Patent Publication Number: US-11656787-B2

Title: Calculation system, information processing device, and optimum solution search process method

Description:
TECHNICAL FIELD 
     The present invention relates to an information processing device, a calculation device, an information processing method, and the like, and relates to a technique for executing an optimum solution search process. 
     BACKGROUND ART 
     PTL 1 discloses “a semiconductor device including a plurality of unit units that include a first memory cell that stores a value representing one spin of the Ising model in three or more states, a second memory cell that stores an interaction coefficient indicating an interaction from another spin that interacts with one spin, and a logic circuit that determines the next state of one spin based on a value that represents the state of another spin and a function that has the interaction coefficient of as a constant or variable.” 
     PTL 2 discloses a method for realizing an optimum solution search by stochastically updating all spins at the same time while satisfying the theoretical background required by the Markov Chain Monte Carlo method for an Ising model having an arbitrary coupling. 
     CITATION LIST 
     Patent Literature 
     PTL 1: JP-A-2016-51314 
     PTL 2: WO-A-2019/216277 
     Non-Patent Literature 
     NPL 1: Okuyama, T., Sonobe, T., Kawarabayashi, K. I., &amp; Yamaoka, M. (2019). Binary optimization by momentum annealing. Physical Review E, 100(1), 012111 
     NPL 2: Botev, Z. I. (2017). The normal law under linear restrictions: simulation and estimation via minimax tilting. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(1), 125-148 
     NPL 3: Neal, R. M. (1998). Suppressing random walks in Markov chain Monte Carlo using ordered overrelaxation. In Learning in graphical models (pp. 205-228). Springer, Dordrecht 
     SUMMARY OF INVENTION 
     Technical Problem 
     Many physical and social phenomena can be expressed by interaction models. An interaction model is defined by a plurality of nodes constituting the model, interactions between the nodes, and further, coefficients that act on each node, if necessary. In the fields of physics and social science, various models including the Ising model have been proposed, but all of them can be interpreted as one form of interaction models. 
     It is important to obtain a node state that minimizes or maximizes an index associated with this interaction model in solving social issues. Examples thereof include the problem of detecting creeks in social networks and the problem of portfolio optimization in the financial field. In the field of operations research, these are roughly divided into unconstrained binary quadratic programming problems and mixed binary quadratic programming problems. 
     The present invention has been made in view of the above background and an object thereof is to provide a technique capable of executing the optimum solution search for a mixed binary quadratic programming problem, including a ground state search for the Ising model at a high speed. 
     Solution to Problem 
     A preferred aspect of the present invention is a calculation system including a variable memory that stores a value indicating a state of a variable of a mixed integer quadratic programming problem; a state transition calculation block that calculates the next state of the value indicating the state of the variable; a nonlinear coefficient memory that stores a nonlinear coefficient of the state transition calculation block; a linear coefficient memory that stores a linear coefficient of the state transition calculation block; a weight input line that receives a weight signal of the state transition calculation block; and a temperature input line that receives a temperature signal of the state transition calculation block. The state transition calculation block includes a difference calculation block that calculates difference calculation by using the weight signal, the nonlinear coefficient, and the linear coefficient, a sampling block that performs random sampling from a probability distribution with an interval constraint by using the weight signal, the temperature signal, and an output value of the difference calculation block, and a next state determination block that calculates the next state of the variable by using the value read from the variable memory. 
     In another preferred aspect, the variable memory stores continuous values as values x 1 , . . . , x N  and y 1 , . . . , y N  indicating the state of the variable. 
     Another preferred aspect of the present invention is an information processing device including the calculation system; and a computer that controls the calculation system. The information processing device includes a storage unit and a variable value reading unit, in which the storage unit stores the domain of the variable of the mixed integer quadratic programming problem, and the variable value reading unit reads a value from the variable memory and transforms at least a part of the continuous value to a binary value based on the domain of the variable. 
     Advantageous Effects of Invention 
     According to the present invention, it is possible to solve an optimization problem referred to as a mixed binary quadratic programming problem at a high speed. Objects, configurations, and effects other than those described above will be clarified by the following description of embodiments for carrying out the invention. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG.  1    is a conceptual diagram illustrating a relationship between a variable array and an objective function value of an optimization problem. 
         FIG.  2    is a diagram providing a description of an embodiment. 
         FIG.  3    is a diagram providing a description of the embodiment. 
         FIG.  4    is a block diagram illustrating a schematic configuration of an information processing device. 
         FIG.  5    is a block diagram illustrating a calculation system. 
         FIG.  6    is a function block diagram illustrating a main function included in the information processing device. 
         FIG.  7    is a flowchart illustrating an optimum solution search process. 
         FIG.  8    is a detailed block diagram of the calculation system. 
         FIG.  9    is a block diagram of a unit constituting the calculation system. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Hereinafter, embodiments will be described in detail based on the drawings. In the following description, the same or similar configurations are denoted by common reference numerals and the duplicated descriptions maybe omitted. When there are a plurality of elements having the same or similar functions, the description may be made with the same reference numerals having different subscripts. If it is not necessary to distinguish between a plurality of elements, the subscripts may be omitted for the explanation. 
     Notations such as “first”, “second”, and “third” in the present specification are provided to identify the components and do not necessarily limit the number, order, or contents thereof. In addition, numbers for identifying components are used for each context, and numbers used in one context do not always indicate the same configuration in other contexts. Further, it does not prevent the component identified by a certain number from having the function of the component identified by another number. 
     One embodiment described below is a calculation system including a variable memory that stores a value indicating a state of a variable in a mixed integer quadratic programming problem, a nonlinear coefficient memory that stores nonlinear coefficient of a state transition calculation block corresponding to the variable memory, a linear coefficient memory that stores a linear coefficient of a state transition calculation block corresponding to the variable memory, a weight input line that receives a weight signal of the state transition calculation block, a temperature input line that receives a temperature signal of the state transition calculation block, a difference calculation block that calculates difference calculation by using the weight signal of the state transition calculation block, the nonlinear coefficient of the state transition calculation block, and the linear coefficient of the state transition calculation block, a sampling block that performs random sampling from a probability distribution with an interval constraint by using the weight signal of the state transition calculation block, the temperature signal of the state transition calculation block, and the output value of the difference calculation block, and a next state calculation block that calculates the next state of the variable by using the output value of the sampling block and the value read from the variable memory. 
     Generally, an integer programming problem refers to an optimization problem that includes integer variables. A case where variables that take integer values and variables that take real numbers are mixed is referred to as a mixed integer programming problem. A mixed integer programming problem that is a quadratic programming problem is referred to as a mixed integer quadratic programming problem. In the present specification, particularly, a mixed integer quadratic programming problem in which variables that take binary values and variables that take real values are mixed is referred to as a mixed binary quadratic programming problem. First, the significance of the mixed binary quadratic programming problem is explained. 
     Depending on the optimization problem desired to solve, binary variables and continuous variables may be mixed. For example, with respect to the problems in the financial field, the purchase ratio of financial products may be 0%, or 10% to 100%. If the product is not purchased, the purchase ratio will be, of course, 0%, and if the product is purchased, the purchase ratio will be 10% or more of the minimum unit. At this time, by using the binary variable x ∈ {−1,1} and the continuous variable y ∈ [−1,1], which indicate whether to purchase or not, the purchase ratio r can be expressed as:
 
 r= {(1+ x )/2}×{0.1+0.9×(1+ y ))/2}.
 
     The continuous variable y can be discretely expressed with a plurality of binary variables, but by making it possible to handle continuous variables, the number of variables is only one. Therefore, by allowing the computer system to handle continuous variables, the number of variables in the optimization problem can be reduced and the scale of the problem that can be handled by the computer resources can be increased. Moreover, when solving a certain problem, a shortening of the calculation time can be expected because the number of variables is reduced. 
     Meanwhile, the problem can be handled only with continuous variables, but with continuous variables, values such as 0.3 are allowed even for variables for which only −1 or +1 is desired to be accepted as a value. In this case, if a constraint of “the variable x is −1 or +1” is added, for example, to the objective function as a penalty function (x 2 −1) 2 , the variable x can be handled as a continuous variable but a quadratic expression cannot be obtained. In addition, there is a problem that the objective function becomes complicated and thus the optimum solution is hardly found. Therefore, in the case of forming a quadratic programming problem, if the domain of a predetermined variable is set to a binary value or a discrete value from the beginning, there is a merit of a configuration in which the problem can be handled with a computer. Hereinafter, when referred to only as an optimization problem in the present specification, it means a mixed binary quadratic programming problem. 
     There are N variables s 1  to s N  of the optimization problem (here, meaning the mixed binary quadratic programming problem). A domain D i  of each variable is either a binary value {−1, +1} or a continuous value [−1, +1]. The value of the domain is determined for each problem. Also, an objective function H of the optimization problem is expressed as Expression 1. That is, the objective function H is represented by a quadratic expression of the variable s. 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
               ⁢ 
               1 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
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                       ⁢ 
                       
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     In Expression 1, s=N-dimensional vector of [s 1 , . . . , s N ], J is an N×N symmetric matrix, and h is an N-dimensional vector. As described above, the domain differs for each variable, and thus the mixed binary quadratic programming problem can be expressed as Expression 2. 
     
       
         
           
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     Here, sets of subscripts Λ b  and Λ c  are defined as in Expression 3. 
     
       
         
           
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     The set S mixed ={s|s i  ∈ D i } is defined. If these notations are used, Expression 2 can also be expressed as Expression 4. 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
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     Hereinafter, for all i ∈ Λ b , the element of the i-th row and i-th column of a matrix J is set to 0. This is because this transformation does not change the optimal solution of Expression 2. 
     If D i ={−1, +1} for all i, this optimization problem is a combinatorial optimization problem referred to as the ground state search problem of the Ising model. In the present embodiment, in the optimization problem including the search for the ground state of the Ising model, an optimum solution or an approximate solution is searched by an algorithm using the Markov Chain Monte Carlo method (hereinafter, referred to as MCMC). 
       FIG.  1    is a conceptual diagram illustrating a landscape of objective function values for a variable array. The horizontal axis of the graph is the variable array s, and the vertical axis is the objective function H(s). MCMC repeats a stochastic transition from a current state s to a certain state s′ near the state s. The probability of transition from the state s to the state s′ is referred to as transition probability P (s, s′). Examples of the transition probability P include the Metropolis method and the heat-bath algorithm. 
     The transition probability has a parameter referred to as temperature, which indicates the ease of transition between states. When MCMC is executed while gradually decreasing the temperature from a large value, it asymptotically converges to the state in which the objective function value is the lowest. A method of obtaining the optimum solution or the approximate solution of the minimization problem by utilizing the above is Simulated Annealing (hereinafter, referred to as SA) or Momentum Annealing (hereinafter, referred to as MA) proposed in NPL 1. 
     In solving the minimization problem presented in Expression 4, solving a minimization problem of Expression 5 is considered instead. Here, the set S relaxed ={s|s i  ∈ [−1, +1]}. 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
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     The optimum solution of Expression 5 is indicated as s*=[s 1 *, . . . , s N *]. Although the proof is omitted, s + =[s 1   + , . . . , s N   + ] obtained by following Expression 6 is one of the optimum solutions of Expression 4. The goal of the examples shown in the present application is to search for the optimum solution of Expression 2. However, even if the transformation of Expression 6 is obtained after solving the optimum solution s* of Expression 5, the desired solution s +  can be obtained. Here, the function sgn is a function that returns +1 if an argument is 0 or more and returns −1 otherwise. 
     
       
         
           
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               Expression 
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     Here, an N-dimensional vector v=[v 1 , . . . , v N ] is introduced to define a function H′ presented in Expression 7.
 
[Expression 7]
 
 H′ ( s, v )= H ( s )+ V ( v )   (7)
 
     Here, the function V(v) is as defined in Expression 8. 
     
       
         
           
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     The matrix W=diag(w 1 , . . . , w N ) is any diagonal matrix, and v i  is a real number that moves [−1, +1]. Instead of the minimization problem of Expression 5, Expression 9 that is a minimization problem of H′ (s, v) is introduced. 
     
       
         
           
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               Expression 
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     Two N-dimensional vectors x=s+v and y=s−v are defined. The objective function of the optimization problem originally desired to solve is only H, but by introducing a function referred to as V here, a new function that can be updated in parallel by MCMC can be obtained. Then, the function H′ can be rewritten as Expression 10. 
     
       
         
           
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     That is, the minimization problem of Expression 5 can be rephrased as a minimization problem of Expression 11. 
     
       
         
           
             
                 
             
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     If the optimum solution of Expression 11 is expressed with x* and y*, the equation of s*=(x*+y*)/2 is established. These arguments are established even if W is a zero matrix. 
     From the above, the optimum solution of the mixed binary quadratic programming problem expressed as Expression 2 can be obtained from the solution of the constrained quadratic programming problem presented in Expression 11. In order to obtain the solution, MCMC is used. 
       FIG.  2    is a graphical model indicating the relationship between the variables of the function G presented in Expression 11. The relationship between the variables of the function G can be represented by a complete bipartite graph. The only variables that can be multiplied by the variable x i  in the function G are y 1 , . . . , y N  and x i . When stochastically updating a variable value, MCMC uses a value of a variable related to that variable. That is, when the value of the variable x 1  is updated, y 1 , . . . , y N , and x 1  are obtained and other variables (here, x 2 , . . . , x N ) are not referred to. The same is also applied to the update of the values of other variables, such as x 2 . Therefore, if the value of the variable array y is constant, the theoretical requirement of MCMC is not violated even if each value of the array x is stochastically updated independently at the same time. 
     Similarly, variables that can be multiplied by the variable y i  are x 1 , . . . , x N  and y i , only. Therefore, while the values of the variable arrays x are constant, each value of the array y can be stochastically updated independently and simultaneously. 
     From the above, by executing MCMC including the procedure of repeating “simultaneous update of x 1 , . . . , x N ” and “simultaneous update of y 1 , . . . , y N ”, while enjoying the advantage of speeding up by parallelization, the arrays x and y that minimize the function G can be searched. 
     Note that in the discussion of this example, there are no constraints on the matrix J. For example, even if all the elements of the matrix J are non-zero, the above argument is established, and thus, a parallel update can be performed. 
       FIG.  3    is an example of a fully connected graph. On the other hand, when MCMC is applied directly to the minimization problem of Expression 2, which is the original problem, since those relating to the variable arrays are expressed by a fully connected graph as illustrated in  FIG.  3   , the probability can be updated for only one variable at a time, and the update is limited to sequential update. 
     From here, the procedure for the stochastic update for each variable will be described. The variable to be updated is set to x i . When the values of the variables y 1  to y N  are constant, the existence probability p(x i ) of the variable x i  in the Boltzmann distribution at the temperature T satisfies Expression 12. 
     
       
         
           
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               12 
             
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     Here, the variable A i  is a value obtained by Expression 13. 
     
       
         
           
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               Expression 
               ⁢ 
               
                   
               
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                   ( 
                   13 
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     Since the variables x i  and y i  are |x i |+|y i |≤2, the range in which x i  can move is −(2−|y i |) or more and (2−|y i |) or less. Therefore, with respect to the variable x i , the next state of x i  only needs to be sampled based on a truncated normal distribution whose domain is −(2−|y i |) or more and (2−|y i |) or less in a normal distribution with the mean of A i /w i  and the variance of T/w i . In this method, the next state is determined regardless of the current state of x i . The same is applied to y i . In the present specification, if the variables of x and y are not distinguished, the variables may be expressed as s. 
     Random numbers that follow the standard normal distribution can be generated by the Box-Muller method. Since the domain is limited here, the algorithm disclosed in NPL 2 may be used. 
     The optimum solution search can be regarded as sampling from the equilibrium state at temperature 0. Therefore, in order to realize a high-quality solution search, it is preferable to converge to an equilibrium state in a short time. In order to improve the convergence to the equilibrium state, various techniques have been proposed by MCMC and these techniques can also be utilized. For example, in NPL 3, an over-relaxation method is proposed. As a candidate for the next state, not only one state but also K states are sampled from the Boltzmann distribution at temperature T. Then, in addition to the sampled K states, the total (K+1) states of the current state are rearranged and expressed as x c   0 ≤x c   r =x i ≤x c   K . That is, the current state is the (r+1)-th from the smallest of the (K+1) values. Then, x c   K+1−r  is employed for the next state. In this method, the next state depends on the current state of x i . 
     Based on the above,  FIGS.  4  to  6    illustrate the configuration of the information processing device that realizes the present invention. 
       FIG.  4    is an example of an information processing device that searches for the optimum solution of a mixed binary quadratic programming problem. As illustrated in the drawing, an information processing device  10  includes a processor  11 , a main storage device  12 , an auxiliary storage device  13 , an input device  14 , an output device  15 , a communication device  16 , one or more calculation devices  20 , and a system bus  5  that communicably connects the devices. For example, the information processing device  10  may be realized by using a virtual information processing resource such as a cloud server of which a part or all is provided by a cloud system. Further, the information processing device  10  may be realized by, for example, a plurality of information processing devices that operate in cooperation with each other and are communicably connected to each other. 
     The processor  11  is configured, for example, by using a central processing unit (CPU) or a micro processing unit (MPU). The main storage device  12  is a device for storing programs or data, and is, for example, a read only memory (ROM), a static random access memory (SRAM), a non-volatile ram (NVRAM), a mask read only memory (mask ROM), a programmable ROM (PROM), a random access memory (RAM), a dynamic random access memory (DRAM), and the like), and the like. The auxiliary storage device  13  is a hard disk drive, a flash memory, a solid state drive (SSD), and an optical storage device (such as a compact disc (CD), a digital versatile disc (DVD)). The programs and data stored in the auxiliary storage device  13  are read into the main storage device  12  at any time. 
     The input device  14  is a user interface that receives input of information from the user, and is, for example, a keyboard, a mouse, a card reader, or a touch panel. The output device  15  is a user interface that provides information to the user, and is, for example, a display device (such as a liquid crystal display (LCD) and a graphic card) that visualizes various kinds of information, an audio output device (speaker), and a printing device. The communication device  16  is a communication interface that communicates with other devices and is, for example, a Network Interface Card (NIC), a wireless communication module, a universal serial interface (USB) module, and a serial communication module. 
     The calculation device  20  is a device that executes a process related to the optimum solution search of the mixed binary quadratic programming problem. The calculation device  20  may take the form of an expansion card to be mounted on the information processing unit  10 , such as a graphics processing unit (GPU). The calculation device  20  is configured with hardware such as a complementary metal oxide semiconductor (CMOS) circuit, a field programmable gate array (FPGA), and an application specific integrated circuit (ASIC). The calculation device  20  includes a control device, a storage device, an interface for connecting to the system bus  5 , and transmits and receives commands and information to and from the processor  11  via the system bus  5 . The calculation device  20  may be, for example, one that is communicably connected to another calculation device  20  via a communication line and operates in cooperation with the other calculation device  20 . The function realized by the calculation device  20  may be realized, for example, by causing a processor (such as CPU and GPU) to execute a program. 
     The calculation device  20  illustrated in  FIG.  4    is described below in  FIG.  5   . One or a plurality of calculation devices  20  can be mounted. 
       FIG.  5    is a diagram for illustrating an operating principle of the calculation device  20  and is a block diagram of a system (hereinafter, referred to as a calculation system  500 ) that configures the calculation device  20 . The calculation system  500  realizes a function of sampling the variable array x 1 , . . . , x N  or the variable array y 1 , . . . , y N  from the Boltzmann distribution (Expression 12) at the temperature T. Hereinafter, an operating principle of the calculation device  20  is described with reference to the same drawing. 
     As illustrated in the drawing, the calculation system  500  includes a variable memory  511 , a nonlinear coefficient memory  512 , a linear coefficient memory  513 , a difference calculation block  514 , a sampling block  515 , and a next state determination block  516 . 
     Information indicating the variables x 1 , . . . , x N  and y 1 , . . . , y N  described above is stored in the variable memory  511  of each of the calculation system  500  (see  FIG.  2   ). 
     The information indicating the matrix J is stored in the nonlinear coefficient memory  512 . The matrix J is generally a symmetric matrix, and the usage amount of the nonlinear coefficient memory  512  can be reduced by using this symmetry. Information indicating the vector h is stored in the linear coefficient memory  513 . 
     As illustrated in the drawing, a control signal EN, a weight signal SW, and a temperature signal TE are input to the calculation system  500 . 
     The signal EN indicates which of the variable arrays x and y is updated, with a signal that periodically repeats the values of high (H) and low (L). For example, when EN is H, it is determined that the variable array x i  s updated, and when EN is L, it is determined that the variable array y is updated. According to the signal EN, the variables x 1 , . . . , x N  are simultaneously updated, and the variables y 1 , . . . , y N  are simultaneously updated. In  FIG.  5   , the signal EN is input only to the sampling block  515  for simplification, but the same is applied to other places, such as the variable memory that require this signal. 
     The signal SW is a signal indicating a vector of N elements representing the diagonal components of the diagonal matrix W. 
     The value of the matrix J stored in the nonlinear coefficient memory  512 , the vector h stored in the linear coefficient memory  513 , the signal SW, and the variable s (x or y) stored in the variable memory  511  are input to the difference calculation block  514 . The difference calculation block  514  outputs (J+diag(w 1 , . . . , w N )) y+h when the signal EN is H, and outputs (J+diag(w 1 , . . . , w N )) x+h when the signal EN is L. This output value corresponds to the above-mentioned A i . 
     The sampling block  515  receives the output of the difference calculation block  514 , the signal SW, a signal TW that stores a value of the temperature parameter, the signal EN, and values of other variables. And the i-th element is randomly sampled and output from the truncated normal distribution represented by Expression 12 with −(2−|y i |) or more and (2−|y i |) or less as the domain when the signal EN is H and −(2−|x i |) or more and (2−|x i |) or less as the domain when the signal EN is L. 
     The next state determination block  516  determines the next state of the variable based on one or more values output from the sampling block  515 . If the MCMC update rule is defined as a simple heat-bath algorithm, the next state determination block  516  may receive only one output value of the sampling block  515  and write the received output value as it is to the variable memory  511 . If a well-known over-relaxation method is used as the MCMC update rule, the next state determination block  516  receives a plurality of values from the sampling block  515  and the current value of the variable to be updated from the variable memory  511 , selects one according to the over-relaxation method, and writes the value on the variable memory  511 . As is well known, in the over-relaxation method, the next state is determined so that the correlation with the immediately preceding state is negative. 
       FIG.  6    illustrates the main functions (software configurations) included in the information processing device  10 . As shown in the drawing, the information processing device  10  includes a storage unit  600 , the model conversion unit  611 , a model coefficient setting unit  612 , a weight setting unit  613 , a variable value initialization unit  614 , a temperature setting unit  615 , an interaction calculation execution unit  616 , and a variable value reading unit  617 . These functions are realized by the processor  11  reading and executing the program stored in the main storage device  12  or by the hardware included in the calculation device  20 . The information processing device  10  may have other functions such as an operating system, a file system, a device driver, and a database management system (DBMS) in addition to the above functions. 
     Among the above functions, the storage unit  600  stores problem data  601 , quadratic programming form problem data  602 , domain data  603 , and a calculation device control program  604  in the main storage device  12  or the auxiliary storage device  13 . The problem data  601  is data, for example, in which an optimization problem or the like is described in a known predetermined description format. The problem data  601  is set by the user, for example, via a user interface (input device, output device, communication device, and the like). 
     The quadratic programming form problem data  602  is data generated by the model transformation unit  611  transforming the problem data  601  into data in a format that matches a format of the quadratic programming problem presented by Expression 4. According to the transformation, the domain of each given variable is written in the domain data  603 . The domain data indicates, for example, whether each variable takes a binary value or a real value. The calculation device control program  604  is a program that is executed when the interaction calculation execution unit  616  controls the calculation device  20  or is loaded by the interaction calculation execution unit  616  on each of the calculation devices  20  and executed on the calculation devices  20 . 
     The model transformation unit  611  transforms the problem data  601  into the quadratic programming format problem data  602 , which is the format of the quadratic programming problem. Therefore, the function of deriving Expression 11 from Expression 1 may be implemented on the model transformation unit  611  as software or hardware. The function of the model transformation unit  611  may not be necessarily implemented on the information processing device  10 , or the information processing device  10  may obtain the quadratic programming format problem data  602  generated by another information processing device or the like via the input device  14  or the communication device  16 . 
     The model coefficient setting unit  612  sets the matrix J of Expression 4 in the nonlinear coefficient memory  512  and sets the vector h in the linear coefficient memory  513  based on the quadratic programming format problem data  602 . 
     The variable value initialization unit  614  initializes the value of each variable stored in the variable memory  511  of the calculation unit  20 . The variable value initialization unit  614  only needs to determine the value of each variable by random sampling uniformly from −1 or more and +1 or less. At this time, care must be taken to satisfy |x i |+|y i |≤2 which is the constraint related to the variable. Also, note that the value of each variable at this time is treated as continuous value. 
     The temperature setting unit  615  sets the temperature T used when the interaction calculation execution unit  616  searches for the optimum solution. 
     The interaction calculation execution unit  616  causes the calculation devices  20  to execute calculation of searching the variable arrays x and y (hereinafter, referred to as interaction calculation) that minimizes the function G represented by Expression 11 for each temperature T set by the temperature setting unit  615 . In the interaction calculation, the interaction calculation execution unit  616  changes, for example, the temperature T from the higher side to the lower side. 
     If the optimum solution search by the interaction calculation execution unit  616  is ended, the variable value reading unit  617  reads the variable arrays x and y stored in the variable memory  511 . Here, the read value is the solution of Expression 11. According to the above discussion, the N-dimensional vector s*=(x+y)/2 is calculated. Then, the domain data  603  is read, a vector s +  obtained by Expression  6  is output to the output device  15  and the communication device  16  as the final solution. That is, sgn(s* i ) is output if the i-th domain is found to be {−1, +1} in the domain data  603 , and s i  itself is output if the i-th domain is [−1, +1]. In this way, a solution according to the defined range is obtained. 
       FIG.  7    is a flowchart illustrating the process performed by the information processing device  10  during the optimum solution search (hereinafter, referred to as an optimum solution search process S 700 ). Hereinafter, the optimum solution search process S 700  is described with reference to the same drawing. In the following, the letter “S” attached before the reference numeral means a processing step. The optimum solution search process S 700  is started by receiving an instruction from the user or the like via the input device  14 , for example. 
     As illustrated in the figure, the model transformation unit  611  first transforms the problem data  601  into the quadratic programming format problem data  602  (S 711 ). In the quadratic programming format problem data, for example, the matrix J and the vector h in the function H expressed by Expression 1 are expressed in an arbitrary format. If the storage unit  600  has already stored the quadratic programming format problem data  602 , the process S 711  is omitted. The process of S 711  and the process of S 712  and the subsequent processes may be executed by different devices, respectively. Further, the process of S 711  and the process of S 712  and the subsequent processes may be executed at different timings (for example, it is conceivable that the process of S 711  is performed in advance). 
     Subsequently, the model coefficient setting unit  612  sets the values of the matrix J and the vector h to the nonlinear coefficient memory  512  and the linear coefficient memory  513  (S 712 ). The value of the memory can also be set or edited by the user via a user interface (realized, for example, by the input device  14 , the output device  15 , and the communication device  16 ). 
     Subsequently, the weight setting unit  613  determines the value of the signal SW. As described in Expression 8 above, the signal SW is allowed to take an arbitrary value in searching for the optimum solution. Therefore, the signal value may always be 0. In this case, the calculation load can be reduced. In addition, as disclosed in Expressions 3 to 5 of PTL 2, the value may be determined from the eigenvalues of the matrix J. Otherwise, the value may be determined from the sum of rows of the matrix J. The calculation of the value calculation of the signal SW may be executed in the calculation device  20  or in the processor  11 . Otherwise, the value may be set by the user himself/herself (S 713 ). 
     Subsequently, the variable value initialization unit  614  initializes the value of each variable stored in the variable memory  511  (S 714 ). The value stored in the variable memory  511  is a continuous value. As described above, the initial value may be random. Hereinafter, the parameter expressing Expression 11 is set. 
     Subsequently, the temperature setting unit  615  sets a series T k  (k=1, 2, 3, . . . ) of the temperature parameters used in the optimum solution search (S 715 ). The above-mentioned subscript k represents the type of temperature T to be set. As a method for setting the temperature T, for example, the method of PTL 1 can be employed. 
     Subsequently, the interaction calculation execution unit  616  executes the stochastic simultaneous update of the variable array by the calculation of the calculation system  500  illustrated in  FIG.  5    (S 716 ). 
     Subsequently, the interaction calculation execution unit  616  determines whether or not a stop condition is satisfied (for example, whether or not the temperature T has reached a preset minimum temperature) (S 717 ). If the interaction calculation execution unit  616  determines that the stop condition is satisfied (S 717 : YES), the process proceeds to S 718 . Meanwhile, if the interaction calculation execution unit  616  determines that the stop condition is not satisfied (S 717 : NO), the process returns to S 716 . 
     In S 718 , the variable value reading unit  617  reads the value of the variable stored in the variable memory  511  and the domain of each variable of the quadratic programming form problem data  602  stored in the domain data  603 . Then, the vector is calculated through the transformation based on Expression 6 and output as the solution of Expression 2 or 4. Here, the optimum solution search process S 700  is completed. 
     As described in detail above, according to the information processing device  10  of the present embodiment, the optimum solution search of the mixed binary quadratic programming problem can be efficiently performed. Therefore, the optimization problem can be solved efficiently. The information processing device  10  (including the calculation device  20 ) has a simple structure and thus can be manufactured inexpensively and easily. 
     As long as the calculation system  500  has a function of executing a calculation for solving the optimization problem described above, the calculation system  500  may be configured with software or may be configured with hardware. Specifically, in the annealing method, not only the hardware mounted by an electronic circuit (digital circuit or the like) but also the method of mounting by a superconducting circuit or the like may be used. Further, hardware that realizes the Ising model other than the annealing method may be used. For example, a laser network method (optical parametric oscillation) and a quantum neural network are known. In addition, although some ideas are different, a quantum gate method in which the calculation performed by the Ising model is replaced with a gate such as the Hadamard gate, a rotating gate, and a control NOT gate can also be employed as the configuration of the present embodiment. 
     As a specific implementation example of the calculation system  500 , an example of being implemented as a complementary metal-oxide semiconductor (CMOS) integrated circuit disclosed in PTL 1 or a logic circuit on a field programmable gate array (FPGA) will be described. 
     In the technology of PTL 1, a large number of units to which the technology of a static random access memory (SRAM) is applied are arranged, and a memory for storing variables and a circuit for updating variables are arranged in each unit . 
       FIG.  8    is a block diagram illustrating a circuit configuration example when the technology of SRAM is applied to the calculation system  500  of the present embodiment. A plurality of units  801  constitute an array unit  802 . Such a configuration can be manufactured by applying the semiconductor manufacturing technology. 
     One unit  801  includes a multi-value memory  901  that stores any one of the variables x 1 , . . . , x N  and y 1 , . . . , y N  and a configuration for updating the value of the multi-value memory  901 . That is, 2N units  801  are prepared. 
     The configuration example of  FIG.  8    will be described with reference to the generalized configuration of  FIG.  5   . The data stored in the nonlinear coefficient memory  512  and the linear coefficient memory  513  is set by the model coefficient setting unit  612 . The nonlinear coefficient memory  512  stores the N×N matrix J, which is commonly used by all units  801 . Further, the N-dimensional vector h is stored in the linear coefficient memory  513  and is commonly used by all the units  801 . In order to reduce the circuit scale, these memories are common to each unit  801 . Therefore, the nonlinear coefficient memory  512  and the linear coefficient memory  513  supply the coefficients J and h to all the units  801 . However, the signal lines for that purpose are omitted in  FIG.  8   . In principle, each unit  801  may individually include the nonlinear coefficient memory  512  and the linear coefficient memory  513 . 
     The vectors of the N elements (w 1 , . . . , w N ) representing the diagonal components of the diagonal matrix W are stored in a weight memory  803 . This data is set by the weight setting unit  613 . Since the i-th unit that stores x i  and y i  uses the i-th component w i , it is required to switch the value of the signal SW for each unit  801 . In  FIG.  8   , the signal line for supplying the signal SW to the unit  801  is omitted. 
     The temperature signal TE supplied from the temperature setting unit  615  is supplied to all the units  801 . The function and configuration of the temperature signal follow the prior art. The signal line that supplies the signal TE to the unit  801  is omitted. 
     An interaction driver  804  alternately inputs signals for allowing the update of the variable x and a signal for allowing the update of the variable y to each unit  801 . As a result, the variables x 1  to x N  are updated at the same time, and the variables y 1  to y N  are updated at the same time. 
     An SRAM interface  805  writes and reads to and from the memory that stores the variables of the units  801  generated by applying the circuit configuration of the SRAM. The variable read after the process is completed in the calculation system  500  is sent to the variable value reading unit  617 . The variable value reading unit  617  obtains a solution to the mixed binary quadratic programming problem by outputting the read variable as a continuous value or a binary value based on the domain data  603 . 
     The controller  806  initializes the calculation system  500  and reports the end of the process according to the instruction of the interaction calculation execution unit  616 . 
       FIG.  9    is a diagram illustrating a circuit configuration example of one unit  801 . The multi-value memory  901  that stores any one of the continuous variables x 1 , . . . , x N  and y 1 , . . . , y N  is included in one unit. 
     The difference calculation circuit  902  realizes the function of the difference calculation block  514 . When the variable stored in the multi-valued memory  901  is any one of x 1 , . . . , x N , the vectors of (y 1 , . . . , y N ) are input to the difference calculation circuit  902 . When the variable stored in the multi-value memory  901  is any one of y 1 , . . . , y N , the vectors of (x 1 , . . . , x N ) are input. These variable vectors are read and generated by an SRAM interface  805  from the multi-value memory  901  of another unit  801 . Further, the N×N matrix J and the N-dimensional vector h, which are coefficients, are input. In addition, the weight w i  is input. The difference calculation circuit  902  outputs the value A i  of the i-th row of (J+diag(w 1 , . . . , w N )) s+h (s is a variable vector of x or y) with respect to these inputs. 
     A sampling circuit  903  realizes the function of the sampling block  515 . The output A i , the signal EN, the signal SW, the signal TE, and y i  if the variable stored in the multi-value memory  901  is x i , or x i  if the variable stored in the multi-valued memory  901  is y i  are input to the sampling circuit  903 . Then, the candidate of the next state of the variable is sampled from the existence probability p(s i ) of the variable s i  based on Expression 12. 
     A state determination circuit  904  determines the next state of the variable based on one or a plurality of candidates output from the sampling circuit  903 . In the state determination circuit  904 , for example, when the over-relaxation method is followed, if a plurality of candidates are obtained from the sampling circuit  903 , a candidate whose correlation with the state immediately before the multi-valued memory  901  is negative is selected, and the next state is determined. The determined next state is stored in the multi-value memory  901 . 
     In the above, the difference calculation block  514 , the sampling block  515 , and the next state determination block  516  are assumed to be hardware such as FPGA. However, for example, software utilizing a large number of calculation devices mounted on a GPU, a vector type computer, or the like can be implemented. By providing a large number of units  801  in this manner, variables can be updated in parallel. 
     Although one embodiment has been described in detail above, it is obvious that the present invention is not limited to the above embodiment and can be variously modified without departing from the gist thereof. For example, the above embodiments have been described in detail in order to describe the present invention for easier understanding and are not necessarily limited to the one including all the described configurations. Further, another configuration can be added, deleted, replaced with respect to a part of the configuration of the above embodiments. 
     Further, each of the above configurations, functional units, processing units, processing means, and the like may be realized by hardware by designing a part or all of them by, for example, an integrated circuit. Further, each of the above configurations, functions, and the like may be realized by software by the processor interpreting and executing a program that realizes each function. Information such as programs, tables, and files that realize each function can be placed in a memory, a hard disk, a recording device such as a solid state drive (SSD), or a recording medium such as an IC card, an SD card, or a DVD. 
     Also, in each of the above figures, the control lines and information lines are illustrated as necessary for explanation, and not all the control lines and information lines in the implementation are necessarily illustrate. For example, in practice, almost all configurations may be considered connected to each other. 
     Further, the arrangement forms of various functional units, various processing units, and various databases of the information processing device  10  described above are only examples. The arrangement forms of the various functional units, the various processing units, and the various databases can be changed to the optimal arrangement forms from the viewpoints of the performance, processing efficiency, communication efficiency, and the like of the hardware and software included in the information processing device  10 . 
     In addition, the configuration of the database (schema and the like) that stores the various data described above can be flexibly changed from the viewpoints of efficient use of resources, improvement of processing efficiency, improvement of access efficiency, improvement of search efficiency, and the like. 
     INDUSTRIAL APPLICABILITY 
     It can be used for information processing devices, calculation devices, information processing methods, and the like. 
     REFERENCE SIGNS LIST 
       10 : information processing device 
       11 : processor 
       12 : main storage device 
       20 : calculation device 
       511 : variable memory 
       512 : nonlinear coefficient memory 
       513 : linear coefficient memory 
       514 : difference calculation block 
       515 : sampling block 
       516 : next state determination block 
       600 : storage unit 
       601 : problem data 
       602 : quadratic programming format problem data 
       603 : domain data 
       604 : calculation device control program 
       611 : model transformation unit 
       612 : model coefficient setting unit 
       613 : weight setting unit 
       614 : variable value initialization unit 
       615 : temperature setting unit 
       616 : interaction calculation execution unit 
       617 : variable value reading unit