalni understand that destined model. Statement of optimization problems. Methods of linear programming. Zagalne that typical tasks for linear programming. Simplex-method of solving problems of linear programming.

Task 1. Know

Task 1 is called up to the release of the system equalization:

that value of another differential:

at the points of rozvyazannya rivnyan (8.3).

If the quadratic form (8.4) is negatively assigned to the point, if it reaches the maximum value, and if it is positively assigned, then the minimum value.

Butt:

The system of equalization may be rozvyazannya:

The point (- 1/3.0) is the maximum point, and the point (1/3.2) is the minimum point.

Task 2. Know

for minds:

Task 2 is broken by the way of Lagrange's multipliers. For which system solution (t + p) rivnyan:

butt. Find the sides of the rectangle with the maximum area inscribed in the colo: . The area A of a rectangle can be written at a glance: A \u003d 4xy also

Task 3. Know:

for minds:

Tse zavdannya ohoplyuє widely kolo zavdanya, which are assigned functions fі  .

Like the stench of the line, the head of the line programming.

manager3 but.

for the minds

It is violated by the simplex method, which, for the help of the apparatus of linear algebra, viroblya goals of straightening enumerating the vertices of a bagatohedron, which is shown (8.13).

Simplex method (Stacked in two stages):

Stage 1. The value of the reference solution x(0) .

The decision support is one point of the bagatohedron (8.13).

Stage 2. Finding the optimal solution.

The optimal solution is to be determined by the last enumeration of the vertices of the bagatohedron (8.13), for which the value of the goal function z on the skin does not change, then:

Okremy type of task of linear programming - this is the name of the transport task.

Transport is not a problem. Let there be warehouses at the points, from which the goods are saved from the quantity of goods. At the checkpoints there are spares, so it is necessary to put the goods in quantity in quantity. Significantly c ij variability of transportation of one unit vantage between points

Doslіdzhuєmo the operation of transportation of goods by the commissaries at the places, sufficient, to satisfy the needs of the compatriots. Significantly through the number of goods that are transported from the point but i y item b j .

In order to please the patient, it is necessary that the size X ij satisfied minds:

At that very hour in warehouse a; it is not possible to bring products from a larger quantity, lower. Tse means that the values ​​are due to the satisfaction of the system of irregularities:

Satisfy the minds (8.14), (8.15), so that you can put together a transportation plan, which can be drunk with comfort, in an infinite number of ways. Sob the last operation of the moment, choose the right decision, to recognize the first decision X ij, may be formulated as a rule of choice, which is used for additional criteria, which reflects our subjective manifestation of the meta.

The problem of the criterion is violated independently in the case of the subsequent operation - the criterion is to blame for the assignments by the party that operates. For whom, one of the possible criteria will be the quality of transportation. Vaughn appears like this:

The same transportation task is formulated as a linear programming task: set the values ​​that satisfy the exchanges (8.14), (8.15) and the functions that deliver (8.16), the minimum value. Exchange (8.15) - ceum's balance; umova (8.14) can be called a method of operation, more than a sense of operation for many poles, in order to ensure the safety of the patients.

Tsі dvі mind to establish, daily, a model of operation. The implementation of the operation is stale according to the criterion, for which the access to the operation will be secured. The criterion can be used in different roles. Vіn mozhe i yak sposіb formalіzії meti yak і principle vyboru dіy v vladі admissible, tbto yakі satisfy obmezhennyam.

One of the most important methods for improving the transport task is the potential method.

At the first stage, the task of solving the tasks is formed, the first plan for the transportation, which satisfies

delimitation (8.14), (8.15). If so (so that the total consumption is not saved from the total stocks of goods in warehouses), then a fictitious storage point or a fictitious warehouse with a variant of transportation, equal to zero, should be introduced. For the new manager, the total number of goods in warehouses is increased due to the total demand. Let's use a method (for example, the smallest element or a pivnіchno-zahіdny kuta) - a cob plan. At the next stage of the procedure for the withdrawn plan, there will be a system of special parameters - potentials. Necessary and sufficient mental optimal plan is yogo potency. The procedure for clarifying the plan is carried out until the time when wine becomes potential (optimal).

Zavdannya 3b. At the beginning of the day, the task (8.10 - 8.11) is called the task of non-linear programming. Let's take a look at the more accepted look:

for the minds

To accomplish this task, there are so called relaxation methods. The process of inducing a sequence of points is called relaxation, as follows:

Descent Method (Scheme). All methods of descent in order to solve the problem of insane optimization (8.17) are distinguished either by choosing a straight descent, or by the method of rushing descent. Methods of descent depend on the upcoming procedure and follow-up { x k }.

As often as possible, a sufficient point is selected x 0 . The following observations will follow such a scheme:

Points x k vibiraetsya straight downhill - s k ;

Know (up to + 1) - e approximation of the formula:

how to choose a value, be it a number that satisfies the inconsistencies

de number - be it such a number, if

For most descent methods, the value of  is chosen to be equal to one. In such a rank, for the purpose of , the task of one-dimensional minimization is brought to the fore.

Gradient descent method. Shards of antigradient - showing directly the most obvious change in function f(x), then natural є displacement from points X k at whom directly. The descent method, which is called the gradient descent method. Yakshcho, then the relaxation process is called the method of the most visible descent.

Direct method. In linear algebra, the method of introduction is like a method of obtaining gradients of decoupling systems of linear equations of algebra AH =b, and also, as a method of minimization of a quadratic function

Scheme method:

Yaxcho = 0, then this circuit is converted to a circuit using the best descent method. Vidpovidny choice of value t k guarantees the efficiency of the method of obtaining directly from the same order as in the methods of gradient descent and ensures the number of iterations of quadratic descent (for example).

Coordinate descent. On the skin iteration like straight downhill - s k vibiraetsya straight uzdovzh odnієї z coordinate axes. The method of maximum speed for the process of minimization is about 0(1/m). Moreover, it won’t lie in the open space.

Scheme method:

de coordinate vector

What's next x k є information about the behavior of the function gradient f(x), for example:

then straight downhill s k you can take the coordinate vector e j . In this way, the speed of the method is n times less, lower with gradient descent.

At the cob stage of the minimization process, it is possible to win the method of cyclic coordinate descent, if the cob descent follows a straight line e 1 , potim for 2 and so on until e P , then the whole cycle is repeated again. The more promising one against the forward one is a coordinate descent, at which a straight descent is chosen by a vertical rank. With such an approach to the choice, one can directly establish a priori estimates that guarantee the function f(x) h Imovirnistyu, scho pragne odinі at , zbіzhnіst process zі svidkіstyu order 0(1/m).

Scheme method:

On the skin cycle of n numbers (1, 2, ..., n) the number j(k) and yak s k select a single coordinate vector e j ( k ) , after what the descent is made:

Vypadkovy descent method.s k, which is ordered on this sphere equal to the equal distribution, and then according to the element calculated at the k-th stage of the process X before signify:

The speed of the slope descent method is n times lower, lower than the gradient descent method, and n times higher, lower than the vertical descent method. The methods of descent of stagnation and to neobov'yazkovo opuklim functions and guarantee їhnyu zbіzhnіst dzhe malih obezhennya (type of presence of local miniums) are examined.

Relaxation methods of mathematical programming. Let's turn to task 36 ((8.17) - (8.18)):

for the minds

In optimization tasks with exchanges, one chooses directly to lower the bandages with the need for a constant re-verification of what is new X k +1 may be just like that, like in front x k Satisfied systems X.

Smart gradient method. In this method, the idea of ​​​​choosing directly descends into the offensive: at the point X before linearize the function f(x), being a linear function and then, minimizing f(x) on faceless X, know the point y k . If someone is respected and farther away, then the descent is straight ahead, so that

In this way, for a direct joke - s k next solve the problem of minimizing a linear function on a multiplier X. Yakscho X one's own calling is given by the linear exchanges, there are the heads of the linear programming.

Possible direct method. Idea of ​​the method: average possible direct directions at the point X before choose those, whichever function f(x) change the most, and then we will change the descent of the narrower directly.

Straight - s at the point X X is called possible, because it is the same number that is for everyone. To know the possible directly, it is necessary to solve the problem of linear programming, or the simplest task of quadratic programming:

For wits:

Come on - the virishennya of this task. Umova (8.25) guarantees that straight away is possible. Umova (8.26) ensure the maximum value (so that the average possible direct s, straight-forward - secure the most recent change in function f(x). Umova (8.27) makes it easier to solve problems. The method of possible straight forward lasting up to possible counting pardons. However, the speed of this life can be assessed in a savage mood smoothly, and the factory is still invincible.

Vipadkovy poshuk method. The implementation of new projects is more of a method of minimization in a slanderous way of being a labor worker, the most simple ways, if an impersonal area can have a simple geometric structure (for example, a rich parallelepiped). In vipadia, the method of vipadkovo poshuku can be even more promising, if you choose the vipadkoviy rank straight down the descent. In case of any misfortune, it is possible to have a low cost of living, but simplicity of choice can directly compensate for the cost of spending money on the top of the task of minimization.

Scheme method:

On the n-world single sphere with the center on the cob of coordinates, a point is selected r k, which follows the order of the sphere equal to the equal distribution, that buv straight down the descent - s k wits,

How often the proximity is chosen. For the point counted on the skin iteration x k will be ( k+ 1)-a point x k +1 :

How to choose whether or not the number z \u003d, which satisfies the inconsistencies:

The feasibility of this method was brought to the case of rather non-hard substitutions for the function f (swelling) that faceless obmezhenie X (vypuklіst and closedness).

Head 1. Know

de x = (x 1 .. x n) e E p

The task is to build up to the top of the system of equalization

that relative value of another differential

at the points (a-|, (*2, a n) the division is equal (7.3).

If the quadratic form (7.4) is negatively assigned to the point, if it reaches the maximum value, and if it is positively assigned, then the minimum value.

Butt:

The system of equalization may be rozvyazannya:

The point (-1; 3.0) is the maximum point, and the point (1; 3.2) is the minimum point.

manager 2. Know

for minds:

The task 2 is checked by the method of Lagrange multipliers, for which you need to know the decoupling of the system (t + p) rivnyan:

butt 2. Find the sides of a rectangle of maximum area inscribed in a column Area L of a rectangle

you can write at the sight: BUT= 4xy, todi

stars

manager 3. Know beyond the drain:

Tse zavdannya oohlyuє a wide range of productions, which are assigned functions f and equal Like the stench of the line, the head of the line programming.

Zavdannya Za.

for the minds

It is violated by the simplex method, which, for the help of the apparatus of linear algebra, determines the purpose of straightening the vertices of a bagatohedron, which is determined (7.13).

Simplex method consists of two stages.

Stage 1. The value of the reference solution x^0). The decision support is one point of the bagatohedron (7.13).

Stage 2. Finding the optimal solution. It is known to be the last enumeration of the vertices of the bagatohedron (7.13), for which the value of the goal function z of the skin does not change, tobto:

Okremy type of task of linear programming is the name of the transport task.

Transport is not a problem. Let’s go to the points a-1, a 2, .... and l there are warehouses, in which goods are stored in the amount of x 1, x 2, ..., X l vіdpovіdno. In paragraphs b-|, b 2,..., b t there are spozhivachі, as it is necessary to put qi goods in quantities y- y 2 , y t obviously. Significantly Cjj variability of transporting one single vantage between points a-| and by.

Doslіdzhuєmo operation transported commodities at the kіlkostakh, sufficient to satisfy the needs of the clientele. Significantly through Hu number of goods to be transported from point a, - point by.

In order to please the patient, it is necessary, x,y values satisfied minds:

At the same hour from the warehouse it is not possible to bring products from a larger quantity, lower there. Tse means that the values ​​are due to the satisfaction of the system of irregularities:

Satisfy your mind (7.14), (7.15), tobto. You can put together a plan for transportation, which will ensure that you ask for help, by an indefinite number of methods. In order for the last operation to be done, choose a penny optimal solution, then. recognize songs Xjj, May be, however, a rule of choice has been formulated, which is used as an additional criterion, which reflects our subjective statement about the meta.

The problem of the criterion is violated independently in the case of the subsequent operation - the criterion is to blame for the assignments by the party that operates. For whom, one of the possible criteria will be the quality of transportation. Won warehouse

The same transportation problem is formulated as a linear programming problem: find the values ​​x, y > O, which satisfy the exchanges (7.14), (7.15) and the functions that deliver (7.16), the minimum value. Exchange (7.15) - ceum's balance; umova (7.14) can be called a method of operation, because the sense of operation in this case is to ensure the drinking of those who are sick.

Vkazіvku dvі mind to establish, daily, the model of operation. The implementation of the operation is stale according to the criterion, for which the access to the operation will be secured. The criterion can be used in different roles. Vіn mozhe buti as a way of formalizing the method, as a principle of selection from the number of admissible, tobto. scho please obezhennyami.

One of the methods used to improve the transport task is the method of potentials, a scheme that seems to be offensive.

At the first stage, the development of tasks is put together the first plan for transportation, which satisfies the exchange (7.14), (7.15). Yakscho

(that is, the total consumption is not saved from the total stock of goods in warehouses), then a fictitious item of storage is introduced into the view or fictitious warehouse

transported with a variant, which costs zero. For the new manager, the total number of goods in warehouses is increased due to the total demand. Then we use the method (for example, the smallest element of the chi pivnіchno-zahіdny kuta) to know the cob plan. At the next stage of the procedure of the adopted plan, a system of special characteristics - potentials will be developed. Necessary and sufficient mental optimal plan is yogo potency. The procedure for updating the plan is repeated until the hour when the plan becomes potential (optimal).

manager 36. In the past, the task (7.10-7.11) is called the task of non-linear programming. Let's look at її at sight

for the minds

For the accomplishment of this task, there are so called relaxation methods. The process of inducing a sequence of points is called relaxation, as follows:

Methods of descent (character scheme). The descent methods used in the crazy optimization problem (7.17) are distinguished either by the choice of direct descent, or by the method of roaming descent. Methods of descent depend on the upcoming procedure and follow-up (HC).

As soon as possible, a sufficient point Xq is selected. The following observations will follow such a scheme:

• point x to vibiraetsya straight downhill - S k ;
• know to (before+ 1) closeness to the formula

de yak value $ to choose whether it is a number that satisfies the inconsistencies

de number X to - whether such a number, if 0 X min f (x k - $ Sk).

For most descending methods, the value X to to be chosen as one. Otzhe, vznachennya (3 ^ be brought to the task of one-time minimization.

Gradient descent method. Oskilki antigradient - G(x to) show directly the most visible change of function f(x), then natural є displacement from points x to to to whom directly. Descent method, in yacomu S k \u003d f "(x k) called the gradient descent method. Yakscho X to= 1, then the relaxation process is called the method of the best descent.

Direct method. IN of linear algebra OH= b, then, as a method of minimization of a quadratic function f(x) =((Dx - b)) 2 .

Scheme method:

Yakscho f k = 0, then this circuit is converted to a circuit using the best descent method. Vidpovidny choice of value t k guarantees the efficiency of the direct fit method with the same order as the methods of gradient descent, it ensures the termination of the number of iterations of quadratic descent (for example,

Coordinate descent. On the skin iteration like straight downhill Sk vibiraєtsya directly uzdovzh odnієї z coordinate axes. The method of maє shvidkіst zbіzhnosti to the process of minimization order 0 (1//77), moreover, it won't lie in the open space.

Scheme method:

de coordinate Vector,

What's next x toє information about the behavior of the function gradient f(x), for example,

then straight downhill Sk you can take the coordinate vector ey. What kind of way is the speed of the method P less, lower for gradient descent.

At the cob stage of the minimization process, you can win the method of cyclic coordinate descent, if the cob descent follows the direct e-|, then - v2, etc. right up to e p, after which the whole cycle is repeated. More promising porіvnjano z description є pokoordinatnyy descent, at which descent is chosen in a vertical rank. With such an approach to the choice, one can directly establish a priori estimates that guarantee the function f(x) z imovirnistyu, scho pragne odinі at zbіzhnіst process zі shvidkіstyu order 0(1 1t).

Scheme method:

On the skin crotch process P numbers (1, 2, ..., P) a number is chosen in a vipadkovy rank j(k) and yak s k choose a single coordinate vector wsh, after what the descent is made:

Vypadkovy descent method. On the n-world single sphere with the center on the cob of coordinates, a vipad point is selected S k , subordered on this sphere equal to the equal subdivision, and then according to the calculated on / s-m krotsі process element x to to be appointed x k+] :

Swidkіst zbіzhnosti method vypadkovy descent P times less, lower in the gradient descent method, but in P razіv more, nizh to the method of vertical coordinate descent. The methods of descent of stagnation and to neobov'yazkovo opuklim functions and guarantee їhnyu zbіzhnіst dzhe malih obezhennya (type of presence of local miniums) are considered.

Relaxation methods of mathematical programming. Let's turn to task 36 ((7.17) - (7.18)):

for the minds

In optimization tasks with exchanges, one chooses directly to lower the bandages with the need for a constant re-verification of what is new x to +" may be just like that, like in front x to, satisfies the system X.

Smart gradient method. IN to this method, the idea of ​​​​choosing directly downhill polgaє in the offensive: at the point x to linearize the function

f(x), being a linear function f(x) = f (x k) + (y "(x k), x-x k), and sweat, minimize f(x) on faceless X, know the point at k. Whom to respect S k \u003d y to - x to and farther away the descent directly Xup to + 1= x to - $to (x to -y to), so, sob g X.

In this rank, for a joke directly Sk next solve the problem of minimizing a linear function on a multiplier of X. As X, in its own way, it is given by linear subdivisions, it becomes the task of linear programming.

Possible direct method. The idea of ​​the method: the middle of the most possible direct directions at the point xk is to choose those that are suitable for the function f(x) change the most, and then we will change the descent of the narrower directly.

straight ahead s at the point X e X is called possible, _ as it is the same number (3\u003e O, that X- (3s e X for all (3 g). a?=> min for the minds

Come on d toі s k- Virishennya tsgogo zavdannya. Umova (7.25) guarantees that directly s k possible. Umov (7.26) ensures the maximum value of (/"( x k), s), tobto. mid-range possible direct s, straight ahead s k secure any change in function f(x). Umova (7.27) makes it easier to solve problems. The method of possible straight forward lasting up to possible counting pardons. However, the speed of this life can be assessed in a savage mood smoothly, but the factory is still untouched.

Vipadkovy poshuk method. The implementation of the earlier methods of minimization in a common way is a labor worker, the most simple ways, if the impersonal border can have a simple geometric structure (for example, a rich parallelepiped). In vipadia, the method of vipadkovo poshuku can be even more promising, if you choose the vipadkoviy rank straight down the descent. If there will be a good program for the security of the economy, then the simplicity of the choice can directly compensate for the amount of money spent on the task of minimization.

Scheme method:

on the i-peace single sphere with the center on the cob of coordinates, a tipping point is selected gu to follow the order on this sphere to the equal rozpodil, and then straight down the descent - s^ wits

Like the cob the proximity is chosen xs e X. According to the calculated skin iteration point X? will be (k + 1)-a point x^+ y:

How to choose be-like the number z what satisfies the nervousness

The feasibility of this method with arc non-hard exchanges on the function / (swelling) and impersonal exchange was brought X(vypuklіst and closedness).