JP2001184334A - Method and device for finding solution of linear scheduling question - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a method and a device for finding solution of linear scheduling question, with which an inner point method can be applied to a linear scheduling question having upper and lower limits without increasing limit conditions by inverting the upper and lower limits by suitably executing a variable conversion in a process for searching a solution. SOLUTION: When a variable having upper and lower limits gets close to the upper limit in the middle of calculating the solution on the basis of the inner point method, because of a limit in the inner point method, the solution can not be continuously calculated as it is. Then, by replacing the variable with a difference between an upper limit value in the definition region of the variable and the variable, the upper and lower limits are inverted and the calculation of the solution is continued while avoiding the limit of the inner point method.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a technique for constraining upper and lower limits of variables in a linear programming problem, and more particularly to a method for solving a linear programming problem having upper and lower bound constraints by an interior point method and a device for solving a linear programming problem.

[0002]

2. Description of the Related Art Conventionally, either a simplex method or an interior point method has been used as a method for solving a linear programming problem in which a variable has only a lower limit. In particular, when the scale of the problem is large and most of the coefficients of the constraint condition are 0 (zero), the performance of the interior point method is considered to be superior to the simplex method. When a variable has both upper and lower limits and has upper and lower constraints, the simplex method is used.

[0003]

However, the prior art has the following problems. The first problem is
When the interior point method is applied to a linear programming problem including a variable having a bounded domain, a calculation time and a necessary storage area increase. The reason is that the interior point method is a calculation method that assumes that only the lower bound is defined as a domain.In order to adopt the upper bound, the upper bound itself must be added as a constraint to the bounded domain. This is because the number of constraints increases by the number of variables having. Furthermore, since it is an inequality constraint, the same number of slack variables must be introduced to convert it into an equality constraint, and the number of variables increases. These two points increase the calculation time and storage area.

[0004] The second problem is that even if the interior point method is more effective than the simplex method due to the problem scale and the nature of the linear constraint, the validity of the interior point method cannot be demonstrated due to the existence of upper and lower limit constraints. is there. The reason is that, as described for the reason of the first problem, the interior point method requires more calculation time and storage area than the simplex method due to the existence of the upper and lower limit constraints.

The present invention has been made in view of such a problem, and an object of the present invention is to appropriately perform variable conversion in a process of searching for a solution to a linear programming problem having upper and lower limit constraints, and It is an object of the present invention to provide a method for solving a linear programming problem and a device for solving a linear programming problem to which the interior point method can be applied without increasing the constraint conditions by reversing the lower limit.

[0006]

The gist of the present invention is that a variable having upper and lower limit constraints and an upper limit value of a domain of the variable are calculated by the interior point method during solution calculation. When approaches the upper limit, the variable is replaced with the difference between the upper limit of the domain of the variable and the variable, the upper and lower limits of the variable are reversed, and the solution is calculated avoiding the limit of the interior point method. To solve a linear programming problem. The gist of the invention described in claim 2 is to minimize an objective function given by a linear function of a variable under linear constraints and upper and lower constraints, and to obtain a value of the objective function at that time by an interior point method. It is a method of solving a linear programming problem characterized by finding. In addition, the gist of the invention described in claim 3 is to apply the interior point method without increasing the constraint conditions by appropriately performing variable conversion in the process of searching for a solution and by inverting the upper and lower limits. It is a method of solving a linear programming problem characterized by the following. Further, the gist of the invention according to claim 4 is that when the variable approaches the upper limit in the middle of the calculation of the solution by the interior point method with respect to the variable having the upper and lower limit constraints and the upper limit of the domain of the variable. Have a means to replace the variable with the difference between the upper limit of the domain of the variable and the variable and to reverse the upper and lower limits of the variable, avoid the limit of the interior point method, and continue the solution calculation A linear programming problem solver characterized by the following. The gist of the invention described in claim 5 is that an objective function given by a linear function of a variable under linear constraints and upper and lower limits is minimized, and the value of the objective function at that time is determined by the interior point method. A linear programming problem solving device characterized by having means for obtaining the linear programming problem. The gist of the invention described in claim 6 is that a variable conversion is appropriately performed in the process of searching for a solution, and the interior point method is applied without increasing the constraints by inverting the upper and lower limits. A linear programming problem solving apparatus characterized by having:

[0007]

DESCRIPTION OF THE PREFERRED EMBODIMENTS In the present invention, for a variable xj having upper and lower limit constraints and an upper limit value uj of the domain of the variable xj, the variable xj approaches the upper limit during the calculation of the solution by the interior point method. Then, the variable xj is replaced with the difference uj−xj between the upper limit value uj of the domain of the variable xj and the variable xj, and the upper and lower limits of the variable xj are reversed to avoid the limit of the interior point method. Continue calculating the solution.

(Problem Format Treated by Interior Point Method) In the interior point method, a problem formulated so that the lower limit of x becomes 0 as described below is handled.

[0009]

(Equation 1)

Where A is an n-dimensional vector x = (x1,
x2,..., xn) m that gives m constraints on T
× n matrix. The general linear programming problem dealt with by the revised simplex method can be transformed into the form of Equation (3.6.1) by translating in the space of x and introducing slack variables. Although there are many types of algorithms classified as the interior point method, the present invention uses a method using affine transformation.

(Search for the Optimal Solution in the Inner Point Method) The simplex method searches for an optimal solution from among the feasible base solutions that correspond to the vertices of the feasible region S. As a starting point, an optimal solution is searched for while changing the solution in a direction to reduce the objective function.

When x (k) is a feasible solution, x
(K + 1) = x (k) + td (k) is executable and c
Vector d satisfying Tx (k + 1) <cTx (k)
Among (k), the direction of the vector d (k) at which the reduction rate of the objective function is the largest is searched. Here, t is called a step size. If you do not consider the constraints,
The direction in which the value c · x of the objective function is reduced most is the direction of −c. Actually, the constraint condition Ax (k + 1) =
Since it is necessary to satisfy A (x (k) + td (k)) = b, as the direction of d (k), −c is set to the subspace of S × x |
What is necessary is just to take what is projected to Ax = 0 °.

(Big M Method) In order to search for an optimal solution by the interior point method, an initial point within the executable region, ie, inside the executable region but not on the boundary, is used as a starting point of the search. is necessary. However, finding such an initial solution x (0) is not obvious. Therefore, in the present invention, the calculation of the feasible solution and the calculation of the optimal solution are performed simultaneously using a method called the Big M method.
First, artificial variables

[0014]

(Equation 2)

Is introduced to formulate a linear programming problem of the following variable n + m.

[0016]

(Equation 3)

[0017]

(Equation 4)

[0018]

[0019]

(Equation 5)

Is clearly feasible for equation (3.6.2)
It is a solution. Set the parameter M to a sufficiently large positive number
For example, the artificial variable x '= (x'1, x'2, ..., x'm) T
(See Equation (3.6.2))
, First, the artificial variable term M
x 'quickly approaches 0. This means that each artificial variable
It means that it approaches 0 rapidly. Each artificial variable is enough
If it is close to 0, the feasible solution of equation (3.6.2)
The part excluding the number is regarded as a feasible solution of equation (3.6.1).
Let When the objective function is further reduced, c^T
Since the term of x decreases, finally the equation (3.6.2)
The part of the optimal solution of excluding the artificial variables is given by Equation (3.6.1)
Is a feasible optimal solution of It should be noted that the equation (3.6.2)
If the artificial variable is not close enough to 0 in the optimal solution
I.e., given a small positive number ε _AAgainst

[0021]

(Equation 6)

If so, it is determined that equation (3.6.1) was not executable. In the following description, the expression (3.
It is assumed that 6.1) already incorporates artificial variables.

(Determination of Search Direction by Affine Transformation) In the above-described method for determining the search direction of the optimal solution, when x (k) is near the center of S, the step size t can be made sufficiently large. Can be greatly reduced, but when x (k) is near the boundary of S, the objective function cannot be sufficiently reduced because the step size cannot be large. Then, the transformation is performed on the variable x so that the current solution x (k) is sufficiently separated from the boundary of the executable region in the transformation destination space, and then the solution is updated so as to greatly reduce the objective function. Inverse transformation is performed on the obtained new solution in the transformation destination space to obtain x (k
+1). For this purpose, a diagonal matrix defined for the current solution x (k)

[0024]

(Equation 7)

Affine transformation

[0026]

(Equation 8)

Is performed. Obviously, x (k) is always moved to (1,1, ..., 1) T in the destination space. The original linear programming problem in the space of y is expressed as follows.

[0028]

(Equation 9)

[0029]

(Equation 10)

The search direction is given by

[0031]

[Equation 11]

Where the search direction d (k) in the original variable is

[0033]

(Equation 12)

In the end, the search direction in the original space is

[0035]

(Equation 13)

Is given by This formula includes the calculation of the inverse matrix, but in the actual calculation, first, the simultaneous linear equations

[0037]

[Equation 14]

To solve for w (k)

[0039]

(Equation 15)

The search direction d (k) can be determined as

(Determination of Step Size) A straight line x given by the search direction d (k) determined as described above
Points on (k) + td (k) (t ≧ 0) are linear constraints

[0042]

(Equation 16)

And the objective function decreases monotonically with increasing step size. However, since it is not allowed to go outside the domain of variables 0 ≦ t ≦ u, the maximum value that the step size t can take is

[0044]

[Equation 17]

Is as follows. Actually, when the boundary of the executable area is reached, the search cannot be continued any more. Therefore, using the parameter α satisfying 0 <α <1,

[0046]

(Equation 18)

Is the step size for x (k), and the new solution is

[0048]

[Equation 19] Given by

(Replacement of upper limit and lower limit of variables) In the interior point method, if any of the variables approaches the lower limit during the search for the optimal solution, the affine transformation is effective, but the variable approaches the upper limit. In such a case, the same method cannot be used. Therefore, in the present invention, when the variable xj approaches its upper limit uj, the following conversion is performed to exchange the upper and lower limits of the variable.

[0050]

(Equation 20)

(Check of Residual for Constraint Condition) As the iterative calculation for searching for an optimal solution is repeated, the residual {Ax-b} for the constraint condition may increase due to a calculation error. Once the residuals are large, subsequent calculations are useless. To prevent this, for a positive number εr given every l iterations,

[0052]

(Equation 21)

Check whether or not
If not

[0054]

(Equation 22)

With the initial solution as the problem, the problem (3.6.2)

[0056]

(Equation 23)

Is recalculated, and the search for the optimum solution is restarted from the beginning.

(Convergence determination)

[0059]

(Equation 24)

When the condition is satisfied, x (k) is regarded as having converged to the optimal solution. εc is a parameter for determining convergence.

Next, embodiments of the present invention will be described in detail with reference to the drawings. FIG. 1 is a functional block diagram for explaining the configuration and operation of a linear programming problem solving device according to an embodiment of the present invention.

Referring to FIG. 1, in this embodiment, in the linear programming problem input unit A1, the number n of variables of the linear programming problem to be solved, the number m of constraints, the coefficient c of the objective function, the constraint c A and constant b, the upper limit value u of the domain of the variable, the parameter M of the Big-M method, the parameter εc of the convergence determination, the parameter εA of the feasibility determination, the parameter α for determining the step size, the maximum number of iterations nev Is input and stored in the storage means B1.

In the initial value determining unit A2, the variable x = (x
1, x2, ..., xn), the upper limit of 0.5
The double value is set and stored in the storage means B2. Also, work information p1, p2, corresponding to variables x1, x2,.
, Pn are all set to 1 and stored in the storage means B2. The value of the number of repetitions k is set to 1 and the storage means B1
To be stored.

In the artificial variable introducing unit A3, the storage means B
1, the number n of variables and the number m of constraints are extracted from the storage means B2, and the variable x = (x1, x2,..., Xn) is extracted from the storage means B2, and the value of the artificial variable xn + i (i = 1, 2,. Are set to 1 and stored in the storage means B2, and the artificial variable xn
+ I (i = 1, 2,..., N), adding m columns to the constraint condition coefficient A = (aij),
To be stored. The value of the part added at this time is

[0065]

(Equation 25)

Is obtained. The value of the part cn + 1, cn + 2,..., Cn + m corresponding to the artificial variable of the coefficient c of the objective function is set to M and stored in the storage means B1.

In the work vector calculation section A4, the number of variables n, the number of constraints m, the coefficient A of the constraint, the coefficient c of the objective function, and the variable x = (x1,
x2,..., xn, xn + 1,..., xn + m)

[0068]

(Equation 26)

The work vector w, which is the solution of the above, is calculated and stored in the storage means B1. Here, D is an (n + m) th-order diagonal matrix in which the diagonal components of the i-th row are the values of the variables xi.

In search direction determining section A5, storage means B
The number of variables n, the number m of constraints, the coefficient A and the constant b of the constraints, the coefficient c of the objective function, and the work vector w are extracted from Equation 1, and the search direction is calculated.

[0071]

[Equation 27]

Is calculated and stored in the storage means B1.

In the maximum step size determining unit A6,
The number n of variables, the number m of constraints, the variables x1, x2,..., Xn, xn + 1, xn + m, the upper limit u of variables (u1, u2,..., Un) and the search direction d = (d
1, d2,..., Dn) and extract the maximum step size tmax

[0074]

[Equation 28]

Then, it is stored in the storage means B1.

In the solution updating unit A7, the number of variables n, the number of constraints, and the search direction d = (d
1, d2, ..., dn + m), maximum step size tma
x, the parameter α for determining the step size is taken out,
The variable x = (x1, x2,..., Stored in the storage means B2.
xn, xn + 1, xn + m)

[0077]

(Equation 29)

The updated variable x 'is stored in the storage means B3.

In the convergence judging unit A8, the number n of variables, the number m of constraints, and the parameter εc of convergence judgment are stored in the storage unit B1, the variable x before the update from the storage unit B2, and the variable x after the update in the storage unit B3. Take out x 'and condition

[0080]

[Equation 30]

If the condition is satisfied, the processing is continued in the solution correcting unit A11. If the condition is not satisfied, the value of the variable x stored in the storage means B2 is replaced with the value of the updated variable x 'stored in the storage means B3. The processing is continued in the upper and lower limit inversion processing section A9.

In the upper / lower limit inversion processor A9, the number n of variables, the number m of constraints, the coefficient A and the constant b of the constraint, and the parameter g of upper / lower inversion are stored in the storage unit B2 as variables x1, x2, ..., xn

[0083]

(Equation 31)

If there is no variable xj that satisfies the condition, the process is continued by the repetition number determination unit A10. Such a variable xj
Exists, the coefficient aij, the constant bj, and the number of iterations k of the constraint condition stored in the storage means B1 are

[0085]

(Equation 32)

As described above, the variable xj and the work information pj stored in the storage means B2 are

[0087]

[Equation 33]

The processing is continued in the artificial variable introducing unit A3 as described above.

The storage means B in the repetition number determination section A10
The number of repetitions k and the maximum number of repetitions nov are extracted from 1 and if the value of the number of repetitions k is equal to the value of the maximum number of repetitions nev,
Before the optimal solution is obtained, the fact that the number of iterations has reached the value of the maximum number of iterations nev is displayed on the output means B4, and the processing is stopped. Otherwise, the value of the number of repetitions k stored in the storage means B1 is

[0090]

(Equation 34)

The work vector calculation unit A4
The processing is continued in.

In the solution correcting unit A11, the number of variables n, the number of constraints m, the coefficient of the objective function c
, Cn, the upper limit is u1, u2,..., Un, the variables x1, x2,..., Xn and the work information p1, p2,. For the variable xj, the value of the variable xj stored in the storage means B2 is

[0093]

(Equation 35)

Is replaced as follows. At the time when this replacement is completed, the variables x1,
.., xn are displayed on the output means B4. The corresponding value of the objective function cTx is displayed on the output means B4.

Next, this embodiment will be described in detail with reference to the drawings. FIG. 2 is a problem example for explaining a method of solving a linear programming problem according to one embodiment of the present invention.

Referring to FIG. 2, the problem (a) is that x = (x1, x2,..., Xn) that minimizes the objective function (3) under the constraint (1) and the upper and lower limit constraints (2). ) Is a problem to obtain the value x *. Problem (b) with the conventional interior point method
Since the upper limit value of the variable can only be handled by incorporating it into the constraint condition (1) as in (4), the constraint condition (1)
Increases. Furthermore, slack variables x6, x7, x8, x9 and x10 are introduced to convert the inequality constraints (2) and (5) into the equality constraints (3) and (6). The number will also increase.

Next, the operation of this embodiment will be described with reference to FIG.
This will be described in detail with reference to FIGS. FIG. 3 is FIG.
4 shows the operation 1 in the method for solving the linear programming problem of FIG.
Shows operation 2 in the method for solving the linear programming problem in FIG.

The problem (a) is a problem for minimizing the objective function (3) under the constraint condition (1) and the upper and lower limit constraint conditions (2). The problem (b) is obtained by rewriting the problem (a) so that it can be handled by the interior point method. First, since only the lower limit can be set as the domain (2) of the variable, the upper limit needs to be added to the constraint (1). In addition, the constraint (1) that gives the upper limit is an inequality, but the constraint (1) that can be handled by the interior point method must be an equation, so that in order to replace this inequality with an equivalent equation, Need to be introduced.

When solving the problem (a) of the present embodiment according to the embodiment, the upper and lower limit inversion processing unit A of the embodiment is used.
If the process of step 9 is omitted, as shown in FIG. 3, after the fifth update of the solution, the variable approaches the upper limit value 1, so that the solution is not updated before reaching the optimal solution. On the other hand, in FIG. 4, the upper and lower limit inversion processing unit A of FIG.
9 shows the progress of the calculation in the case where the processing of No. 9 is performed.
After updating the solution for the fourth time, the upper and lower limits of x1 are inverted, and after updating the solution for the fourth time, the upper and lower limits of x4 and x5 are inverted to reach the optimal solution by the twentieth update. Here, x6, x7, and x8 are introduced from the initial solution that does not always satisfy the constraint condition (1) for the calculation of the Big-M method for reaching the optimal solution that satisfies the constraint condition (1). This is an artificial variable.

As described above, according to the present embodiment, when the variable xj having the upper and lower limit constraints approaches the upper limit value uj of the domain of the variable xj during the calculation of the solution by the interior point method, In order to solve the problem that the calculation of the solution cannot be continued as it is at the limit of the point method, the difference uj between the upper limit value uj of the domain of the variable xj and the variable xj is determined.
By replacing the variable xj with −xj, the upper and lower limits are reversed, and the calculation of the solution is continued while avoiding the limit of the interior point method. In other words, in a method for solving a linear programming problem having upper and lower bound constraints by the interior point method, variable transformation is appropriately performed in the process of searching for a solution, and the upper and lower limits are reversed, thereby increasing the inner constraints without increasing the constraints. The point method can be applied. As a result, there is an effect that the calculation can be continued while avoiding the limit of the interior point method.

It should be noted that the present invention is not limited to the above embodiment, and it is apparent that the above embodiment can be appropriately modified within the scope of the technical idea of the present invention. Further, the number, position, shape, and the like of the constituent members are not limited to the above-described embodiment, and can be set to numbers, positions, shapes, and the like suitable for carrying out the present invention. In each drawing, the same components are denoted by the same reference numerals.

[0102]

Since the present invention is constructed as described above, for a linear programming problem in which variables have upper and lower limits and the coefficients which provide the coefficients of the constraints, that is, the coefficient matrix is a large-scale and sparse matrix, The advantage is that the optimal solution can be calculated effectively using the interior point method. The reason is that the interior point method is more effective than the simplex method when the coefficient matrix is large and sparse, and furthermore, the present invention allows the upper limit of variables to be introduced without increasing constraints. is there.

[Brief description of the drawings]

FIG. 1 is a functional block diagram for explaining the configuration and operation of a linear programming problem solving device according to an embodiment of the present invention.

FIG. 2 is an example of a problem for explaining a method for solving a linear programming problem according to an embodiment of the present invention.

3 is an operation 1 in the method for solving a linear programming problem in FIG. 2;
Is shown.

4 is an operation 2 in the method for solving the linear programming problem in FIG. 2;
Is shown.

[Explanation of symbols]

 A1: Linear programming problem input unit A2: Initial value determination unit A3: Artificial variable introduction unit A4: Work vector calculation unit A5: Search direction determination unit A6: Maximum step size determination unit A7: Solution update unit A8: Convergence determination unit A9 ... Upper / lower limit inversion processing unit A10: Iterative number determination unit A11: Solution correction unit B1, B2, B3, C1 ... Storage means B4 ... Output means

 ────────────────────────────────────────────────── ─── Continuing on the front page (72) Inventor Kenichi Takada 3-2-1 Sakado, Takatsu-ku, Kawasaki-shi, Kanagawa F-term in NEC Information Systems Co., Ltd. 5B056 AA00 BB37 BB92 HH00

Claims (6)

[Claims] 1. A variable having upper and lower limits, and an upper limit of a domain of the variable when the variable approaches an upper limit during calculation of a solution by an interior point method with respect to an upper limit of a domain of the variable. A method for solving a linear programming problem, characterized in that the variable is replaced by the difference between the variable and the variable, the upper and lower limits of the variable are reversed, and the calculation of the solution is continued while avoiding the limit of the interior point method. 2. A linear program characterized by minimizing an objective function given by a linear function of a variable under a linear constraint condition and upper and lower limit constraints and obtaining a value of the objective function at that time by an interior point method. How to solve the problem. 3. A linear programming problem characterized by applying an interior point method without increasing constraints by performing variable transformations as appropriate in the process of searching for a solution and inverting upper and lower limits. Solution method. 4. An upper limit value of a variable having an upper / lower limit and an upper limit value of a domain of the variable when the variable approaches an upper limit during calculation of a solution by an interior point method with respect to an upper limit value of a domain of the variable. Solving the linear programming problem by replacing the variable with the difference between the variable and the variable, inverting the upper and lower limits of the variable, and avoiding the limit of the interior point method and continuing the calculation of the solution. apparatus. 5. A method for minimizing an objective function given by a linear function of a variable under linear constraints and upper and lower limits, and obtaining a value of the objective function at that time by an interior point method. To solve linear programming problems. 6. The method according to claim 1, further comprising the step of executing variable conversion as needed in the process of searching for a solution and inverting the upper and lower limits to apply the interior point method without increasing constraints. Planning problem solver.