Patent Publication Number: US-2006010091-A1

Title: Solution search apparatus and initial value setting method thereof

Description:
BACKGROUND OF THE INVENTION  
      1. Field of the Invention  
      The present invention relates to a solution search apparatus and its initial value setting method. For example, the present invention relates to a calculation method of automatically searching values of a plurality of parameters used by a physical or chemical model included in a numerical software, by using a genetic algorithm, in order to make a discrepancy between a calculated value and given data be minimum.  
      Further, the solution search apparatus of the present invention is used for enhancing prediction ability of a simulator by reducing a discrepancy between the simulator and a measured result, and concretely, by setting a plurality of parameters of a transistor model included in a semiconductor circuit simulator in order to reduce a discrepancy between a calculated value and measured current voltage data (I-V data), or fitting a plurality of parameters of ion implantation model included in a semiconductor process simulator to reproduce a result of SIMS measurement.  
      2. Description of the Related Art  
      Generally, when searching for a plurality of parameter values of a model having a plurality of parameters and strong nonlinear behavior, if a discrepancy between a calculated value based on the parameter values and measured data is expressed as a function of these parameters, it is experientially known that there are many local-minimum values.  
       FIG. 10  shows the case a model function with four parameters, called a Pearson4 function, is fitted to target measured data (denoted by mark ◯). As shown in  FIG. 11 , which is displaying the fitting errors with respect to two parameters, there appear two almost flat valleys (right and left). This means that there exist a plurality of parameter combinations which gives almost the same small fitting errors.  
      In such a case, seeking a parameter set with the minimum discrepancy is equivalent to seeking a parameter set being the lowest valley among many valleys. Such a problem is called a “multi-valley problem.” If a search method utilizing a gradient of discrepancy given by a parameter set is applied to such a problem, the finally obtained parameter set will strongly depend on an initial parameter value given when search starts. Then, a genetic algorithm (GA) which can practically search a solution with a small discrepancy even in a multi-valley problem is noticed as a predominant technique in recent years.  
      In a genetic algorithm, a search range, expressed by a minimum value and a maximum value for each parameter, is given in advance for each parameter to be searched. Then, a value of each parameter is set up at random by using pseudo-random numbers uniformly distributed in the search range.  
      For example, when each parameter value is represented by 10-bit data composed of 1 or 0, in the case of searching for four parameters, forty bits of 1 or 0 data is used. This data is called a “gene.” With respect to one gene, a value of discrepancy between a calculated model using the parameter set and measured data is arranged to be corresponding to the gene. This discrepancy value is called a “fitness value.” 
      In this way, a sufficient number of pairs, for instance, one hundred of genes with their fitness values are generated using pseudo-random numbers.  
      The Related Arts of the present Invention are: 
      [Patent Document 1]
        Japanese Unexamined Patent Publication No. JP2003-85526    
        [Non-Patent Document 1]
        “Examination of Parameter Extraction Method using Genetic Algorithm”, by Wada and Kanemura, the Japan Society of Applied Physics, Silicone Technology Subcommittee, No. 32, STM 01-07, pp. 37-42, 2001    
       

      In the search stated above, it is necessary to specify a search range of a parameter. And an initial value of the parameter is set in the specified range. For example, when a parameter p 0  is specified to be equal to or greater than 1 and equal to or less than 20, initial values of the parameter p 0  are set by random numbers uniformly distributed between 1 and 20 for each gene. However, this method of setting an initial value does not work well for parameters whose search range covers several digits, due to the following reasons.  
      For example, searching for a true solution, being 20 in this case, of a parameter whose search range of p 0  is wide, such as from 10 to 100000, by using one hundred genes will be examined. As shown in  FIG. 12 , if an initial distribution of p 0  is made by using uniformly distributed random numbers between 10 and 100000, in the most case, we will get a value of several ten-thousands, and values equal to or less than 100, for example, will rarely be obtained as an initial value of p 0 . Therefore, after the search starting, the possibility of searching the range equal to or less than 100, for example, falls very much for the following reasons. In order to make p 0  become a value in a specific range in a GA search, it is necessary to perform an operation of “crossover” and “mutations” based on a parameter value of a selected parent individual. In the case of the initial value of p 0  being equal to or greater than 100 in all the gene individuals, since a bit sequence of 1 and 0 corresponding to the true solution being about 20 is not included in any gene individual, a bit sequence corresponding to a value about 20 is not produced by a crossover operation. Therefore, it cannot but wait for such a bit sequence to be generated by “mutation.” Then, if the generation probability of mutation is highly set up, the probability of desirable bit sequence information narrowed down by crossover operation being destroyed will increase, and consequently the efficiency of search will fall.  
     SUMMARY OF THE INVENTION  
      The present invention has been contrived to solve the above problems. One of objects of the present invention is to increase efficiency of searching a solution by making an effective initial value and an initial distribution in solution searching.  
      These and other objects of the embodiments of the present invention are accomplished by the present invention as hereinafter described in further detail.  
      According to one aspect of the present invention, a solution search apparatus which searches for a solution by a predetermined algorithm, using a parameter having a predetermined search range, includes: 
          a parameter search range storing part to store a search range of a parameter;     a parameter storing part to store a plurality of parameters;     a parameter setting part to refer to the search range of the parameter stored in the parameter search range storing part, generate parameter values so that logarithmic values of the parameter values for at least a part of the search range of the parameter may be predeterminedly distributed, and store the parameter to which generated parameter values are set, in the parameter storing part; and     a search part to read the parameter to which the generated parameter values are set by the parameter setting part, from the parameter storing part, and search for the solution by the predetermined algorithm.        

      According to another aspect of the present Invention, the parameter setting part includes a uniform distribution calculation part to generate the parameter values so that the logarithmic values of the parameter values for at least a part of the search range of the parameter may be uniformly distributed.  
      According to one aspect of the present Invention, a solution search apparatus which searches for a solution by a predetermined algorithm, using a parameter having a predetermined search range, includes: 
          a parameter search range storing part to store a search range of a parameter;     a parameter storing part to store a plurality of parameters;     a parameter setting part to refer to the search range of the parameter stored in the parameter search range storing part, generate parameter values so that distribution of logarithmic values of the parameter values for a part of the search range of the parameter may become denser compared with distribution of logarithmic values of parameter values for other part of the search range, and store the parameter to which generated parameter values are set, in the parameter storing part; and     a search part to read the parameter to which the generated parameter values are set by the parameter setting part, from the parameter storing part, and search for the solution by the predetermined algorithm.        

      According to another aspect of the present Invention, the parameter setting part includes a normal distribution calculation part to generate the parameter values so that the distribution of the logarithmic values of the parameter values for a part of the search range of the parameter may become a normal distribution.  
      According to another aspect of the present Invention, the parameter setting part includes a uniform distribution calculation part to generate parameter values so that distribution of logarithmic values of the parameter values for other part of the search range of the parameter may become a uniform distribution.  
      According to one aspect of the present Invention, an initial value setting method of a solution search apparatus, which searches for a solution with updating a parameter value stored in a storing part, by using a genetic algorithm, includes: 
          acquiring a search range of a parameter and storing the search range of the parameter in a storing part; and     setting initial values of a parameter so that logarithmic values of parameter values may be predeterminedly distributed for the search range of the parameter stored in the storing part, and storing the initial values in the storing part.        

      According to another aspect of the present Invention, the initial values of the parameter are set so that distribution of the logarithmic values of the parameter values for at least a part of the search range of the parameter may become a uniform distribution.  
      According to another aspect of the present Invention, the initial values of the parameter are set so that the distribution of the logarithmic values of the parameter values for at least a part of the search range of the parameter may become a normal distribution.  
      Further scope of applicability of the present invention will become apparent form the detailed description given hereinafter. However, it should be understood that the detailed description and specific examples, while indicating preferred embodiments of the invention, are given by way of illustration only, since various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
      A complete appreciation of the present invention and many of the attendant advantages thereof will be readily obtained as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings, wherein:  
       FIG. 1  shows an example of an outward view of a solution search apparatus  100  according to Embodiment 1;  
       FIG. 2  shows an example of a hardware configuration of the solution search apparatus  100  according to Embodiment 1;  
       FIG. 3  shows an example of a logical structure of the solution search apparatus  100  according to Embodiment 1;  
       FIG. 4  shows a logarithmic distribution of an initial value of a parameter p 0  according to an initial value setting step of a parameter setting part  12  of Embodiment 1;  
       FIG. 5  shows explanation of a genetic algorithm as a method of searching for an optimum value by a search part  15 ;  
       FIG. 6  shows a result of executing GA by setting initial values of a parameter value according to the method of the present Embodiment;  
       FIG. 7  shows a result of executing GA by setting initial values of a parameter value according to a conventional simple uniform distribution;  
       FIG. 8  shows fitting errors with respect to a model function including a parameter whose search range changes within several digits;  
       FIG. 9  shows a logarithmic distribution of an initial value of a parameter p 0  according to an initial value setting step of a parameter setting part  12  of Embodiment 2;  
       FIG. 10  shows a calculation example (curve) of fitting a Pearson4 function to target data (denoted by mark ◯);  
       FIG. 11  shows a result of representing fitting errors corresponding to two of four parameters in the result of  FIG. 10 , on the vertical axis; and  
       FIG. 12  shows an initial distribution uniformly distributed in the search range (from 10 to 100000) of the parameter p 0 . 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
      Reference will now be made in detail to the present preferred embodiments of the invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals indicate like devices through out the several views.  
     Embodiment 1  
       FIG. 1  shows an example of an outward view of a solution search apparatus according to Embodiment 1. In  FIG. 1 , a solution search apparatus  100  is equipped with a system unit  910 , a CRT (Cathode Ray Tube) display  901 , a keyboard (K/B)  902 , a mouse  903 , a compact disk drive (CDD)  905 , a printer device  906 , and a scanner device  907 , and these are connected through cables. Furthermore, the solution search apparatus  100  is connected to a facsimile machine  932  and a telephone  931  through cables, and to the Internet  940  through a local area network (LAN)  942  and a web server  941 .  
       FIG. 2  shows an example of a hardware configuration of the solution search apparatus  100  according to Embodiment 1. In  FIG. 2 , the solution search apparatus  100  includes a CPU (Central Processing Unit)  911  which executes a program. The CPU  911  is connected to a ROM  913 , a RAM  914 , a communication board  915 , the CRT display  901 , the K/B  902 , the mouse  903 , a FDD (Flexible Disk Drive)  904 , a magnetic disk drive  920 , the CDD  905 , the printer device  906 , and the scanner device  907 , through a bus  912 .  
      The RAM  914  is an example of a volatile memory, and the ROM  913 , the FDD  904 , the CDD  905 , and the magnetic disk drive  920  are examples of nonvolatile memories. These are examples of storage devices or storing parts.  
      The communication board  915  is connected to the facsimile machine  932 , the telephone  931 , the LAN  942 , etc. The communication board  915 , the K/B  902 , the scanner device  907 , the FDD  904 , etc. are examples of input parts. The communication board  915 , the CRT display  901 , etc. are examples of output parts.  
      It is also acceptable to directly connect the communication board  915  to the Internet  940  or a WAN (Wide Area Network), such as ISDN, without being connected to the LAN  942 . When the communication board  915  is directly connected to the Internet  940  or the WAN, such as ISDN, the solution search apparatus  100  is connected to the Internet  940  or the WAN, such as ISDN. Then, the web server  941  becomes unnecessary.  
      In the magnetic disk drive  920 , an operating system (OS)  921 , a window system  922 , a program group  923 , and a file group  924  are stored. The program group  923  is executed by the CPU  911 , the OS  921 , and the window system  922 .  
      In the program group  923 , programs which execute functions of what is described as a “—part” in Embodiment below are stored. The programs are read and executed by the CPU  911 .  
      In the file group  924 , what is described as a “judgment result of—”, “calculation result of—”, and “processing result of—” in Embodiment below are stored as a “—file”.  
      Arrows in the flowchart explained in Embodiment below mainly show inputs or outputs of data. Then, in order to be inputted or outputted, the data is recorded in the magnetic disk drive  920 , a FD (Flexible Disk), an optical disk, a CD (compact disk), a MD (mini disk), a DVD (Digital Versatile Disk), and other recording medium, or transmitted through a signal line or other transmission medium.  
      Moreover, it is also acceptable to execute what is described as a “—part” in Embodiment below, by firmware stored in the ROM  913 , or to execute it by software only, hardware only, a combination of software and hardware, or a combination of software, hardware and firmware.  
      Furthermore, it is also acceptable to store the programs for executing Embodiment described below, by using the magnetic disk drive  920 , a FD (Flexible Disk), an optical disk, a CD (compact disk), a MD (mini disk), a DVD (Digital Versatile Disk) and a recording apparatus made of other recording medium.  
       FIG. 3  shows an example of a logical structure of the solution search apparatus  100  according to Embodiment 1. The solution search apparatus  100  in  FIG. 3  is an apparatus which searches for a solution by a predetermined genetic algorithm, with using a parameter having a predetermined search range.  
      The solution search apparatus  100  includes a parameter search range storing part  21  which stores a search range of a parameter, and a parameter storing part  22  which stores genetic data (henceforth, called a gene or an individual) having a plurality of parameters. In  FIG. 3 , a gene is a model function which has four parameters p 0 , p 1 , p 2 , and p 3 .  FIG. 3  shows the case where one hundred genes are stored.  
      The solution search apparatus  100  includes a parameter search range input part  11  which inputs a search range of a parameter from an input apparatus and stores the search range of the parameter in the parameter search range storing part  21 .  
      The solution search apparatus  100  also includes a parameter setting part  12  which refers to a search range of a parameter stored in the parameter search range storing part  21 , generates parameter values so that logarithmic values of the parameter values for at least a part of the search range of the parameter may be predeterminedly distributed, and stores the parameter to which the generated parameter values are set, in the parameter storing part  22 .  
      The solution search apparatus  100  also includes a search part  15  which reads the parameter to which the parameter values are set by the parameter setting part  12 , from the parameter storing part  22 , and searches for a solution by the genetic algorithm.  
      The parameter setting part  12  includes a uniform distribution calculation part  13  which generates parameter values so that logarithmic values of the parameter values for at least a part of the search range may be uniformly distributed. A normal distribution calculation part  14  will be explained in Embodiment 2.  
      An initial value setting method of the solution search apparatus, which searches for a solution with updating a parameter value stored in the storing part, by using a genetic algorithm will be explained. The initial value setting method of the solution search apparatus according to Embodiment 1 is characterized by a setting method of an initial value of the parameter p 0  which can be with several digits.  
      For example, the case of fitting a model function having four parameters p 0 , p 1 , p 2 , and p 3  including the parameter p 0  which can be a value from 10 to 100000, to given measured data will be described. Moreover, the case of searching for a parameter by GA using one hundred individuals will be described.  
      The following operations are performed in a search range storing step. The parameter search range input part  11  inputs search ranges of the parameters p 0 , p 1 , p 2 , and p 3  from the input apparatus, and stores the search ranges in the parameter search range storing part  21 . The search range herein is represented by a minimum value and a maximum value.  
      It is herein assumed that the parameter p 0  can be a value from 10 to 100000. That is, the minimum value of the parameter p 0  is p 0 min=10 and the maximum value is p 0 max=100000.  
      It is also assumed that the search ranges of the parameters p 0 , p 1 , p 2 , and p 3  were input as follows:  
      The minimum value of the parameter p 1  is p 1 min, and the maximum value is p 1 max.  
      The minimum value of the parameter p 2  is p 2 min, and the maximum value is p 2 max.  
      The minimum value of the parameter p 3  is p 3 min, and the maximum value is p 3 max.  
      The following operations are performed in an initial value setting step. The parameter setting part  12  refers to the search ranges of the parameters p 0 , p 1 , p 2 , and p 3  stored in the parameter search range storing part  21 , sets up initial values of the parameters, and stores them in the parameter storing part  22 .  
      The uniform distribution calculation part  13  generates one hundred parameter values so that logarithmic values of the parameter values for the search range of the parameter p 0  may be uniformly distributed, sets the generated one hundred parameter values to the parameter p 0  of the individual of the parameter storing part  22 , and stores them. That is, in the case of searching for a parameter by GA using one hundred individuals, one hundred p 0 s are calculated for each individual by p 0 =10ˆ{log(p 0 min)+R*(log(p 0  max/p 0 min))}(ˆindicates power) regarding R as a random number uniformly distributed between [0, 1].  
       FIG. 4  shows a logarithmic distribution of the initial value of the parameter p 0  according to the initial value setting step of the parameter setting part  12  of Embodiment 1. The horizontal axis showing a logarithmic scale indicates the logarithmic value of p 0 , namely log (p 0 ). The vertical axis indicates generation probability of a parameter. Assuming that the minimum value of the parameter p 0  is p 0 min=10 and the maximum value is p 0 max=100000, the logarithmic value of p 0 , namely log (p 0 ), will be calculated as follows. The base is supposed to be 10. 
 log( p   0 )=log10ˆ{log( p   0 min)+ R *(log( p   0 max/ p   0 min))} log( p   0 )=log10ˆ{log(10)+ R *(log(100000/10))} log( p   0 )=log10ˆ{log(10)+ R *(log(10000))} log( p   0 )=log10ˆ{1 +R* 4} log( p   0 )=1 +R *4           where R denotes a random number uniformly distributed between [0, 1], the minimum value of R is 0 (minimum value=0) and the maximum value of R is 1 (maximum value=1)     when R=0, log(p 0 )=1+R*4=1, and     when R=1, log(p 0 )=1+R*4=5        
      Therefore, the minimum value of the logarithmic value of p 0  is 1, (minimum value=1) and the maximum value is 5 (maximum value=5). Since R is a random number uniformly distributed between [0, 1], the logarithmic value of p 0  is also distributed uniformly. Then, as shown in  FIG. 4 , the logarithmic value of p 0  is uniformly distributed between 1 and 5.  
      On the other hand, the parameter setting part  12  calculates one hundred parameters of p 1 , p 2 , and p 3  respectively for each individual by the following calculation, regarding R as a random number uniformly distributed between [0, 1]. 
 
 p   1 = p   1 min+ R *( p   1 max− p   1 min) 
 
 p   2 = p   2 min+ R *( p   2 max− p   2 min) 
 
 p   3 = p   3 min+ R *( p   3 max− p   3 min) 
 
      By dint of the above, p 1 , p 2 , and p 3  are uniformly distributed between their own maximum values and minimum values.  
      The following operations are performed in a searching step. The search part  15  reads an individual to which parameter values have been set by the parameter setting part  12 , from the parameter storing part  22 , and searches for an optimum solution by a genetic algorithm.  
      With reference to  FIG. 5 , the genetic algorithm will be explained as a method of searching for the optimum solution by the search part  15 .  
      When each parameter value is represented by a 10-bit sequence composed 1 or 0, in the case of four parameters, the gene becomes a sequence expressed by 1 or 0 of forty bits.  
      With respect to one gene, a value of discrepancy between a value calculated by a model function by using a parameter set representing the gene and measured data is arranged to be corresponding to the gene. This discrepancy value is called a “fitness value.” 
      The search part  15  selects two genes A and B by a selection method in which a gene with a desirable fitness value, i.e. a smaller fitness value, can be easily selected at random than other genes. Each of Genes A and B is called a “parent.” 
      With respect to each 40-bit sequence composed of 1 or 0, the search part  15  crosses the two genes at the place corresponding to a number equal to or less than 40 selected at random in the sequence of 40 bits in order to create two new genes C and D composed of 1 or 0. Each of Genes C and D is called an “offspring.” 
      For example, Gene C has the same bit order from the beginning to the 13th as that of Gene A, the same bit order from the 14th to the 37th as that of Gene B, and the same bit order from the 38th to the end as that of Gene A. Gene D has the same bit order from the beginning to the 13th as that of Gene B, the same bit order from the 14th to the 37th as that of Gene A, and the same bit order from the 38th to the end as that of Gene B. This operation of the search part  15  is called “crossover.” 
      Next, the search part  15  reverses the values of 1 and 0 of Genes C and D at a predetermined rate, for example 5%, and calculates fitness values of Genes C and D respectively. This operation of the search part  15  is called “mutation.” In this example, the above-mentioned operation of the search part  15  is repeated fifty times to create one hundred new genes. When the number of new genes becomes the same as the number of the genes created first, only a gene with the smallest fitness value in the genes created first is left and other ninety-nine in the genes created first are replaced with ninety-nine new created genes except for a gene with the greatest fitness value in the new created genes.  
      The search part  15  repeats the above-mentioned operation. As the above-mentioned operation is repeated, the value of a parameter of a gene with the minimum fitness value in the one hundred genes approaches the minimum value (the optimum solution) gradually. By repeating selection of a parent, crossover, and mutation at the searching step after an initial gene having been generated at the initial value setting step, the search based on information about a parameter value can be effectively forwarded.  
      In Embodiment 1, as mentioned above, the solution search apparatus  100  has the function of reading a model function specified by the user, information about the number of a plurality of parameters and their search ranges, a target measured data file to be fitted, and information about a file for outputting a calculation result, the function of performing calculation processing to search for a minimum value by using a genetic algorithm, and the function of writing a search result of a parameter into a specified file.  
      Furthermore, the solution search apparatus  100  has the function of reading a search range of a parameter specified by a user and estimated values of the parameter specified by the user, and the function of setting a parameter so that logarithmic distribution of the parameter may be uniformly distributed in the search range of the parameter in the initial setting operation of the parameter, based on the specification contents of specifying the initial value.  
      Effects acquired according to the present Embodiment will be described below.  FIG. 6  shows a result of setting initial values of a parameter so that logarithmic values of the parameter may be uniformly distributed by the method according to the present Embodiment, and of performing fitting by GA using two hundred individuals for the given measured data (marked ◯), with assuming that Lorentz function having five model parameters is a model function and the range of one of the five parameters is between 10 and 100000.  
       FIG. 7  shows a result of setting an initial value by a conventional method which uses random numbers making all the parameters be uniformly distributed. Comparing  FIG. 6  and  FIG. 7 , it turns out that data is more consistent in the present Embodiment.  
       FIG. 8  shows fitting errors in repeating extraction ten times with changing a random number series for the above-mentioned case. Relating to a random number generated inside a computer, an integer being a “seed” is given first, and a number obtained by performing calculation for the integer is used as a random number. “To change a random number series” herein means to change a value of the seed given first. If a random number series is changed, a result will minutely change in the case of GA. Therefore, by performing GA several times with changing a random number series, the stability of the present Embodiment (calculation program) can be verified. That is, in the present Embodiment, it can be confirmed difference between generated random numbers and difference between random number series do not have a large influence on the accuracy of the fitting errors.  
      The first column shows a trial number. The second column shows a fitting error of a result of the conventional method in which initial values of a parameter of each individual are set by random numbers all uniformly distributed and of the calculation performed by using the initial values having been set. The third column shows a fitting error of a result which is given by the method according to the present Embodiment, i.e. setting an initial value so that logarithmic values of parameters having wide search ranges may be uniformly distributed.  
      Since random numbers are used in calculation of GA, calculation results generally differ slightly each time. Thus, it is important to try repeatedly in order to confirm the stability of the present Embodiment (calculation program).  
      When comparing the stability of the conventional method and the present Embodiment, as shown in the graph of  FIG. 7  and the values of the second column of  FIG. 8 , the fitting errors according to the conventional method are large, and as shown in the graph of  FIG. 6  and the values of the third column of  FIG. 8 , the fitting errors according to the method of the present Embodiment are smaller than those of the conventional method in almost all the cases.  
     Embodiment 2  
      A different point of Embodiment 2 from Embodiment 1 will be explained with reference to  FIG. 3 . The solution search apparatus  100  according to Embodiment 2 has the following features: the parameter setting part  12  refers to a search range of a parameter stored in the parameter search range storing part  21 , generates parameter values so that distribution of logarithmic values of the parameter values for a part of the search range of the parameter may become denser compared with distribution of logarithmic values of the parameter values for other part of the search range, and stores the parameter to which the generated parameter values are set, in the parameter storing part  22 .  
      The normal distribution calculation part  14  generates parameter values so that distribution of logarithmic values of the parameter values for a part of the search range of the parameter may become a normal distribution. Operations of the initial value setting step of the parameter setting part  12  of the solution search apparatus  100  according to Embodiment 2 will be explained.  
      It is also assumed that the value of the parameter p 0  can be with several digits, and an estimated value p 0 ′ of the parameter p 0  is to be given. The parameter search range input part  11  inputs the estimated value p 0 ′, the maximum value p 0 max, and the minimum value p 0 min of the parameter p 0  from the input apparatus, as information on an initial value, and stores them in the parameter search range storing part  21 .  
      When the estimated value p 0 ′, the maximum value p 0 max, and the minimum value p 0 min of the parameter p 0  are given, with respect to genes of 30% of all the genes for example, the normal distribution calculation part  14  distributes initial values of the parameter p 0  so that they may be distributed as Gaussian distribution, whose average value is p 0 ′ and the width of the Gaussian distribution is (p 0 max−p 0 min)/5, on the logarithmic axis. With respect to remaining 70% of individuals, the uniform distribution calculation part  13  sets the value of p 0  so that logarithmic values of the p 0  may be uniformly distributed, by the method according to Embodiment 1. In this way, the given information on the initial value can be effectively used.  
       FIG. 9  shows a logarithmic distribution of the initial value of the parameter p 0  according to the initial value setting step of the parameter setting part  12  of Embodiment 2. The horizontal axis showing a logarithmic scale indicates the logarithmic value of p 0 , namely log (p 0 ). The vertical axis indicates generation probability of a parameter.  
       FIG. 9  shows the case, as an example, the uniform distribution calculation part  13  generates seventy parameter values so that logarithmic values may be uniformly distributed, and the normal distribution calculation part  14  generates thirty parameter values so that they may be distributed as the Gaussian distribution, whose average value is p 0 ,′ on the logarithmic axis. As shown in  FIG. 9 , by estimating the value of solution of p 0  to be p 0 ′ and adding parameter values distributed as the Gaussian distribution whose average value is p 0 ′ to parameter values uniformly distributed, probability of generating the parameter values adjacent to the estimated value p 0 ′ increases. If the parameter values generated by the above-mentioned Gaussian distribution becomes equal to or less than p 0 min or equal to or greater than p 0 max, by discarding the value and repeating the setup operation again, all the initial values can certainly be within the search range.  FIG. 9  shows the case in which the portion where the parameter values on the Gaussian distribution become equal to or less than p 0 min is not used.  
      Relating to the rate of a parameter value to be generated by the Gaussian distribution, the reason for 30% being desirable will be explained.  
      An estimated value set up by the user is sometimes very much different from an actual solution. If parameter values are spread only near an estimated value of the parameter concerned for all individuals, because of the reason mentioned above, the probability of searching up to a far search range becomes low. Therefore, the likelihood of search failure becomes high. Then, some are arranged to thoroughly search near the estimated value, and others are arranged throughout the search range as many places as possible so as to prepare for failure of the estimated value. If the estimated value is proper, it becomes short to search for a solution, and even if the estimated value is not proper, since the whole parameter values are covered, the solution search will be hard to fail. This is the reason for setting the rate 30%. However, 30% has no reason in strict meaning, and is an experiential numerical value.  
      Moreover, in order to “net” densely near the estimated value, it is desirable to use the Gaussian distribution, in which the parameter generation rate becomes higher in proportion to coming near the estimated value.  
      Next, the reason for (p 0 max−p 0 min)/5 being desirable as a width of the Gaussian distribution will be explained. If random numbers (random numbers with a narrow distribution width) whose parameter values are too close to the initial value are generated, compensation will be large when failing in estimating the estimated value. Therefore, being approximately ⅕ (one-fifth) of the whole is considered to be a range for thoroughly searching near the initial value.  
      In addition, 30% and ⅕ are mutually related in the following meaning. For example, with respect to 30% ( 3/10) of the individuals, if initial values of a parameter are generated in the range of 30% ( 3/10) of the whole search range of the parameter, it becomes almost the same case as initial values uniformly generated in the whole search range. That is, if the number of parameter values to be generated and the range of parameter values to be generated are set to be the same, it might be meaningless. This is the reason for setting the number having a different rate from the range. Then, it is set that the rate of the number is 3/10, the rate of the range is ⅕= 2/10, and the rate of the number&gt;the rate of the range. That is, many numbers are generated in a narrow range, and the number and the range have correlation in this sense.  
      As mentioned above, the initial value setting operation of Embodiment 2 is characterized by the function of distributing logarithmic values of a target parameter to be searched, to be dense near a specified estimated value based on the specification of a user, and distributing logarithmic values almost uniformly at the other part of the range.  
      According to the present Embodiment, by repeating selection of a parent, crossover, and mutation at the searching step after an initial gene having been generated at the initial value setting step, the search based on information known beforehand about a parameter value can be effectively forwarded.  
      In this example, the case has been described that the uniform distribution calculation part  13  generates seventy parameter values so that logarithmic values may be uniformly distributed, and the normal distribution calculation part  14  generates thirty parameter values so that they may be distributed as the Gaussian distribution, whose average value is p 0 ′, on the logarithmic axis. However, it is also acceptable that the uniform distribution calculation part  13  do nothing and only the normal distribution calculation part  14  generates all the one hundred parameter values so that they may be distributed as the Gaussian distribution, whose average value is p 0  , on the logarithmic axis.  
      Moreover, it is also acceptable that the parameter setting part  12  generates another distribution other than the uniform distribution and the normal distribution. It is also acceptable the search method of an optimum parameter solution according to the function of the solution search apparatus  100  is realized by a program executed by a computer. Furthermore, it is also acceptable to use not a genetic algorithm but other solution search algorithm.  
      Having thus described several particular embodiments of the present invention, various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be part of this disclosure, and are intended to be within the spirit and scope of the present invention. Accordingly, the foregoing description is by way of example only, and is not intended to be limiting. The present invention is limited only as defined in the following claims and the equivalents thereto.