SU1765830A1 - Device for finding extremum of multivariable additive function - Google Patents

Abstract

The invention relates to the field of computing and can be used in the development of specialized equipment to find the extremum of additive functions. The purpose of the invention is to expand the functionality by taking into account the restrictions on the sum of the arguments of the objective function. The essence of the invention is that the device contains three groups of keys, a key, delay lines, three registers, the first group of function value calculation blocks, a group of multiplication blocks, accumulating adder, coefficient setting block, ring counter. New in the invention is the introduction of a trigger, a fourth group of keys, two groups of adders, a second group of units calculating the value of a function, two groups of subtractors, a subtractor, a unit for setting increments of arguments, a block for setting limits, a block of division, a block for setting the number of arguments with corresponding links . 1 il. sl C

Description

The invention relates to computing and can be used to find the extremum (minimum) of the additive function of many variables.

A device for finding extrema is known, comprising a block for setting function parameters, a clock pulse generator, two address counters, three memory blocks, five comparison circuits, eight multipliers, three registers, two groups of OR elements, two quadrs, an adder, three accumulating adders, nine switches, two delay elements, two subtractors, a dividing unit, a square root extraction unit 1. The device is designed to find the extrema of functions and implements a method of optimizing the second order, which entails The main drawback is the necessity of calculating at each step of the iterative process of solving not only the value of the objective function, but also its derivatives of the first and second order. In addition, there is a device for finding extrema containing a function setting parameter block, a clock pulse generator, six comparison elements, three memory blocks, a logarithmic converter, two address counters, a group of OR elements, seven multipliers, four registers, eleven AND blocks two exponential converters, five accumulating adders, three delay elements, three subtractors, five division blocks, a reverse magnitude calculator, two exponentiation blocks, the inverse logarithmic cue prev VJ SL SL 00 WITH

ABOUT

equalizer 2. The device is designed to find the extremes of functions for arbitrary starting points; the disadvantage of it is that it solves the problem of geometric programming, but without taking into account the restrictions on the arguments.

A device for optimizing functions of several variables is known - a prototype containing a group of blocks for generating linearly independent functions, registers, three groups of keys, four keys, a pulse generator, a control unit, a block for specifying coefficients, a delay line, a ring counter, two groups of multiplication blocks, a block multiply, two adders, accumulating adder and two counters 3.

The device allows finding the minimum of the function of several variables in the case when single implementations of non-stationary random processes arrive at the device input.

The disadvantage of this device is that it does not provide a solution to the extremal problem of convex programming with constraints.

F (X) F (xi, x2xn) §f (x,) - min0)

i 1

at

J

I 1

x, M,

The aim of the invention is to expand the functionality of the device by taking into account the restrictions on the sum of the arguments of the objective function.

The proposed device implements a recurrent minimization procedure using the gradient projection method 4. The gradient projection method is based on the relaxation process of building successive X vectors, where m is the step number m 1, M, such that each successive vector belongs to the restriction region (2) and each subsequent value of the objective function does not exceed the previous

The next optimization step includes:

1. Finding the direction of motion from point X to the minimum of the objective function, i.e., calculating the gradient value of the function at a given point

VF F (Xrm)) gradF (X (m))

Finding partial derivatives of the objective function is proposed by the method of divided differences.

Яfrri) f (xЈm) h-Ah,) - f (xЈm))

-chgt- xi xN.

F X |

xf

t - (3)

As an initial approximation of the solution at the so-called zero step

it is possible to accept any values of variables, including zero, i.e., x ° | O, i - 1, n), satisfying constraints (2).

2. Finding the subsequent vector of the solution without taking into account the restrictions (step in

antigradient direction)

(t + 1) | HN .. a.ZH. I (t) ate)

 dXi X | X |

where JS is the coefficient determining the step length in the selected direction.

3. Accounting for constraints on variables by projecting a vector on the region of constraints by the formula

)

Jm + 1) - (m + 1) Xi - Xi

i 1

(five)

where N is the limit on the sum of the arguments. The decision process is terminated after

2Q a specified number of steps M.

Thus, the proposed device allows to optimize the distribution of homogeneous resources, the total amount of which is limited, and the effect of

25 their use is described by a smooth unimodal function (1).

The goal is achieved by the fact that in a known device containing three groups of keys, a key, a delay line, an input

3Q of which is connected to the key output, a pulse generator, the output of which is connected to the key information input, three registers, a group of function value calculation blocks, a group of multiplication blocks,

35 accumulative adder, coefficient setting block, ring counter, trigger entered, the installation input in one state of which is connected to the reset input of the ring counter, to the control inputs of the first group of keys and the external start input of the device, the reset input of which is connected to the output ring counter, the counting input of which is connected to the fourth output of the line

45 delay, the trigger output is connected by a key-guiding input, the fourth group of keys, the control inputs of which are connected to the third output of the delay line, the outputs of which are connected to

50 corresponding to the outputs of the first group of keys and the inputs of the first register, the information inputs of the first group of keys are the inputs of the initial values of the device decision vector, the outputs

55 of the first register are outputs of the optimal values of the device's decision vector, connected to the corresponding inputs of the first group of function value calculation blocks and to the information inputs of the second group of keys, the control inputs of which are connected to the first output of the delay line, and the outputs are connected to the corresponding inputs of the second register, the first group of adders, the inputs of the first term of which are connected to the outputs of the second register, the block for setting the increment of the arguments, the output of which is connected to the inputs of the second slug the first group of adders, the second group of units for calculating the value of the function, the inputs of which are connected to the corresponding outputs of the first group of adders, the first group of adders-subtractors, the inputs of which are deductible are connected to the corresponding outputs of the first group of units for calculating the value of the function; calculating the value of the function, and the outputs are connected to the corresponding inputs of the second factors of the multiplication unit group, the inputs of the first factors of which are connected The coefficients are added to the output of the block, the second group of adders-subtractors, the inputs of which are subtracted are connected to the outputs of the group of multipliers, the inputs of which are decremented are connected to the corresponding outputs of the second register, and the outputs are connected to information inputs of the third group of keys whose control inputs are connected to the second the output of the delay line, and the benefits are connected to the corresponding inputs of the third register, whose outputs are connected to the inputs of the accumulating adder, the adder-subtractor, the input is subtracted which is connected to the output of the accumulating adder, a restriction setting block, the output of which is connected to the input of the decremented totalizer-subtractor, a division block whose input is divisible is connected to the output of the totalizer-subtractor, a block specifying the number of arguments, the output of which is connected to the splitter input of the division the second group of adders, the inputs of the first addends of which are connected to the corresponding outputs of the third register, the inputs of the second addends are connected to the output of the division unit, and the outputs are connected to the corresponding and formational inputs of the fourth group of keys.

Comparative analysis with the prototype shows that the proposed device is distinguished by the presence of new blocks: a trigger, a group of keys, two groups of adders, two groups of blocks calculating the value of a function, a group of adders-subtractors, a totalizer-subtractor, a block for setting increments of arguments, a block of tasks constraints, a division block, a block specifying the number of arguments and their relation to the rest of the circuit elements.

Thus, the claimed device meets the criteria of the invention of novelty.

A comparison of the claimed solution with other technical solutions shows that these blocks are known 1,2,3.

However, when introduced in the specified

0 communication with the rest of the circuit elements in the inventive device for finding the extremachadditive function of many variables with constraints on the sum of the arguments it exhibits new properties,

5 that provides a solution to the extremal convex programming problem with constraints. This makes it possible to conclude that the technical solution meets the criterion of significant differences.

0 The drawing shows a block diagram of the device.

The device contains a trigger 1, keys 2, a delay line 3, a pulse generator 4, registers 5, a block 6 for setting increments

5 arguments, groups of blocks 7 for calculating the value of a function, a group of blocks of multiplying 8, a cumulative adder 9, groups of adders 10, subtractors 11, block 12 for specifying coefficients, a ring counter 13,

0 block 14 of the task of restriction, block 15 of the task of the number of arguments, division block 16.

The design of the proposed device is based on the use of known elements and does not represent technical difficulties for implementation.

The device works as follows

The device starts up with a Start signal. This signal triggers trigger 1, which permits the passage of pulse generator signals 4 through key 2 to the input of delay line 3, returns to the initial state (resets) the ring counter 5 and 13, and also ensures that the initial values of the vector of arguments are entered into the first register 5 °, i 1, p) through the first group of keys 2.

Trigger 1, Key 2, Delay Line 3,

0 pulse generator 4, a ring counter of 13 M iterations is a control device that provides the implementation of M iterative steps.

5 solution optimization process.

Each step of the iterative process is as follows.

The initial values of the arguments vector x i, i 1, n from the first register 5 are fed to the inputs of the first group of function value calculation units 7, the calculated values

the functions f (), i 1, p, are fed to the inputs of the first group of subtractors 11 that are being subtracted.

The signal C1 from the first output of the delay line 3 arriving at the control inputs of the second group of keys 2, the values of the vector arguments i 1, n from the first register 5 are copied to the second register 5. From the outputs of the second register 5, the values go to the inputs of the first term of the first group of adders 10, to the inputs of the second addend which receive single increments from the output of the argument setting increment block 6. From the outputs of the first group of adders 10 values of the arguments that have been incremented X0n) | + i Ax, 1, i 1, p are fed to the inputs of the second group of blocks calculating the values of function 7, from whose outputs the values f (+1) are fed to the inputs of the decreasing first group of subtractors 11. From the outputs of the first group of subtractors 11, the increments of

f (xЈm) + 1) -frfm))

one

as values of the gradient function vector

iL (m) Ah, x -

are fed to the inputs of the second multipliers of the multiplication block group 8, where they are multiplied by the step size / 3 coming from the coefficient setting block 12, and fed to the inputs of the subtracted second subtractor group 11. The inputs of the second subtractor group 11 are supplied with the values of the argument vector, i 1, n from the outputs of the second register 5. By the time the signal C2 arrives from the second output of the delay line 3, the information outputs of the third group of keys 2 contain the following values of the vector of arguments 1 1 1, i 1, n without considering restrictions. According to the signal C2, arriving at the control inputs of the third group of keys 2, the value Zst + 1 | rewritten into third register 5.

From the output of the third register, 5 values of the vector of arguments x t + H are fed to the inputs of accumulating adder 9, from the output

om + 1) | arrives at

whose value is 2J i 1

input of the subtracted adder-subtractor 11, the input from which is being reduced is the signal from the block 14 of the restriction setting N. From the output of the subtractor 11

the value of N i is fed to the input

the divisible division block 16, to the input of the divider which receives a signal from the block 15, specifying the number of arguments

)

The magnitude of the output of the division block 16 is fed to the inputs of the second term of the second group of adders 10,

to the inputs of the first addendum of which the post-rjaiOT values of the argument vector) jt j 1 | P by the time the SZ signal arrives from the third output of the delay line 3, the information inputs of the fourth group of keys 2 contain the values of the arguments vector at the next step

N-§x1m + 1)

x (m + 1) (m + 1)

According to the SZ signal, which arrives at the control inputs of the fourth group of keys 2, the calculated values of the argument vector are entered into the first register 5.

At this next step approaching

ends with the fourth output of the delay line, one is added to the ring counter 13, the pulse generator 4 generates the next pulse and the next step is repeated.

When the M steps are performed by a signal from the ring counter 13, the trigger 1 is reset and the solution is terminated. From this point in time, the first register 5 contains the optimal value of the vector of arguments, that is, the solution of the problem.

Thus, the proposed device allows one to find the extremum of an additive function of many variables with a restriction on the sum of the arguments.

Claims (1)

Invention Formula A device for finding the extremum of an additive function of many variables, containing three groups of keys, a key, a delay line whose input is connected to the key output, a pulse generator whose output is connected to the key information input, three registers, the first group of function value calculation blocks, a group of multiplication blocks accumulator, coefficient setting unit, ring counter, characterized in that, in order to expand the functionality of the device by taking into account the restrictions on the sum of the target function arguments Functions, it contains the fourth group of keys, the group of adders, the block for setting the increment of the argument, the second group of blocks for calculating the value of the function, the group of subtractors, the subtractor, the block for specifying the limit, the block for blocking, the block for setting the number of arguments and the trigger; to the reset input of the ring counter, to the control inputs of the keys of the first group and is the external start input of the device, the reset input of the trigger is connected to the output of the ring counter, the counting input of which is connected with the fourth output of the delay line, the trigger output is connected to the control key input, the control inputs of the keys of the fourth group are connected to the third output of the delay line, and the outputs are connected to the corresponding outputs of the keys of the first group and the inputs of the first register, the information inputs of the keys of the first group are the inputs of the initial values of the device decision vector, the outputs of the first register are the outputs of the optimal values of the device decision vector and connected to the corresponding inputs of the calculation units The first function of the first group and the information inputs of the keys of the second group, the control inputs of which are connected to the first output of the delay line, and the outputs are connected to the corresponding inputs of the second register, the inputs of the first adder of the first group are connected to the outputs of the second register, the output of the increment argument block is connected to the inputs of the second term of the adders of the first group, the inputs of the units for calculating the value of the function of the second group are connected to the outputs of the corresponding adders of the first group, the inputs The deductible subtractors of the first group are connected to the outputs of the corresponding blocks calculating the value of the function of the first group, the inputs of the reduced ones are connected to the outputs of blocks calculating the values of the function of the second group, and the outputs are connected to the inputs of the first multipliers of the corresponding multiplications blocks of the second factors of the multipliers the inputs of the deductible subtractors of the second group are connected to the outputs of the corresponding multiplication units of the group, the inputs of which are decremented are connected with the corresponding outputs of the second register, and the outputs are connected to the information inputs of the keys of the third group, the control inputs of which are connected to the second output of the delay line, and the outputs connected to the corresponding inputs of the third register, the outputs of which are connected to the inputs of the accumulating adder, the input of the deductible subtractor is connected to the output of the accumulating adder, the output of the restriction task unit is connected to the input of the decremented subtractor, the input of the dividend divider is connected to the output of the subtractor, the output of the number of arguments parameter is connected to divider input the division unit, the inputs of the first components of the adders of the second group are connected to the corresponding outputs of the third register, the inputs of the second components are connected to the output of the division unit, and the outputs are connected with information inputs of the corresponding keys of the fourth group.