JP2001117773A - Method for searching discrete optimal solution - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a method and a device to solve a discrete optimization program so that an optimal solution does not fall into a local solution when the optimal solution on a discrete set is searched. SOLUTION: In this searching method to find a solution of the discrete optimization problem by performing random search, plural searching vicinities of low-order and high-order and defined, the solution is searched by using the vicinities, a low-order vicinity is intensively searched when possibility that many improved solutions exist in the vicinity of the present solution is high and a wide area is searched when the possibility that the improved solutions exist in the vicinity of the present solution is low. In this method, the vicinities to be searched are found by defining distance in a searching space, the vicinities to be searched of the solution are switched depending on success/failure information on the search, the high-order vicinity is searched when the search of the solution is successfully performed, the high-order vicinity is searched when the search of the solution fails, the search of the low-order vicinity is focused on at a point of time of an early stage of the search, after completion of the early stage, the search of the low-order vicinity and the high-order vicinity are switched depending on the success/failure information on the search of the solution.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention relates to an adaptive random search method (Ra) for a discrete value for finding an optimal solution on a discrete set.
ndom Search method with Intensification and Divers
certification-Discrete: hereinafter referred to as RasID-D).

[0002]

2. Description of the Related Art For a possible solution x obtained by some method, a neighborhood N (x) (that is, a set of solutions obtained by making a small change) is defined, and a possible solution in N (x) is defined. If there is a solution that can improve the objective function value, a method of replacing the approximate solution with the solution is called a neighborhood search method. Here, N (x) is defined as the current solution x and other solutions x ′
And a distance d (x, x) appropriately defined according to the problem.
x ′) based on the l-order neighborhood N ^l (x) = {x ′ | d (x, x ′) = l} (Equation 1) is often used.

An algorithm that repeats this neighborhood search method until no improvement can be obtained is called a local search method. That is, for the current solution x, a solution x ′ selected from its neighborhood N (x)
∈N (x) is generated and its solution is better than the current solution f
If (x ′) <f (x) (in the case of minimization), the solution is improved (x = x ′), and the process is repeated until the solution in the vicinity cannot be improved. In addition, this improvement includes a first improvement in which a solution that is slightly better than the current solution is found, and a transition to that solution as soon as the solution is found, and a transition to the best shape when all the neighbors are searched. There is best improvement. The algorithm is as follows.

[Step 0 (initial solution)] An initial feasible solution x is obtained. k = l ₀ [Step k (improvement)] Search for a possible solution y having an objective function value better than x in the neighborhood N (x) of x (neighborhood search). If such a y is found, x = y, k = k + 1
And return to step k. Otherwise, output the current x as a solution and stop.

FIG. 1 shows the progress of the local search method using the vicinity of a circle. However, in this figure, x _k is the k-th possible solution. When the search stops, the solution x _{k at} that time is the neighborhood N
In the sense that (x _k) there is no better solution is x _k in, it can be said to be a local optimal solution (quasi-optimal solution).

[0006]

In this local search method, since a search is made within the neighborhood, generally a good local optimal solution tends to be obtained by starting from a good initial solution. Also, with the local search method, the possibility that a better solution remains in the region of the last search cannot be denied. As a countermeasure for this, various initial solutions may be tried, or a stochastic operation may be introduced into the search.

The problem to be solved by the present invention is to provide a method for solving a local set when searching for an optimal solution on a discrete set, and a search device for executing a search program using this method. It is to be.

[0008]

In order to solve the above-mentioned problems, a method for searching for a discrete optimal solution according to the present invention includes a number of improved solutions near a current solution when finding an optimal solution on a discrete set. If the probability is high, the search for the neighborhood is intensively performed, and if the possibility that an improved solution exists near the current solution is low, the solution falls into a local solution by performing a wide area search. Is to avoid.

[0009] RasID-D is an optimization method for random values (Random Search method with Intensification and
Diversification (hereinafter referred to as RasID). First, the basic concept of RasID will be described. Ras
ID is an optimization method that handles search centralization and diversification in a unified framework. If the search is successful, it is considered that there is a high possibility that an improved solution exists in the vicinity, and the search range is limited, and the search is performed with focus on the direction in which the search was successful (intensive search). In this way, the parameter search is finely performed to improve the evaluation index, and the search speed is improved. On the other hand, if the search is unsuccessful, it is considered that there is a low possibility that an improved solution exists in the vicinity, and the search is performed randomly over a wide area (wide area search). As described above, it is possible to surely improve the evaluation index, improve the search speed, and escape from the local minimum. R
asID-D extends this concept to discrete events,
This is a local search method for solving the objective function f (x) of the discrete optimization problem of (Equation 2) → maximum or minimization constraint x∈X⊆Z ⁿ (Equation 2). As a feature, using the l-th order neighborhood N ^l (x) = {x ′ | d (x, x ′) = l} defined by Equation 1, (1) many improvements to the neighborhood of the current solution When there is a high possibility that a solution exists, the search for low-order neighborhoods with small l is intensively performed. (2) When there is a low possibility that an improved solution exists near the current solution, l Search for a high-order neighborhood having a large value, in other words, search for a wide area. That is mentioned. In other words, at first, it works intensively near the existing solution and there are many improved solutions nearby, but after a while it is naturally difficult to find an improved solution near the existing solution. In such a case, the search range is enlarged, a new improved solution is found, and a search near the solution is intensively performed. In the local search method, the neighborhood of one solution x _k is fixed at N (x _k ), whereas the search of RasID-D expands the neighborhood of one solution.

As described above, the RasID-D has a basic configuration of a centralized search and a wide area search for improving the search efficiency. One of the points that can be improved by using RasID-D is to escape from “local minimum”. FIG. 2 shows the method. The escape from the local minimum is performed by the following procedure. A: Intensive search near the current solution B: Because the solution has not improved because it has fallen to the local minimum, C: A new solution is found when performing a wide area search, D: Intensive search near that solution again E: Settle down to the global minimum.

By using such a method, it is difficult to escape from the local minimum in FIG. 2 using only the centralized search D. However, if the wide area search C is added to this, it is possible to escape from the local minimum. It will be closer to the global minimum E. As a result, even if the solution has not been improved anymore because of being caught by the local minimum, it will escape from the local minimum and further improve the solution.

[0012]

DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described with reference to a simple flowchart shown in FIG. RasID-
The flow of the algorithm of D is shown below.

[Step 100] Initial conditions are determined.・ Determine the total number of searches.・ Determine the maximum order in the vicinity.・ Determine the number of times to search one neighborhood. -Find the initial solution by some method.

[Step 101] The lowest neighborhood of the solution is searched. (A) If a solution is found in step 102, step 1
At 03, the solution is returned to step 101 as a new solution. (B) If a solution is not found even if the number of times of searching for one neighborhood is exceeded in step 105, go to step 106.

[Step 106] A similar search is performed by increasing the order in the vicinity by one. (A) When a solution is found, the solution is set as a new solution and the process returns to step 101. (B) If a solution is not found even if the number of times to search for one neighborhood is exceeded, go to step 106. (C) If it exceeds the maximum degree in the vicinity determined first in step 105, the process returns to step 101. If the number of searches exceeds the number of searches initially determined in step 105, the process proceeds to step 104 and ends. Because of this simple algorithm,
The algorithm using RasID-D can be realized by programming the algorithm of RasID-D and using a notebook personal computer.

[0016]

DESCRIPTION OF THE PREFERRED EMBODIMENTS As an embodiment of the present invention, a search for a restoration system in a power system will be described. In this embodiment, when an accident occurs in the power system and an overload of the equipment or a supply failure (power failure) occurs, a system in which these are eliminated is searched for. The accident recovery system is searched using RasID-D with the supply disturbance amount and the like as the objective function as the constraint condition.

FIGS. 4 and 5 are flowcharts specifically showing a search method for finding x (optimal solution) for minimizing the evaluation function f (x). Step 300: The total number of searches (calculations), the maximum order of neighbors, and the maximum number of searches for neighbors of the same order are R,
Input to L and Q Step 301: Set the count value of the total number of searches to r and substitute 1 Step 302: Initially select a value randomly selected from the entire search range X, and compare the value obtained by the evaluation function with the value to be compared Step 303: Substitute 1 for the search range (order of neighborhood) as l. Step 304: Determine whether an improved solution is found by the search process. There is an improved solution when Flag = 1, and there is no improved solution when Flag = 0. Step 305: Substitute x2 for x1, and exchange an improved solution. Step 306: Search number check Step 307: One is added to the search range (order in the vicinity) l. Step 308: Check neighborhood degree Step 309: Assign 0 to the improvement solution flag Flag. Step 310: 1 is added to the counter q of the same-order neighborhood search.
Substitute. Step 311: A solution is randomly selected within the range represented by the degree l, and substituted into x2. Step 312: 1 is added to the entire search number counter r. Step 313: Calculate the evaluation function f (x2) using x2 and substitute it for the evaluation value f2. Step 314: Previous evaluation value f1 and current evaluation value f2
Compare. improved solution if f (x2) <f (x1), f
(X2) ≧ f (x1), no improvement solution. Step 315: Substituting 1 for the improved solution flag Flag Step 316: 1 for the counter q in the same-order neighborhood search
Addition Step 317: Check the number of times of searching for the vicinity of the same order Step 318: Output x1 as the optimal solution.

[0018]

As described above, according to the present invention,
The following effects are obtained. (1) If there is a high possibility that there are many improved solutions near the current solution, the search for low-order neighborhoods is intensively performed, and there is an improved solution near the current solution. If it is unlikely that the best solution will be obtained, a wide area search can be performed to prevent the optimum solution to be obtained from falling into a local solution, and a further improved solution can be obtained. (2) Since the algorithm is simplified, the capacity of the program can be reduced, and it can be realized on a small computer such as a notebook personal computer. (3) By using the present invention as described above, in a field to which a method for obtaining an optimal solution on a discrete set is applied, that is, various wirings such as a base wiring that handles several hundred points, a delivery plan of a transportation company, scheduling, traffic monitoring, and the like. Practical effects in various industrial fields.

[Brief description of the drawings]

FIG. 1 is an explanatory diagram showing the progress of a local search.

FIG. 2 is an explanatory diagram showing a method of escaping from a local minimum.

FIG. 3 is a flowchart of RasID-D.

FIG. 4 is a flowchart for obtaining x (optimum solution) for minimizing an evaluation function f (x).

FIG. 5 is a flowchart of a search process in the flowchart of FIG. 4;

Claims (5)

[Claims] In a search method for finding a solution of a discrete optimization problem by random search, a plurality of low-order and high-order search neighborhoods are defined, a solution search is performed using these neighborhoods, and a current solution is searched for. If there is a high possibility that there is a possibility that many improved solutions exist in the vicinity of, search for lower-order neighborhoods is intensively performed,
A method for searching for a discrete optimal solution, characterized by performing a wide area search when there is a low possibility that an improved solution exists near the current solution. 2. The method according to claim 1, wherein the search neighborhood is obtained by defining a distance in a search space. 3. The method for searching for a discrete optimum solution according to claim 1, wherein the search neighborhood of the solution is switched according to the success / failure information of the search. 4. The discrete optimal solution according to claim 1, wherein if the solution search is successful, a lower-order neighborhood is searched, and if the solution search fails, a higher-order neighborhood is searched. Search method. 5. A low-order neighborhood search is intensively performed at an initial point of the search, and after the end of the initial point, the low-order neighborhood search and the high-order neighborhood search are switched based on success / failure information of the solution search. 2. The method for searching for a discrete optimal solution according to claim 1, wherein