JPS59157761A - Parallel data processing system for linear programming problem calculation - Google Patents

Abstract

PURPOSE:To get a solution in a high speed by performing calculations for reduction of a search region and resetting of the advance direction of all particles in a single host computing processing device and performing the computing for tracking of the motion of one particle in each of many low order calculation processing devices. CONSTITUTION:The host processing device stops computing of low order processing devices at intervals of a proper time to decide the convergence and stores final results in a low-speed storage device 34 if cmputings are converged. If they are not converged, the host processing device selects intermediate best points on a basis of collision points stored in high-speed storage devices 3-1-3-p and information of objective function values at these points and reads out the number of a restricted plane, which is active concerning intermediate best points, and its coefficient information from a high-speed storage device and a low-speed storage device respectively and computes new advance directions of all particles and stores results in high-speed storage devices corresponding to individual particles. Low order processing devices 2-1-2-p reads out information concerning initial advance directions and control conditions stored in corresponding high- speed storage devices in accordance with the computing start indication of the host processing device and compute the motion of particles.

Description

Detailed Description of the Invention [Field of Application of the Invention] The present invention relates to the calculation of linear programming problems that often appear in the planning and control of real systems, and particularly to large-scale problems with more than 104 variables. This invention relates to a parallel data processing method for calculating linear programming problems suitable for.

[Prior art]

The most common solution method for linear programming problems is the revised simplex method [Sekine: Mathematical Programming (Iwanami)], and this method is said to be the most efficient when calculating using a single processing device. It recognized.

On the other hand, when performing parallel calculations using multiple processing devices for the purpose of high-speed calculation, a method has been proposed in which the parallelizable parts of the revised simplex method are extracted and distributed to multiple processing devices for execution. (Kinyen: Linear programming calculation and tridiagonalization calculation of real symmetric matrix using parallel processing system:
Information Processing Paper Vo 1.19. AI). However, this method is composed of a large number of different processing stages, and each stage has a different degree of parallelism, so it seems difficult to utilize multiple processing devices efficiently.

[Purpose of the invention]

An object of the present invention is to provide a parallel five-column data processing method that can execute large-scale linear programming calculations at high speed by using a large number of inexpensive calculation processing devices (microcomputers) with relatively low functionality. be.

[Summary of the invention]

In order to achieve the above object, the method for solving linear programming problems used in the present invention is completely different from the revised simplex method. For example, when considering the motion of particles within a constrained region, It is characterized by finding the optimum point. More specifically, when a particle collides with a constraint surface, it reflects based on a predetermined law and moves around within the constraint area, so it memorizes the collision point and the objective function value there, and stores all past collision points. The point with the best objective function value is constantly updated, and as the particle motion progresses, the objective function value is improved toward the optimal point. Basically, it is based on the above idea, but it is expected that it will take a lot of time to find the optimal point if a single computing device is used to track the movement of a single particle. By using a calculation processing device and having each of them track the movement of one leap of particles, calculations for tracking the movement of a large number of particles are executed in parallel, thereby speeding up the solution. This tracking calculation is a simple process;
It can be executed by an inexpensive calculation processing device with relatively low functionality. Therefore, there is a high possibility that a large number of processing devices can be used, and it is suitable for highly parallel processing.

In addition to parallelizing tracking calculations, the search area is reduced and the traveling direction of each particle is reset at appropriate timing in order to speed up the solution. The search area can be reduced by adding a plane that has the same gradient as the objective function plane and passes through the best point among the past collision points as a new constraint plane (the direction in which the objective function decreases is the permissible direction). conduct. Regarding resetting the traveling direction of the particles, the best point among the past collision points is used as the starting point for all particles, and the traveling direction of each particle is set so that the particles are scattered in all directions in the reduced area. shall be determined.

As for the device configuration, a single upper-level processing unit performs calculations for reducing the search area and resetting the traveling direction of all particles, and a number of lower-order processing units each perform tracking calculations for the movement of L 1 particles. shall be equivalent.

[Embodiments of the invention]

First, an embodiment of the solution algorithm according to the present invention will be described with reference to a simple example (two-dimensional, two-particle case) shown in FIG. 1, and then an embodiment of the apparatus configuration will be described according to FIG.

Consider the following linear programming problem.

Ax ku b ・・・・River・(1)C
Tx-+-...Group (2) Here, A: mxn matrix (suppose that the row vector is normalized to length 1), X: n vector, C: n vector, b2 m vector, T represents transpose. Note that the normally set constraint X > O shall be included in equation (1).

(1) Example of solution algorithm The two particles starting from 10 in FIG. 1(a) each travel straight, and when they collide with the constraint surfaces (1 to 7), they are reflected according to the given law and travel straight again. As a result of performing such exercise for a suitable period of time, an intermediate best score of 20 is obtained.

Constraint surface 1 generated from the objective function is raised to a position where it passes through 20 (1' in (b)), and the search area is reduced. Even in the reduced search area in (b) (
The same particle movement as in a) is performed, and the next intermediate best point 3o is obtained. The constraint surface 1' is raised to 1, and the search area is further reduced (1" in (C)). The above processing is carried out in the following calculations (1) to 4ψ.

(1) Simulation of particle motion/discovery of collision constraint surfaces Ll is the traveling direction vector (length 1) of the particle, aj is the row vector of A (J-1+ 2'+...4 m), and the starting point is Assuming Xo, b4-ajTx6 1■Im Sj= MI・ □ ・・・・・・・・・
(3) m+1>j>1 m+1>j>l a
Find I u (j=m-1-1 represents the constraint generated from the objective function). j=j, which satisfies the equation (3), gives a constraint surface aJIX"")jI on which particles collide, and the collision point is Xo+5JILl.

- Determining the direction of reflection after collision Colliding particles are assumed to be reflected in the permissible normal direction of the constraint surface. That is, the reflection direction U' is u""ajl
......(4).

(11) Let X be the best set intermediate point of the constraint surface generated by the objective function. Constraint surface CTx CTx ・・・・・・・・・
(5) is added, and the search area is reduced.

011) Determining the traveling direction of particles from the intermediate best point (11)
When the process is performed, all the particles are collected at the intermediate best point X, and their traveling direction is newly set as follows. X
A perpendicular line is drawn from x1 to the hyperplane spanned by the permissible direction normal vector of a constraint (active constraint) that includes ``''. (direction) - and let this be the traveling direction of one particle.This can be calculated as follows (Figure 2).

U-1Σ αral ・・・・・・・・・
(6) EAa Σ P=1 ・・・・・・・・・(
7) LIT(at a4 )=0 (1, J εkc
) --... (8) Here, Ac is Acttve
represents the subscript set of the constraint surface. Regarding residual particles,
In order to provide diversity in the search direction, the subspace a'''
"' 0 (a part of the orthogonal system included in 1" A cl is randomly arranged and the number is equal to the number of particles), and these are taken as the traveling direction of each particle. For generation of orthogonal system, 5chrn
The orthogonalization method of tdt [Satake two matrices and determinant (Shokabo)] can be used.

ψ Convergence Judgment Convergence is judged by evaluating the degree of improvement of intermediate improvement points. The process ends if the following formula holds true.

l CTX+'C"X' K: Here, XI"l x2 are the previous intermediate sharp point and the current intermediate best point, respectively, and ε is a constant for determining convergence.

(2) Example of device configuration FIG. 3 shows an example of the device configuration. The upper processing device 31 is in charge of the above-mentioned steps t(ii) to ψ, and also controls the starting and stopping of the lower processing devices 2-1 to 2-p. Each of the lower processing devices 2-1 to 2-p performs the calculation (1) for one particle.

The higher-level processing device stops the total output q of the lower-level processing device 1t at appropriate time intervals to determine convergence, and if it has converged to 1, stores the final result in the low-speed storage device 31.

If it has not converged, perform the following processing. The intermediate best point is selected from the collision points stored in the high-speed storage devices 3-1 to 3-p and the information on the internal function values there, and the number of the constraint plane that is ACtive regarding the intermediate best point is determined from the high-speed storage device. The coefficient information is read out from the low-speed storage device, new fc travel directions of all particles are calculated, and the results are stored in the high-speed storage device corresponding to each particle.

Lower processing devices 2-1 to 2. -p is the corresponding high-speed storage device 3 to 1 according to the calculation start instruction from the higher-level processing device.
The information regarding the initial traveling direction and constraint conditions stored in ~3-p is read out, and the particle motion is calculated. Regarding the constraint conditions, it is unwise to store all the coefficient information @ in each of the high-speed storage devices 3-1 to 3-p. It is assumed that a partial report is to be transferred to a high-speed storage device.

Figure 4 shows a flowchart of the processing contents of the upper processing device.
FIG. 5 shows a 7O-chart of the processing contents of the lower processing unit.

Next, we will discuss the results of evaluating the calculation time using the required number of multiplications. The number of variables is 11 the number of constraints is m1 the number of particles is p
,, Let r be the number of ACtiye constraints, (1) ~ (
The multiplication 1 required for processing ψ is roughly as follows (
For each treatment, only the highest order term is shown fC).

(1) cannot be executed in parallel by 9 processing units, so the calculation time must be considered as 1/p, but in the numerical example listed in the right column, even in the case of 1/J) L7 still predominant among treatments (2mn=108). Therefore, we will focus on and evaluate only the process (1). The amount of multiplication described above is required for one particle collision calculation, and if S is the number of collisions until the optimum point is reached, the total amount of multiplication is 2 nmS. On the other hand, it is said that calculations using the revised simplex method require a multiplication amount on the order of m2n on average for id [Iri et al.: Calculation Efficiency and its Limits (Nippon Hyoronsha)], so the improvement ratio is as follows. become.

In the numerical example, it can be seen that improvement is possible if S is on the order of approximately 103 or less.

The comparison with the revised simplex method, which uses a single processing device, may seem unfair, but the processing device in this case is a large computer, whereas the processing devices used in large numbers in the present invention are microprocessors. Since it is a computer, it can be fully competitive in terms of cost.

〔Effect of the invention〕

As explained above, according to the present invention, large-scale linear programming can be executed at high speed by using a large number of microcomputer-level processing devices in combination, which is highly effective.

[Brief explanation of the drawing]

Fig. 1 is a model diagram explaining the parallel solution algorithm according to the present invention based on a two-dimensional, two-particle example;
4 is a block diagram of a parallel data processing device for calculating linear programming problems, FIG. 4 is a flowchart of processing in the upper processing device, and FIG. 5 is a flowchart of processing in the lower processing device. 4- Figure 5th Ward

Claims (1)

[Claims] In a parallel data processing method for calculating linear programming problems that calculates linear programming problems based on the simulation of the motion of particles p (p: a positive integer) in a constrained region, the traveling direction of all particles and the region to be searched are a higher-level processing device that reconfigures p
P particles are tracked and calculated based on predetermined constraint expressions and objective functions, and A parallel data processing method for calculating linear programming problems, which is characterized by tracking and calculating the motion of