JPH01243176A - System for searching minimum and maximum values - Google Patents

Abstract

PURPOSE:To obtain the minimum value of a function in a short time by deciding a temperature so that a time, when the function is changed from an initial condition up to a condition to give the minimum value, can be made minimum. CONSTITUTION:At the time of obtaining the minimum value of a function E to have many extreme values, the maximization of exp(-E/T) is considered instead of the function E. A parameter T is called temperature, it is introduced in order to generate a random noise and to permit a probability-like management, and when an attainment to the minimum value of the E is executed, the T is brought to O, and it needs to be remain at the minimum value without an error. Then, the temperature T is decided so that the time from the initial condition up to the condition to give the minimum value which is a final goal can be made minimum. Thus, the minimum value can be surely and speedily obtained.

Description

[Detailed Description of the Invention] [Industrial Application Field] The present invention relates to finding the minimum (or maximum) value of a given function, and particularly to minimizing (or maximizing) a function that has a large number of extreme values. This invention relates to a minimum/maximum value frame search method suitable for.

[Conventional technology]

When finding the minimum value (maximum value) of a given cost function E, if the cost function has many extreme values,
In the conventional deterministic hill-climbing method, it is generally difficult to find this minimum value. This is because when a value near a certain extreme value is given as an initial value, the local minimum value falls because it is deterministic, and it is impossible to escape from there.

Therefore, the minimum cannot be determined. Traditionally, simulated annealing (Simultaneous annealing) has been used to solve this problem.
A deterministic method of mountain climbing called a-ted annealjng has been proposed. Simply put, it attempts to reach the final destination by allowing not only to climb a mountain but also to descend a mountain with a certain probability.

The most commonly used method is as follows, taking the minimum value problem of E as an example. First, instead of considering the cost function E directly, we consider the Boltzmann distribution P −exp(−E
/T). The parameter T introduced here is called temperature and was introduced to generate random r9) noise and allow stochastic handling. Therefore, when the minimum value of E is reached, T
It is necessary to bring it to O and stay at its minimum value without error. Deciding on a cooling schedule for how to lower T to a low temperature is the biggest challenge for the simulator indoor 21 ring.

The Geman brother schedule that has been commonly used is IEEE Transactj, on on P.
AtternAnalysis and Machine
eIntelligence, vol, 6゜pp,
721. -741. (1985), the states are generated according to the Boltzmann distribution, and TO is a positive constant, T(t)=To/Qog(t+1)
That is. Here, t corresponds to the number of Monte Carlo simulations, and is defined as time here. Of course, if t is increased, 1.27' T(t) becomes 0. Although some cases in which this method works successfully have already been reported, there are many cases in which it is not always applicable. Also, recently Physics Letters.

vol, 123. pp, 157-161. (1987
In order to increase the degree of convergence to the maximum value of P, we use the Lorentz distribution with a large distribution instead of the Boltzmann distribution, as discussed by Szu and Hartley in ”To/
A schedule of (t + 1-) has also been proposed. However, a common drawback of these schedules is that they do not take into account the functional form of the cost function to be minimized. What kind of barrier (difference in cost between the minimum value and the nearby maximum value) is climbed and when the final value is reached (when T is set to O) is reflected in T (t, ). It has not been. Numerically, it has been proven that both of them will definitely reach the desired maximum value of P after an infinite amount of time. However, in reality, although there are cases in which the minimum value is reached within a finite time in which simulation can be executed, there are many cases in which this is not the case, so their utility value is not necessarily high.

[Problem to be solved by the invention]

A common drawback of the above schedules is that the temperature T does not reflect the functional form of the cost function to be minimized. Therefore, in reality, the minimum value is very often not found within the finite time in which simulation can be executed. In order to overcome these problems, the temperature T is changed to the time t
In addition, it also depends on the function E. In other words,
Let T=T(t, E). Some kind of guideline is required to determine the E dependence of temperature, but here it is required to minimize the time from the initial state to the state that provides the final target minimum value.

With tl as the final time, set E so that this tz is the minimum.
The purpose is to determine the

[Means to solve the problem]

Taking the problem of maximizing the Boltzmann distribution exp(-E/T) in a one-dimensional space as an example, the basic procedure of the simulated annealing method is first-order (FIG. 2). First, exp(-E/T)=exp(-f(E(
Rewrite it as x)/T)'dx). Here, the symbol ' represents differentiation with respect to the spatial variable, and f...dx represents integration.

If only T h<t is taken as a function, this equation is just an equation. Now, a certain state x (201) and the next state x
'(202) The difference in cost is ΔE=E(x')
-E(X)"FE(X)' (203), X'
The probability of transitioning to (204,) is maX (1+exp
(-Δ(E/T))). Whether or not to allow transition from state X to state X' is determined by comparing this value with a uniform random number η (205) from O to 1. Therefore,
If 8E<O, it will definitely shift to X' and ΔE≧O
In other words, even if the cost is higher, Δ
The transition is allowed with a probability determined by the value of E (206).
(207).

The dynamic process from a given initial state to the maximum value is defined by the following time-evolving differential equation.

d x(t)/d t=-rH' (t, x)+ξ(x
)...(1) Here, H(t, x)=E(x)/T(t, E) is defined, and it is assumed that temperature generally depends on both t, E, or X. Also, the positive parameter "is.

Let it be the variance value of additive Gaussian noise ξ(X) with a mean value of 0. It can be easily seen as follows that this dynamic equation gives the Boltzmann distribution exP(-H(t, x)) at least in the steady state. If the probability that the state takes the value X at a certain time is P (t, x), then the stochastic differential equation for equation (1) is aP (t,,
a(H'(t,x)P)/ax+"a"P/ax2 (2). The distribution Ps in the steady state is clearly exp
It is proportional to (-H). Therefore, the method of searching for the maximum value uses the Boltzmann distribution in the vicinity of the maximum value of P, similar to simulated annealing, but the dynamic process leading to that point uses the following method. To,
In a sense, it uses a more efficient probability distribution.

As a policy for determining the target H(t,,x), first, a search is required in the minimum time. In other words, minimize . Here, to is the initial time and tl is the final time. Since tl is not known in advance, it is an unknown number here. Intuitively speaking, in order to minimize tl, if the temperature T is made as high as possible, it can easily reach the vicinity of the maximum value. However, the problem is that the fluctuation there increases in proportion to 5. That is, although it is possible to quickly reach the vicinity, T must be made smaller in order to suppress it to the true maximum value. When this trade-off relationship is expressed by a cost function J to be minimized, for example, using a positive constant tl, the appropriate function is J=/(L/2T2+1)dτ.

It's urgent. Since equation (1) contains H instead of T, use the given costs E and H.

...Expand to (4). Here, <...> means the average at probability P(t, x). If T does not depend on X, then clearly it is equal to the cost mentioned above.

To summarize the problem, for the dynamic equation according to equation (1), 1
The purpose is to determine the optimal function H* for minimizing the cost of equation (4).

[Effect]

It is known that the formal solution to this problem is as follows. First, the optimal function H11 of H is -aV (t + x)/at = min(-rH'δ\/ax+Y'a2V/δx2+
L/2(H'/E')")H'...(
5) is determined to give the minimum value on the right side.

H'zero=L-"FE'"aV/ae-(6
) In general, it is known that temperature has a relationship with additive noise as T=O(r), so a V / a x=
o (r ”).Here, the symbol O(...) is...
...indicates the order of the values. Substituting this into equation (5), the equation according to ■ becomes -a V(t, x)/at =-1/2L-1F2E'''(aV/ax)2+Y'
a”V/ax2...(7).In addition, the initial conditions are:
) is given by Since this initial condition gives a value at the final time, in order to rewrite it into an easy-to-handle initial value problem, it is converted to a new time τ with τ=t1−t.

Then, the above formula is aV(τ,
azV/a x”...(9) v(O5x)=0...(10
). This is the final equation that determines ■, that is, H*.

In the end, the problem is to solve equations (9) and (10) and find ■. Of course, a numerical solution is possible, but it takes CPU time and is not practical. Therefore, an analytical solution is desired. Here, using the singular perturbation method with r as a small parameter,
Obtain an approximate analytical solution. Let state X that gives the maximum value of P be an α umbrella.

Although this value itself is unknown, the behavior of the solution in its vicinity can be investigated. Therefore, aV/ax=o('
-2), and estimate the size of the right-hand side of equation (9). Now, state X is in a neighborhood (internal region; Then, the first term is o (r”) and the second term is O (“”/2), so
The second term is important. On the other hand, if state ”) f
Note, the first term is the dominant term. Therefore, it is sufficient to obtain solutions for these regions separately and to smoothly combine the two. In the following, approximate solutions up to o < J7 are constructed in each region.

(1) Internal region; can be abbreviated to Letting the solution in this region be Vi, the approximate equation up to ○(E) is a Vi (τ, X)/aτ= “a”V i /
a x”...(11) The solution that satisfies the condition of equation (11) can be easily found, and Vi(t, x)=r'texp(-(x-α*)z/2
1”τ)...(12) When τ→0, clearly vi(0,x)→O.

Here, since α inside is unknown, this formula is not actually used as is, but its meaning will be clarified later.

(2) External domain depth L-2F2E' 2(a V o/
a x)2...(13). This solution is found using the variable separation method. Now, prepare a function A (τ) of only τ and a function B (x) of only X, and get V o (τ, x) = A (τ) B (X)
=-(14). Substituting this into equation (13), dA(τ)/dτ/A2=-1/2
Obtain L-"r2E'"B'"/B2・-(15). Both sides are functions of independent variables, so both sides must be constants. With C as a constant, both sides are expressed as 1/2L-' for convenience. Let it be r”C. Then, for each function, dA(τ)/dτ=-1/2L-'r"CB' Z=C
We obtain the differential equation B''/E' Z ・(16). When these solutions are found, we get the following.

A(τ)=1/(1/Am+1/2L-'17"Ct)
B(x)=C/4(71/E' d x+ct)” −
417) Here, Am and Cz are integral constants, which are determined from the connection conditions of internal and external solutions (12) and (17) described below.

(3) Connection condition for solutions In order to smoothly connect the solution (12) in the inner region and the solution (17) in the outer region and find a −-like solution in the entire region,
At the boundary position xb=α+O(~/i), the function value of the solution in each region and its spatial differential value may be made equal.

Vib, xb) = Vo(τ, xb) δVj, (τ, xb)/ax = δVo(t, xb)/
ax - (18) First, determine Am. Since Equation (12) describes the state near the maximum value of P, it is an equation that originally holds true in a small area. However, in order to connect with the solution in the external region, if we expand the region and find the asymptotic form of the solution in the region where τ is large, then V i (τ, xb) ~ r/2
(-1+2). Similarly, the asymptotic form of the solution in a small τ region of equation (17) that holds in a large τ region is V
o (t, x b) ~ -Am (-1+1/2L
'2CAmτ)B(xb). Therefore, by comparing both sides, A m = 4 L Im-'C
-”. Therefore, A(τ)=4L17 “C-’/(1+2r)~2Lr
"C"/τ...(19) is determined. Here, 1 is omitted by taking advantage of the fact that τ is large. Formula (1
2) and (17) into equation (18).

xb I'τexp(-I'/2T) =A(τ)C/4・(
/ 1/E'dx+ct)"xb - "τexp(-"/2τ)=A(τ)C/2・ge 1
/E'dx+Cz)/E' (xb) (20,) When these equations are solved, the constant C1 and the time τ at the time of connection are determined at the same time.

xb Ct=2τ/E' (xb) f 1/E' dx
1/τ=-2+4LP-8exp(r'/2τ)/(E
'(xb))''...(21) When these values are used for the solution in the external domain, Vo(
τ, x) = LI'-"/2τ・(f1/E' dx-2
t/E' (xb)Pb (22) Here, since the desired quantity is the differential of ■ with respect to X, the last term 2τ/E'(xb) in the above equation can be omitted.

V o (τ, x)=L I'-"/2 τ(/1/
E' d x)''...(23) is obtained.

From equation (6), the E' umbrella to be found is Hi'*=-L-'I"(E'(x))" (x-α
t)exp(-(X-αz/2'τ) Ho' fist=(r τ)-E'(x)/1/E'
d x ...(24).

Considering ease of use, H' = (E/T)' = E
It is more convenient to define the temperature Topt as ' / Topt. This is because it is only necessary to calculate the change in cost E without directly calculating the change in H. If the above equation is rewritten using this temperature, the following equation is obtained in each region.

Tiopt=-Lr'-"((x-a umbrella)E' (x
))・exp((x-asu”/2r’t)Toopt=
Fτ/(f 1/E' d x)
...C25) In terms of order, the last equation is E'20 (~r positive) in the internal region, Tiopt =
O(r).

Toopt=○(r), which does not contradict the definition of temperature.

This is considered to be because the temperature is increased and the magnitude of the additive noise is increased and adjusted in order to exceed the local maximum value in the external region. Here, it is assumed that the temperature constraint Topt>O. The -th equation of equation (25) is always negative because it becomes E(x) to (X-α*)z near the α umbrella. Therefore, in the internal region Tiopt=Q
shall be. This requirement means reducing fluctuations at extreme values, especially in the state that gives the minimum value.
It means eliminating fluctuations. Furthermore, also in terms of time dependence, τ=O in that state. Therefore, both areas can be combined and written as follows.

Topt= rτe [:1/(f1/E' d x)
) -(26) Here, the function θ introduced here is a function whose value becomes 0 when its argument is positive and 0 when it is negative.

Here, since 11 is unknown, τ=ti−t cannot be calculated directly. However, if the fact that r is small is actively utilized, the change in Fτ will be slight, so it seems that Topt will work effectively even if this quantity is treated as a constant.

〔Example〕

Here, we will use a simple one-dimensional cost function as an example to confirm the effectiveness of the new schedule Topt proposed here. Let k be a positive constant and be the differential E' of the cost function.

E' (x)=kn(x-αi) =42
Consider 7). To perform the integration of equation (26), E
Rewrite the reciprocal of ′ as follows.

17E'(x) = Σyi/(x-αi) −(28
) Then, the integration can be easily performed and Topt can be found as follows.

γ1 Topt=1' τke (1/IT(Qogl x-
αi l +q))yi (29) Here, q=-QogIxb-ail. Although this equation still includes an undetermined constant q, these are treated as parameters. In actual calculations, sufficiently good results were obtained even when q=Q.

Further, rτ = 1.0.

As an example of the above cost function, E(x)=β(-(7)X)"-1/3 ・(1)
x)"+ 1/4 ・Cp
1=2/ρ. Among these, α5 gives the minimum value.

The cost barrier to be overcome is 5β/12. T
The parameters required to calculate opt are γ1=-ρ2/2
．． γ2=ρ2/6. γ8=ρ2/3.

Figure 3 shows the conventional method T(t)=To/Qog(t+1)
; Comparison of simulation results performed using To=3 and Topt is shown. First of all, unlike the conventional method,
Almost no fluctuation occurs at the minimum value. Moreover, convergence to the minimum value is fast (a) to (d).

In extreme examples (e) and (f), even if the conventional method cannot reach the minimum value, the present method can reach it. Note that this result shows the average value obtained by executing the simulation 100 times.

Apply to more complex examples. In this case, the expression (
Since the integral of 26) cannot be directly obtained, an approximate optimal schedule as shown below is used. The major feature of the schedule shown in the above example is that it takes a large value when approaching the cost barrier to be exceeded, and takes a value of ○ near the extreme value. From this, it is thought that the essence of the optimal schedule can be captured by setting it to the maximum value when the second derivative 2E of the cost function becomes negative. Therefore, Topt=Φ[2E(X)) −(3
1) We propose an approximate schedule. Here, Φ is
When the argument is positive, a sufficiently large positive constant Φm is set; when the argument is negative, it is set to 0. In other words, only when the second-order differential is negative, the temperature is increased and the additional noise is increased so that the maximum value can be easily exceeded. An overall conceptual diagram of the simulated annealing algorithm in this case is shown in FIG.

First, an initial state (101) is set, a next state (103) is determined by simulated annealing (102), and if the end condition (104) is No, these processes are repeated. Optimal temperature in annealing (107)
is determined from the differential equation (31) of the given function (105). In order to confirm the effectiveness of this approximate schedule, we apply it to the following problem. Variable Xi (Kokoni, j=
1.2. −, N (N=100×100, where i is defined as a point on a two-dimensional plane grid) take two values of ±1. As the cost function E that should be the minimum value, E((x))=Σ(-ΣX i Xj + h X i
)...(32) Assume that i is given. The first Σ represents the sum of all variables, the second Σ represents the sum of variables adjacent to the first variable on the plane, and 6h is a positive constant.
Here, it is set to 0.1. Further, it is assumed that the initial state of Xi is randomly given. The difficulty of this problem lies in the fact that since the variable Xi takes only two values, there are many local minimum values. Minimization is possible even with binary variables, but here we use Hopfield and T
ank.

Biologica], Cybernetics,
vol, 52. p, 142. (1985)), the binary values are converted into continuous values and the simulation is performed. For this reason, the original variable Xi is changed to X i = tanh(F j, / B )
-(33). Obviously, Fi is one
It becomes a continuous quantity from to 10, and Fi = ±ω is X i =
Corresponds to ±1. Here, in order to improve convergence, constant B was set to 0.01.

For comparison, the minimum value of the function defined by equation (32) was determined using three different schedules shown below.

(A)T (t)” To/fl og(t +
1) (B) T (t) = constant (C) Topt = Φ [2E], Φm = 5
- (34) The number of Monte Carlo simulations is represented by the symbol t, and the maximum value t max of t is 1000 times.

Also, for comparison, the minimum value of Topt is not 0, but (A
) is the value of T at t'may, 1/n oH(t
max+1). Figure 4 shows the simulation results. In the conventional method (A), To=1.
Simulations were performed with 0 and 4.0. Although they show slightly different values at the initial time, both have almost the same cost at tmax -=-0,63
give. However, in (C) we were able to obtain −0.90, which is clearly a much lower cost than the other methods. However, although method (C) seems to have achieved low cost simply by increasing the temperature, the circumstances are completely different. To see this, in (B), T(
The simulation was performed at a high temperature of 1:)=5.0. The results were even worse than the conventional method due to large noise.

FIG. 5 schematically shows an outline of an image processing system using the present invention. The electrical signal obtained by imaging the object to be observed 5 with the ITV camera 52 is sent to an image processing device 60 having at least a processor and a memory, and is processed as an analog signal.
It is converted into digital image data by the digital conversion and quantization device 53, and the file 5 is converted as original image data.
It is stored in 4. This original image data contains noise due to various factors in addition to errors caused by the nonlinear characteristics of the ITV camera 52. The original image data is then read from the file 54 and sent to, for example, a probabilistic image processing device 55. In this process, processing such as noise removal without edge blunting is generally achieved by minimizing the energy of the image comprised of the original image data. This minimization is achieved by the minimization according to the present invention.
This is performed by the maximum value search device 56. The processing is carried out by the iterative method according to Figures 1 and 2, and the intermediate results are in file 5.
7 is stored. The processed image data is file 5.
8 and then read out as needed, subjected to other image processing, or digital-to-analog conversion 59
After being sent to the device.

It is displayed on the display device 60.

〔Effect of the invention〕

When finding the minimum (maximum) value of a function that has many extreme values, a stochastic hill-climbing method called simulated annealing has been proposed, considering the maximization of exp(E/T) instead of the function E. ing. The parameter T introduced here is called temperature, and was introduced to generate random noise and allow stochastic handling. Therefore, when the minimum value of E is reached, it is necessary to bring T to 0 and stay at that minimum value without error. Deciding how to lower T to a low temperature is the biggest challenge in simulated annealing.

For this reason, the temperature was determined so as to minimize the time from the initial state to the state giving the maximum value, which is the ultimate goal. As a result of simulation experiments, it was confirmed that the method can obtain lower minimum values than conventional methods even for complex nonlinear functions with many independent variables. This makes it possible to quickly and reliably find the minimum value.

[Brief explanation of the drawing]

Fig. 1 is an overall conceptual diagram of the algorithm of the minimum-maximum value search device which is an embodiment of the present invention, Fig. 2 is a calculation method of simulated annealing, and Figs. 3 and 4 are applications of the present invention. Give an example. FIG. 5 shows an image processing system in which the present invention is utilized for image processing. 102...Simulated annealing, 107
...Optimal temperature (schedule).

Claims (1)

[Claims] 1. When finding the minimum value of a function E or (-E) that has many extreme values using a stochastic hill-climbing method called simulated annealing, the temperature T introduced for stochastic handling is used to calculate noise. and the given function E
Instead, consider maximizing exp(-E/T), consider at least the shape of E, and determine the temperature T so as to minimize the time from the initial state to the state that gives the minimum value. A minimum/maximum value search method that is characterized by setting it to 0 when it reaches , and stopping it there without error.