Method, computer program, system and communication device for optimizing communication channel capacity

Abstract

The invention relates to a method for optimizing the capacity of a communication channel in a communication system, the communication system comprising at least a transmitter (10), a receiver (11), and a communication channel (12) between the transmitter and the receiver. The transmitter (10) uses a finite set of symbols Ω = {ω1, ..., ω N} to send a message including at least one symbol on the communication channel (11). The communication channel (11) is represented by a conditional probability distribution p Y|X (y|x), where y is the symbol received at the receiver (12) and x is the symbol sent by the transmitter. More specifically, for each possible transmitted symbol x, the conditional probability distribution p is obtained by using a mixture model of the probability distribution represented by an exponential function. Y|X (y|x). The optimized input distribution p is calculated based on the parameters of the mixture model x (x) to define the optimized symbol positions and probabilities to be used at the transmitter to optimize the capacity of the channel.

Description

Technical Field

[0001] The present invention relates to the field of telecommunications and, more particularly, to the problem of optimizing the capacity of a communication channel. Background Art

[0002] The optimization may be realized by computer means and more particularly for example by artificial intelligence and may be based on observations of whether messages sent from a sender via a communication channel are received well at a receiver.

[0003] In particular, it is difficult to solve the case of mixed channels where the optimal input distribution cannot be obtained theoretically. The probability distribution can be decomposed in terms of the basis elements of the function.

[0004] The basic characteristics of a memoryless communication channel can be expressed by the conditional probability distribution p of the output Y given an input X. Y|X Here are some examples of well-known communication channels:

[0005] ● an additive white Gaussian noise channel, where Y=X+η and η is a Gaussian distribution model, the additive white Gaussian noise channel corresponds to a wired communication subject to disturbances caused by thermal noise at the receiver,

[0006] A fading channel Y = a.X + η, where a follows a fading distribution (such as a Rayleigh distribution), which models transmission over a narrowband wireless channel in a radio propagation environment involving rich scattering,

[0007] More complex channels may involve nonlinear effects. This is for example the case for optical channels where the Kerr effect cannot be ignored and reduces the channel capacity when the transmission power is increased too much in wavelength division multiplexing (WDM transmission), due to the nonlinear Schrödinger effect. Equation driven.

[0008] Once the conditional probability distribution p is known accurately Y|X (y|x), the communication system can be optimized as follows:

[0009] ● The design of the input signal so as to maximize the mutual information between the input and output of the channel,

[0010] ●The design of the optimal receiver usually depends on the likelihood probability p Y|X Processing of (y|x).

[0011] In general, there still remains a need for a solution to the problem of optimizing the transmission strategy, preferably by designing the transmission probability and optionally the position of each symbol of the constellation (e.g. QAM, PSK, etc.). The main challenges are:

[0012] ● Obtain the channel-conditional probability distribution of the channel decomposed on the basis elements of the function.

[0013] ●Optimize input distribution.

[0014] The present invention aims to improve this situation. Summary of the invention

[0015] To this end, a method for optimizing the capacity of a communication channel in a communication system is proposed, wherein the communication system comprises at least a transmitter, a receiver, and a communication channel between the transmitter and the receiver, wherein the transmitter uses a finite set Ω = {ω1, ..., ω N}, a message including at least one symbol is sent on the communication channel, and the communication channel is represented by a conditional probability distribution p Y|X (y|x), where y is the symbol received at the receiver and x is the symbol sent by the transmitter.

[0016] More specifically, for each possible transmitted symbol x, the above conditional probability distribution p is obtained by using a mixture model of the probability distribution represented by an exponential function Y|X (y|x), and calculate the optimized input distribution p based on the parameters of the mixture model x (x) to define the optimized symbol positions and probabilities to be used at the transmitter to optimize the capacity of the channel.

[0017] Therefore, a decomposed representation of the channel conditional probability distribution in terms of the basis elements of the exponential distribution function is used in order to limit the computational complexity of computing the optimal input distribution. By improving the input signal probability distribution based on channel knowledge, the channel capacity is greatly improved.

[0018] The above-mentioned optimized symbol positions and probabilities may be obtained at the transmitter, but in certain embodiments may also be obtained at the receiver.

[0019] In an embodiment, the transmitter may transmit the signals corresponding to the symbols ω1, …, ω N The messages transmitted by a finite set of signals are each associated with an input distribution p according to the optimization x (x) is associated with the transmission probability of the corresponding optimized input signal probability. Then, in this embodiment, the transmitter will:

[0020] - the message to be sent, and

[0021] -Optimize the probability distribution of the input signal as input,

[0022] And output the transmission signal on the communication channel.

[0023] In this embodiment, the communication channel takes a transmitted signal as input and outputs a received signal to be processed at a receiver (usually in order to decode the received message at the receiver), and the conditional probability distribution p Y|X (y|x) is therefore related to the probability of outputting a given signal y when the input x is fixed.

[0024] Preferably, in this embodiment, the conditional probability distribution p Y|X (y|x) is defined over a continuous input / output alphabet as a probability density function.

[0025] With conditional probability distribution p Y|X The estimated value of (y|x) is used as input to output the optimized input signal probability distribution p to be obtained at the transmitter (and in embodiments at the receiver). x (x), the conditional probability distribution estimate is used to calculate the optimized input signal probability distribution, and the conditional probability distribution estimate is approximated by the mixture model.

[0026] In an embodiment, the receiver receives the signal and optimizes the input signal probability distribution p x (x) and the channel conditional probability distribution p Y|X An estimated value of (y|x) is taken as input and an estimation of the message conveyed in the received signal is performed.

[0027] Therefore, in this embodiment, the channel condition probability distribution p can be estimated from Y|X The optimal input signal probability distribution p of (y|x) x (x), the receiver can perform enhanced determination of the transmitted message.

[0028] In an embodiment, the above hybrid model follows a conditional probability distribution p that can be decomposed into basic elements of the probability distribution exponential function g(y|x; θ) Y|X (y|x), where θ is the set of parameters such that:

[0029]

[0030] Among them, K is a predetermined parameter, and the set {θ j}、{w j} are the parameters representing the mean vector coordinates and the covariance matrix parameters respectively.

[0031] In this embodiment, the derivative of the probability distribution exponential function g(y|x; θ) is given by g(y|x; θ)=h(y, θ)exp(x T ya(x,θ)), where h(y,θ) is a function of y and θ, and a(x,θ) is the moment generating function, x and y are vectors,

[0032] So that the derivative is given by:

[0033]

[0034] By determining p Y|X The analytical observations of (y|x) are consistent with those given by The parameter set {θ j}、{w j}, the distribution p is approximated by a finite set of continuous functions that minimize the metric defined by the Kullbeck–Leibler divergence Y|X (y|x).

[0035] Input distribution p x (x) can be represented as a list of N constellation positions {(x1,π1),…,(x N ,π N )}, where x i and π i denote the constellation position and probability weight respectively,

[0036] And the input distribution p x (x) is estimated by solving the optimization problem given by:

[0037]

[0038] Constrained by

[0039]

[0040] 0<π i <1, for i=1,…,N

[0041] in:

[0042] - I(x,π) is the position vector x = [x1,…,x N ] T and weight vector π=[π1,…,π N ] T The mutual information of the function,

[0043] - the best values ​​are marked with a superscript *, and

[0044] -P indicates the total transmit power.

[0045] Mutual information can be expressed as:

[0046]

[0047] in,

[0048] And the variable y i,m is from the distribution p Y|X (y|x i ) samples.

[0049] In this embodiment, the alternating optimization may be performed iteratively to calculate p x (x) and Both, in order to derive the optimal position π from the calculation (t) , the optimal position π (t) The description from the previous iteration t-1 to the current iteration t is as follows:

[0050] - First, for the symbol position x (t-1) A fixed set of and the previous position value π (t-1) Optimize position π (t) ;

[0051] - Then, for the π thus determined (t) and the previous value x (t-1) Optimize symbol position x (t) ,

[0052] And these two steps are repeated alternately until the stopping condition appears for the mutual information I(x,π).

[0053] Refer to the following Figure 2 Describing this embodiment in detail, Figure 2 An example of the steps involved in the gradient descent algorithm for particles representing positions on a telecommunication constellation (QAM, PSK, etc.) is shown.

[0054] The invention also relates to a computer program comprising instructions which, when such instructions are executed by a processing circuit, cause the processing circuit to carry out a method as described above.

[0055] Comments below Figure 2 An algorithm of such a computer program may be exemplified.

[0056] The present invention also relates to a system, which comprises at least a transmitter, a receiver and a communication channel between the transmitter and the receiver, wherein the transmitter is at least configured to implement the above method.

[0057] The invention also relates to a communication device comprising a processing circuit configured to perform an optimization method as presented above. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Further details and advantages of possible embodiments of the invention will be presented below with reference to the attached drawings.

[0059] [ Figure 1 ]

[0060] Figure 1 is a system overview of an example according to an embodiment of the present invention.

[0061] [ Figure 2 ]

[0062] Figure 2 Possible steps of an optimization method according to an embodiment of the present invention are shown.

[0063] [ Figure 3 ]

[0064] Figure 3 The processing circuit of the communication device for executing the optimization method of the present invention is schematically shown. DETAILED DESCRIPTION

[0065] Reference Figure 1 , the system according to the present invention includes a transmitter 10, a receiver 12, a transmission channel 11 and an input signal probability distribution optimizer 13 in the example of implementation.

[0066] The transmitter 10 transmits a message conveyed by a signal belonging to a finite set of signals, each signal being associated with a transmission probability according to an (optimized) input signal probability distribution. The transmitter 10 takes as input a message and an (optimized) input signal probability distribution and outputs a signal to be transmitted on a channel.

[0067] The channel 11 takes as input a transmitted signal and outputs a received signal which is processed at the receiver 12 to decode the transmitted message. The channel is characterized by a channel-conditioned probability distribution of the probability of outputting a given signal when the input is fixed. Probability distributions can generally be defined over discrete or continuous input and / or output alphabets. Here, as an example, an alphabet of continuous outputs is considered, and in this case the probability distribution is called a probability density function.

[0068] The input signal probability distribution optimizer 13 takes the conditional probability distribution estimate as input and outputs an optimized input signal probability distribution to the transmitter 10 and the receiver 12 .

[0069] It is worth noting that the optimizer 13 can be the same module as part of both the transmitter and the receiver. Alternatively, it can be a module that is part of a scheduling entity (e.g., a base station, etc.) in a telecommunication network that links the transmitter and the receiver through a communication channel. More specifically, a communication device such as a transmitter 10, a receiver 12, or any device 13 capable of performing the optimization method may include such a module, which may actually have such Figure 3The structure of the processing circuit shown. Such a processing circuit may generally comprise: an input interface IN linked to a processor PROC cooperating with a memory unit MEM (storing at least the instructions of the computer program according to the invention) to receive data (at least data enabling the estimation of the conditional probability distribution); and an output OUT transmitting the result of the optimization calculation.

[0070] More specifically, the optimized input signal probability distribution is calculated using the conditional probability distribution estimate at the input signal probability distribution optimizer 13. Specifically, it is shown below that the optimization is more efficient when the conditional probability distribution estimate is approximated by a mixture of exponential distributions.

[0071] The receiver 12 takes as input the received signal, the optimized input signal probability distribution and the estimated channel condition probability distribution and estimates the message conveyed in the received signal.

[0072] In the following, the transmission channel 11 is represented by a model that follows a conditional probability distribution p(y|x;θ) that can be decomposed into basic elements of a probability distribution function p(y|x;θ). Y|X (y|x), where θ is the set of parameters. For example, the distribution function is an exponential family, and the parameters are essentially the mean and variance in the scalar case, and more generally the mean vector and covariance matrix in the multivariate case, such that:

[0073]

[0074] where K and the set {θ j}、{w j} is a parameter.

[0075] For example, three channel examples following this model can be cited below.

[0076] A channel can have random discrete states when it fluctuates randomly in time according to discrete events such as:

[0077] - interference caused by bursts that vary the signal-to-noise ratio from one transmission to another,

[0078] - the shadowing effect of changing the received signal power,

[0079] - approximation of a random fading channel by a discrete distribution, where the channel coefficient α is random and follows where α j is the n possible values ​​of the channel coefficient α with probability w j A value appears, and δ(.) is the Kronecker function. Therefore, when the variance is In the case of Gaussian noise, this fading channel leads to a probability distribution denoted by

[0080]

[0081] In the case of impaired channel estimation (usually when the transmission channel is not perfectly known), the residual self-interference is obtained on the received signal. Typically, the channel model is obtained as This yields:

[0082]

[0083] Therefore, it is shown here that according to any known continuous distribution p Y|X (y|x), this distribution can be approximated by a finite set of continuous functions.

[0084] The approximation is done by minimizing an indicator. A related indicator is the Kullback-Leibler divergence which allows to obtain a measure of the difference between the two distributions. Thus, when p is known analytically Y|X (y|x) can be found to make p Y|X The parameter set {θ that minimizes the Kullbeck-Leibler divergence between (y|x) and the approximate expression of the form of equation (E) given above is j}、{w j}.

[0085] According to p Y|X The estimated histogram of (y|x) can be approximated by a finite set of continuous functions in the same way as known continuous distributions, using the Kullback–Leibler divergence as an indicator.

[0086] Function p Y|X (y|x) is a two-variable function with variables x and y, usually spanning a continuous domain.

[0087] In the following, the focus is on the finite alphabet Ω = {ω1,…,ω N} on the symbol x.

[0088] Further assume that the derivative of the probability distribution function g(y|x; θ) is known. For example, when g(y|x; θ) is from the exponential family, it can be written as:

[0089] g(y|x;θ)=h(y,θ)exp(x T ya(x,θ)),

[0090] where h(y,θ) is a function of y and θ, and a(x,θ) is the moment generating function, in the general case where x and y are vectors. Thus,

[0091]

[0092] For example, in the scalar Gaussian case, the probability density function is decomposed as follows:

[0093]

[0094] The input signal distribution optimizer 13 relies on an estimate of the channel probability distribution in the form of equation (E). When the function basis chosen for channel estimation is an exponential family, a closed form expression can be derived and the algorithm converges to an optimal solution.

[0095] For some channels, the capacity close to the input is considered discrete. For the case of achieving continuous capacity of the input (which is the case for more general channels), the input distribution p X (x) can be represented as a list of N particles as follows:

[0096] {(x1,π1),…,(x N ,π N )},

[0097] where x i and π i denote the position (i.e., represented by a set of coordinates or by complex numbers in the two-dimensional case) and the weight, respectively. The optimization problem in the transmitter can be written as

[0098]

[0099] Constrained by

[0100]

[0101] 0<π i <1, for i=1,…,N (4)

[0102] in:

[0103] - I(x,π) is the position vector x = [x1,…,x N ] T and weight vector π=[π1,…,π N ] T The mutual information of the function,

[0104] - the best value is indicated by a superscript *, and

[0105] -P represents an arbitrarily set total transmit power constraint. Typically, this value is defined by the transmitter's power budget related to the physical limitations of the power amplifier or to the maximum radiated power allowed by regulations.

[0106] Constraint (2) sets the total probability of a particle to 1. Constraints (3) and (4) ensure that the total emission power is less than or equal to P and that the magnitudes of the particle probabilities are positive values ​​less than 1, respectively. Involves the integration of a continuous random variable, but can be approximated by Monte-Carlo integration (the main principle of which is to generate samples that are the average of the realized values ​​and obtained values ​​of the random variable in place of the expectation function usually involved in the integration):

[0107]

[0108] where M represents the sample size (i.e., the number of realized values ​​of the random variables generated from their probability distribution), and thus, where

[0109]

[0110] represents the conditional probability p Y|X (y|x) is decomposed into j The basic elements of the function g().

[0111] The variables in (5) come from the distribution p Y|X (y|x i ) samples.

[0112] In the following, an alternating optimization method is proposed, from iteration t-1 to t as follows:

[0113] -First, for particle x (t-1) A fixed set of and prior values ​​π (t-1) Optimizing π (t) ;

[0114] -Then, for the obtained π (t) and the prior value π (t-1) Optimize x (t) .

[0115] These two steps are denoted as S1 and S2 in the following. They can be inserted after the initialization step S0 of the algorithm described below.

[0116] Step S1: For particle x (t-1) A fixed set of and prior values ​​π (t-1) Optimizing π (t)

[0117] The optimization in (1) is concave with respect to π for fixed values ​​of x. Thus, for a given x (t-1) , by writing the Lagrangian as follows and solving for π for i = 1,…,N i , to solve (1) for π:

[0118]

[0119] in,

[0120]

[0121] Here, the expression is based on the random variable y i The mathematical expectation of By the Monte Carlo integration mentioned above (i.e., by i The distribution of M samples is used to perform the approximation. Advantageously, this can be replaced by numerical integration or closed form expressions when available.

[0122] In (7), β represents the Lagrangian multiplier, which can be determined by replacing (7) in (3) with the equation for the maximum total transmitted power P and obtaining the following nonlinear equation:

[0123]

[0124] The nonlinear equation (8) can be solved using different tools, for example, gradient descent-based methods such as Newton-Raphson, or by choosing several values ​​of β, calculating the left-hand side of the equation in (8) and retaining the one with the absolute value closest to 0. And from (7) we obtain The value of .

[0125] Step S2: For a fixed π (t) and first x (t-1) Optimize x (t)

[0126] With a given weight vector π (t) The Lagrangian used for optimization in (1) can be given by:

[0127]

[0128] Obtain a position vector x such that the Kullback-Leibler divergence D(p Y|X (y|x i )||p Y (y)) is maximized. Thus, after each update of the position and weight vector, the Lagrangian The value of (i.e., the penalized mutual information) is greater than or equal to the previous value. This is achieved through a gradient ascent-based method, namely:

[0129]

[0130] Among them, the step size λ t is a positive real number.

[0131] In the above-mentioned gradient ascent method, it is necessary to convert the term D(p Y|X (y|x i )||p Y The derivative of (y)) is calculated as:

[0132]

[0133] in,

[0134] Using formula (6), we can obtain:

[0135]

[0136] Therefore, when g(y|x;θ j ) is known in closed form and its derivative is also known in closed form, the equation can be evaluated.

[0137] Finally, we get x (t) The value can be continued until a stop condition is met. The stop condition is, for example, the execution time, or if I(x (t) ,π (t) )-I(x (t-1) ,π (t-1) ) is below a given threshold which is usually small.

[0138] Reference Figure 2 , an example of the algorithm is detailed below.

[0139] Step S0: Initialization step

[0140] -Step S01: Get input parameters

[0141] ○P, the power limit of the constellation

[0142] ○ Initial constellation of N symbols

[0143] ○ Stopping criterion threshold ε

[0144] -Step S02: Get The channel conditional probability distribution is in the form of j is a scalar parameter, and θ j is a parameter set, where The expression for is known.

[0145] Step S02: Set t=0; Set all Set all according to the initial constellation C0 Setting I (-1) =0; set t=1

[0146] Step S1: Iteration step t

[0147] Step S10: Sample generation

[0148] -S101: For all i in [1,N], from the distribution Generate M samples y i,m

[0149] -S102: For all i in [1,N], for all j in [1,N], calculate

[0150] Step S11: Calculate the stop condition

[0151] -S111: Calculation

[0152] -S112: If I (t-1) -I (t-2) <ε, stop the iterative algorithm (S113). Otherwise, go to S121.

[0153] Step S12: Update probability

[0154] -S121: For all i in [1,N] and all m in [1,M], calculate

[0155] -S122: Calculate β by solving the following equation:

[0156]

[0157] o For example, by using Newton-Raphson descent, and / or

[0158] ○ By using a line search strategy (take several values ​​of β, evaluate the above expression and choose the one closest to 0);

[0159] -S123: For all i in [1,N], calculate

[0160] Step S2: Use the new and the previous Update symbols Location

[0161] -S21: For all i in [1,N], for all j in [1,N], calculate This is done by using y i,m Replace y and x i Substitute x, from the known expression Obtained

[0162] -S22: For all i in [1,N],

[0163] calculate in is for function Value

[0164] The next step S3 is to increment t to loop, for the next iteration, to step S101.

[0165] Therefore, artificial intelligence can be programmed with this algorithm to optimize the capacity of a given communication channel (one or several communication channels) in a telecommunications network.

Claims

1. A method for optimizing the capacity of a communication channel in a communication system, the communication system comprising at least a transmitter, a receiver, and the communication channel between the transmitter and the receiver, The transmitter uses a finite set of symbols with corresponding positions on the constellation diagram Ω = {ω1, ..., ω N } to send a message comprising at least one symbol on the communication channel, The communication channel is represented by the conditional probability distribution p Y|X (y|x), where y is a signal received at the receiver and x is a signal sent by the transmitter, and x includes at least one symbol in the finite set Ω; in, For each possible transmitted signal x, the conditional probability distribution p is obtained by using a mixture model of probability distribution represented by an exponential function Y|X (y|x), and calculate the optimized input distribution p based on the parameters of the mixture model x (x) to define optimized symbol positions and probabilities to be used at the transmitter to optimize the capacity of the communication channel; Among them, the conditional probability distribution p obtained using the mixed model Y|X (y|x) can be decomposed into the basic elements of the probability distribution exponential function g(y|x; θ), where θ is the parameter set such that: Among them, K is a predetermined parameter, and the set {θ j }、{w j } are the parameters representing the mean vector coordinates and the covariance matrix parameters respectively.

2. The method according to claim 1, wherein: The optimized symbol positions and probabilities are obtained at the transmitter and the receiver.

3. The method according to claim 1 or 2, wherein: The transmitter sends the signals corresponding to the symbols ω1,…,ω N The messages transmitted by a finite set of signals are each associated with an input distribution p according to the optimization x (x) is associated with the transmission probability corresponding to the optimized input signal probability distribution, And the transmitter takes the message to be sent and the optimized input signal probability distribution as input, and outputs a transmission signal on the communication channel.

4. The method according to claim 3, wherein: The communication channel takes the transmitted signal as input and outputs a received signal to be processed at the receiver, and the conditional probability distribution p Y|X (y|x) is therefore related to the probability of outputting a given signal y when the input x is fixed.

5. The method according to claim 4, wherein: The conditional probability distribution p Y|X The estimated value of (y|x) is taken as input to output the optimized input signal probability distribution p to be obtained at least at the transmitter. x (x), a conditional probability distribution estimate is used to calculate the optimized input signal probability distribution, and the conditional probability distribution estimate is approximated by the hybrid model.

6. The method according to claim 4 or 5, wherein: The receiver converts the received signal, the optimized input signal probability distribution p x (x) and the channel conditional probability distribution p Y|X An estimated value of (y|x) is taken as input and an estimation of the message conveyed in the received signal is performed.

7. The method according to claim 1, wherein: The derivative of the probability distribution exponential function g(y|x;θ) is given by g(y|x;θ)=h(y,θ)exp(x T ya(x,θ)), where h(y,θ) is a function of y and θ, and a(x,θ) is the moment generating function, x and y are vectors, So that the derivative is given by:

8. The method according to claim 1 or 7, wherein: By determining that p Y|X The analytical observations of (y|x) are consistent with those given by The parameter set {θ j }、{w j }, the distribution p is approximated by a finite set of continuous functions that minimize the metric defined by the Kullbeck–Leibler divergence Y|X (y|x).

9. The method according to any one of claims 1, 2 and 7, wherein: The input distribution p x (x) is represented as a list of N constellation positions {(x1,π1),…,(x N ,π N )}, where x i and π i denote the constellation position and probability weight respectively, And wherein the input distribution p x (x) is estimated by solving the optimization problem given by: Constrained by 0<π i <1, for i=1,…,N in: - I(x,π) is the position vector x = [x1,…,x N ] T and weight vector π=[π1,…,π N ] T The mutual information of the function of , - the best value is marked with a superscript *, and -P indicates the total transmit power.

10. The method according to claim 9, wherein: The mutual information is expressed as: in, And the variable y i,m is from the distribution p Y|X (y|x i ) samples.

11. The method according to claim 10, wherein: Iteratively perform alternating optimization to calculate p x (x) and Both, in order to derive the optimal position π from the calculation (t) , the optimal position π (t) The description from the previous iteration t-1 to the current iteration t is as follows: -First, for the symbol position x (t-1) A fixed set of and prior position values ​​π (t-1) Optimize position π (t) ; - Then, for the π thus determined (t) and the previous value x (t-1) Optimize symbol position x (t) , And repeat these two steps alternately until a stopping condition appears with respect to the mutual information I(x,π).

12. A computer program product comprising a computer program including instructions which, when such instructions are executed by a processing circuit, cause the processing circuit to implement the method according to any one of claims 1 to 11.

13. A communication system, comprising at least a transmitter, a receiver and a communication channel between the transmitter and the receiver, wherein: The transmitter is at least configured to implement the method according to any one of claims 1 to 11.

14. A communication device comprising a processing circuit configured to perform the optimization method according to any one of claims 1 to 11.

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