DE10327287A1 - Method of designing a technical system - Google Patents

Abstract

The invention relates to a method for designing a technical system, in which the technical system comprises an objective function (f (x)) which depends on at least one parameter (x) in a variable space in which target function values are determined iteratively by a plurality of calculation steps, in each calculation step a development of a stochastic variable which is dependent on the objective function (f (x)) and is based on a stochastic differential equation is calculated, in which the development of the stochastic quantity depends on a step size (sigma (x)) chosen in such a way that the development with increasing number of calculation steps converges to a stationary stochastic distribution, in which, after a number of calculation steps carried out, a local optimization method for determining an optimal function criterion optimal function value is performed, wherein as the starting value of the local optimization ngsverfahrens one of the calculated according to the number of performed calculation steps target function values is used, and in which the determined with the local optimization method optimal target function value and the associated parameters are used for the design of the technical system.

Description

The The invention relates to a method and an arrangement for the design a technical system and a corresponding computer program product.

For the design of a complex technical system are already in an early stage Planning phase significant sizes of this Systems, e.g. costs for the production or efficiency during the term, of great interest. A realization of the system is often based on empirically known become parameter. Furthermore, often operating parameters for the operation of the existing technical system. This adaptation is mostly based on special expertise.

The technical system can often be described by an objective function which covers several targets and which depends on the above parameters. Examples of such targets are e.g. the cost and efficiency of a system.

It there is a need for the Objective function of a technical system to find the optimal parameters. These parameters are, for example, the parameter values, where the objective function takes a minimum. Includes the objective function several outcomes, will as an optimal solution usually the so-called Pareto-optimal solution Are defined. This Pareto-optimal solution is characterized by it that the corresponding parameters of the solution are no longer changed can, without thereby causing a deterioration for at least one target size of the objective function follows.

at a variety of technical systems are the optimal parameters then if the objective function of the corre sponding technical system not just a local minimum or maximum, but a global one Minimum or maximum. To solve such a global one Optimization problem is already the method of the document [1] known. With this method is by means of a stochastic differential equation solved the global optimization problem numerically, with sufficient to numerical solution that The Euler method known from the prior art is used. The method gives good results, however it is by use of the Euler method numerically very complex.

task The invention is therefore a method for the design of a technical To provide system, which compared to known methods the prior art a lower numerical effort with it brings.

at the method according to the invention is considered a technical system, whose objective function of depends on at least one parameter in a variable space. It become iteratively new objective function values through several calculation steps determined, wherein in each calculation step one dependent on the target function and development based on a stochastic differential equation a stochastic size, in particular a Markov chain. The evolution of stochastic Size depends on this from a step size chosen such that the development with increasing number of calculation steps against a stationary stochastic Distribution converges. Numerical simulations have shown that by such a choice of step size the global optimum of Target function can be achieved without having to take into account whether with the chosen one Step the development of stochastic size too indeed a good approximation of the solution represents the differential equation. After a number of calculation steps can in the process according to the invention be assumed that at least one of the calculated Target function values nearby There was a global optimum of the objective function. Therefore, after a number of calculation steps a local optimization method for determining an optimal target function value with respect to an optimality criterion carried out, wherein as starting value of the optimization method one of the after predetermined number of calculation steps determined target function values is used. It can be assumed that the with the function value also determined by the local optimization method global optimum of the objective function is, so this optimal functional value and the associated ones Parameters are used for the design of the technical system.

In a particularly preferred embodiment of the method, the parameters on which the objective function depends are continuous parameters and the objective function is a scalar-valued function f: R ⁿ -> R.

für alle x ∊ R \ {x ∊ Rⁿ : ||x||₂ ≤ r}, wobei ∇f(x) den Gradienten der Zielfunktion f : Rⁿ → R an der Stelle x bezeichne; und
es existiert ein r' ∊ R, r' > 0, und eine stetige Funktion σ : Rⁿ → R, σ(x) > 0 für alle x ∊ Rⁿ, so dass für alle x ∊ Rⁿ die Matrix A_σ,x : = E_n + σ(x)∇²f(x) invertierbar ist und für alle x ∊ Rⁿ mit ||x||₂> r' gilt

wobei σ(x) die Schrittweite ist.In a further preferred embodiment of the method, the objective function fulfills the following prerequisites:
There exists an ε> 0 such that there is an r> 0 with

for all x ∈ R \ {x ∈ R ⁿ : || x || ₂ ≤ r}, where ∇f (x) denotes the gradient of the objective function f: R ⁿ → R at x; and
there exists a r 'ε R, r'> 0, and a continuous function σ: R ⁿ → R, σ (x)> 0 for all x ε R ⁿ , so that for all x ε R ⁿ the matrix A _{σ, x} : = E _n + σ (x) ∇ ² f (x) is invertible and for all x ∈ R ⁿ with || x || ₂ > r 'applies

where σ (x) is the step size.

These Requirements make sure that the objective function is outside a ball does not just rise enough around the zero point but that this slope is also sufficiently uniform. The Inventors were able to prove that with these conditions always guaranteed is that the evolution of stochastic size with increasing number Calculation steps against a stationary stochastic distribution converges. The requirement does not impose any significant restriction the objective function, since in the case that the function is the prerequisite not fulfilled, This can always be exchanged for an auxiliary function, the meets this requirement and has the same global optimum.

wobei f(x) die Zielfunktion und σ: Rⁿ -> R eine Schrittweite ist,
{Φ_t ^ε'^x0(ω)} ein stochastischer Prozess ist und
{B_t : t ≥ 0} eine Brownsche Bewegung ist.In another particularly preferred embodiment, the stochastic differential equation is as follows:

where f (x) is the objective function and σ: R ⁿ -> R is a step size,
{Φ _t ^ε ' ^x 0 (ω)} is a stochastic process and
{B _t : t ≥ 0} is a Brownian motion.

wobei ξ eine in einem vorangegangen Berechnungsschritt ermittelte Realisierung von

ist.Further, the development of the stochastic size is preferably as follows:

where ξ is a realization, determined in a preceding calculation step, of

is.

In Another particularly preferred embodiment will after each Calculation step determined function value with that in the previous Calculation step determined function value compared and the Increment is increased when the determined function value is less than or equal to the one in the previous one Calculation step determined function value is. Is not this In the case, the step size is reduced. It has become at the Implementation of the method shown that thereby the term shortened the procedure can be.

In a further preferred embodiment After a predetermined number of calculation steps, the Difference between the maximum and the minimum function value of the previous ones Calculation steps determined and the step size is reduced, if it is less than a predetermined value. It has become the implementation of the method shown that thereby a Stagnation of the process in a local minimum repealed faster can be.

wobei ε > 0 und Φ σ / i die Entwicklung der stochastischen Größe im i-ten Berechnungsschritt ist und k die Anzahl der bereits durchgeführten Berechnungsschritte ist.In a further embodiment of the invention, the number of calculation steps carried out is determined by the following termination criterion:
Convergence of the expression:

where ε> 0 and Φ σ / i is the evolution of the stochastic quantity in the ith calculation step and k is the number of calculation steps already performed.

The Inventors were able to prove that the objective function values in the previous Calculation steps were close to the global optimum at least once, if the convergence of the above expression is ensured. This was the term chosen so that greater weight is placed on smaller functional values.

Next the method described above for the design of a technical system the invention further relates to an arrangement for the design of a technical System with a processor unit that is set up in such a way that the inventive method feasible is. Furthermore, the invention comprises a computer program product, which has a storage medium on which a computer program is stored is, wherein the inventive method carried out is when the computer program runs on a computer.

embodiments The invention will be described below with reference to the accompanying drawings described.

It demonstrate:

1A to 1C a diagram illustrating the sequence of the method according to the invention; and

2 a schematic representation of the processor unit, with which the inventive method is feasible.

The inventive method to design a technical system is based on a procedure for global optimization with stochastic integration, which in Reference [1] is described. To understand the invention it is first necessary, some mathematical foundations from probability theory and the theory of stochastic differential equations.

The The following description of the basics of stochastics comes up the documents [2], [3] and [4] return:

Notation:

^N B denote for n ε N is the Borel-σ-algebra in R ^n, that is, from the open volumes in R ⁿ generated σ algebra, B _{[a, b],} a, ε R b with a <b, the Borel σ algebra on the interval [a, b] and λ ⁿ the n-dimensional Lebesgue measure. Furthermore, the amount forms B : = {A ∈ P ( R ): A ∩ R ε B} is a σ-algebra R ,

In the following, only a special probability space - the Vienna area - is considered. This is the space Ω ^{n of} all continuous functions F: R + / 0 → R ⁿ with the σ-algebra B (Ω ⁿ ). In this case, B (Ω ⁿ ) designates the smallest of all σ-algebras S over Ω ⁿ , so that the projections p _t : Ω ⁿ → R ⁿ , F → F (t), t ∈ R + / 0, SB ⁿ -measurable , In order to be able to define a measure - the Viennese measure - in this measuring room, we need some definitions.

Definition 1 (P-almost everywhere, P-almost sure)

It let (Ω, S, P) a measure room. One property applies to P-almost everywhere, if there is a set A of dimension 0, leaving the property outside from A applies. Is (Ω, S, P) is a probability space, a property P-fast holds sure, if for a lot of measure 1 applies.

Definition 2 (numeric Function)

A function g: A → defined on a non-empty set A ⊆ Ω ⁿ R is called numeric function.

Definition 3 (n-dimensional Random variable)

Let n ∈ N. A function X: Ω ⁿ → R ⁿ is called an (n-dimensional) random variable if it is B (Ω ⁿ ) -B ⁿ measurable.

Becomes run a random experiment, so a random variable shows the outcome of this experiment as n-dimensional real vector. This is called a realization the random variable X.

One central term for the inventive method is the stochastic process.

Definition 4 (stochastic Process)

Let X _t : Ω ⁿ → R ^{n be} an n-dimensional random variable for each t ∈ I, where I is an index set. Then the set {X _t : t ∈ I} is called a stochastic process.

Definition 5 (Path of a stochastic process)

Let {X _t : t ≥ 0} be a stochastic process. Then the function X ^{ω is called} : R + / 0 → R ⁿ , t ↦ X _t (ω), ω ε Ω ⁿ , path of the stochastic process {X _t : t ≥ 0}.

The Continuity of the stochastic process is considered path by path.

Definition 6 (continuity a stochastic process)

Let {X _t : t ≥ 0} be a stochastic process. Then {X _t : t ≥ 0} is called continuous if every path of {X _t : t ≥ 0} is continuous.

are all information from the past in the current σ-algebra contain, so receives an ascending sequence of σ-algebras, which is called filtration.

definition 7 (Filtration)

Let (Ω, S, P) be a probability space, U ⊆ R and {F _s : s ∈ U} a set of σ-algebras on Ω with F _s ⊆ S for all s ε U. Let F _s ⊆ F _r for all s, r ε U with s ≤ r. Then {F _s : s ε U} is called a filtration in S.

Definition 8 (adapted)

Let (Ω, S, P) be a probability space U ⊆ R and {X _s ε U} a stochastic process and {F _s : s ∈ U} a filtration in S. Then the stochastic process {X _s : s ε U} of the filtration {F _s : s ε U} adapted if for each s ε U, the random variable X _s F _s -B ^n- measurable.

The canonical filtration of a stochastic process {X _s : s ε U} is the filtration {S _s : s ε U}, given by S _s : = σ (X _t : t ≤ s) for all s ε U. Here denote σ (X _t : t ≤ s) is the smallest of all σ-algebras S, so that all random variables X _t , t ≤ s, SB ^{n are} measurable.

Theorem 9 (integration of continuous stochastic processes)

stetig.Let n ∈ N and {X _t : t ≥ 0} with X _t : Ω ⁿ → R ⁿ , t ∈ R + / 0, be a continuous stochastic process. Then the stochastic process {Y _t : t ≥ 0} is given for every continuous function g: R ⁿ → R ⁿ

steadily.

Now an important stochastic process - the Brownian motion - is defined. First, consider them in general spaces, then a sentence follows that says that it is a measure of (Ω ⁿ , B (Ω ⁿ )) gives, so that a Brownian movement exists.

Definition 10 (Brownian motion on (Ω, S, P))

• i) X₀ = 0 P-fast sicher,
• ii) Für jedes k ∊ N und für jedes Tupel (t₁, t₂, ..., t_k)^T ∊ R^k mit 0 ≤ t₁ < t₂ ... <t_k sind die Zufallsvariablen
  
  tochastisch unabhängig.
• iii) für alle s, t ∊ R + / 0, s < t, ist die Zufallsvariable X_t – X_s N(0, (t-s)E_n)- normal verteilt, wobei E_n die n-dimensionale Einheitsmatrix bezeichne,
• iv) alle Pfade von {X_t : t ∊ R + / 0} sind stetig.

Let (Ω, S, P) a probability space and X _t: Ω → R ⁿ for every t ε R + / 0 SB ⁿ -measurable. Then the stochastic process is called {X _t : t ∈ R + / 0} n-dimensional Brownian motion (Ω, S, P), if applicable

• i) X ₀ = 0 P-almost sure
• ii) For each k ε N and for each tuple (t ₁ , t ₂ , ..., t _k ) ^T ε R ^k with 0 ≤ t ₁ <t ₂ ... <t _k are the random variables
  
  tochastically independent.
• iii) for all s, t ∈ R + / 0, s <t, the random variable X _t - X _s N (0, (ts) E _n ) - is normally distributed, where E _{n is} the n-dimensional unit matrix,
• iv) all paths of {X _t : t ∈ R + / 0} are continuous.

The following sentence states that there is a measure W ⁿ in the Vienna area, so that a Brownian motion exists.

Sentence and definition 11 (Wiener measure, Wiener space)

• i) B₀ = 0 Wⁿ-fast sicher,
• ii) für alle t₀, t₁, ... t_m, m ∊ N mit 0 = t₀ < t₁ < ... < t_m sind die Zufallsvariablen
  
  stochastisch unabhängig.
• iii) für alle s, t ∊ N mit 0 ≤ s < t ist die Zufallsvariable B_t-B_s N(0, (t-s)E_n)- normal verteilt, wobei E_n die n-dimensionale Einheitsmatrix bezeichne.

Let (Ω ⁿ , B (Ω ⁿ )) be the space of all continuous functions F: R + / 0 → R with the σ-algebra B (Ω ⁿ ). Here B (Ω ⁿ ) denotes the smallest of all σ-algebras S over Ω ⁿ , so that the projections p _t : Ω ⁿ → R ⁿ , F ↦ F (t), t ∈ R + / 0, SB (Ω ⁿ ) are measurable. Let {B _t : t ≥ 0} be a stochastic process with B _t (ω) = ω (t) for all t ≥ 0 and ω ε Ω ⁿ . Then for every n ∈ N there exists exactly one measure W ⁿ on (Ω ⁿ , B (Ω ⁿ )), called the Viennese measure, with the following properties

• i) B ₀ = 0 W ⁿ -most sure
• ii) for all t ₀ , t ₁ , ... t _m , m ∈ N with 0 = t ₀ <t ₁ <... <t _m are the random variables
  
  stochastically independent.
• iii) for all s, t ε N with 0 ≤ s <t, the random variable B _t -B _s N (0, (ts) E _n ) - is normally distributed, where E _{n is} the n-dimensional unit matrix.

The probability space (Ω ⁿ , B (Ω ⁿ ), W ⁿ ) is called Viennese space. Thus, the following definition is mandatory.

Definition 12 (Brownian movement (Ω ^n, B (Ω ^n), W ⁿ⁾⁾

The stochastic process {B _t : t ≥ 0} with B _t (ω) = ω (t) for all ω ε Ω ⁿ and t ε R + / 0, n ε N, is called the n-dimensional Brownian motion on (Ω ⁿ , B (Ω ⁿ ), W ⁿ ).

Now that the measure space (Ω ⁿ , B (Ω ⁿ ), W ⁿ ) has been defined, the distribution and the expected value of a random variable can be defined on this measure space.

Definition 13 (distribution the random variables)

Let X: Ω ⁿ → R ^{n be} a random variable. Then the measure μ on the measuring space (R ⁿ , B (R ⁿ )) is called
μ (A): = W ⁿ ({w ε Ω ⁿ : X (ω) ∈ A})
for all A ε B ⁿ the distribution of X.

Definition 14 (expected value a random variable)

existiert, wird es Erwartungswert der Zufallsvariablen X genannt und mit E(X) bezeichnet.Let X: Ω ⁿ → R ^{n be} a random variable. If the integral

it is called the expected value of the random variable X and denoted by E (X).

Next the just defined foundations of stochastics is to understand the Invention also requires the definition of the stochastic Ito integral, since this integral in the stochastic used in the process according to the invention Differential equation is used. For a more detailed account of the stochastic integrals is to be found in references [4] and [5] directed.

Definition 15 (elementary stochastic process)

• i) {X_s : s ∊ [a, b]} ist der Filtration {F_s : s ∊ [a, b]} adaptiert.
• ii) Es existieren t₀, ..., t_k ∊ [a, b] mit a = t₀ < t₁, ..., < t_k = b derart, dass
  
  für alle s ∊ [t_i, t_i+1 [, i ∊ {0, ..., k –1} und alle ω ∊ Ω.
• iii)
  
  .

Let (Ω, S, P) be a probability space, [a, b] a closed interval with a, b ∈ R + / 0, a <b and {F _s : s ∈ [a, b] a filtration in S. Then the stochastic process is called {X _s : s ε [a, b]} elementary stochastic process with respect to {F _s : s ε [a, b]} on (Ω, S, P) if the following conditions are fulfilled:

• i) {X _s : s ε [a, b]} is adapted to the filtration {F _s : s ε [a, b]}.
• ii) There exist t ₀ , ..., t _k ε [a, b] with a = t ₀ <t ₁ , ..., <t _k = b such that
  
  for all s ∈ [t _i , t _{i + 1} [, i ∈ {0, ..., k -1} and all ω ∈ Ω.
• iii)
  
  ,

elementary Stochastic processes are - figuratively speaking - such stochastic processes whose paths are staircase functions.

For elementary stochastic processes define the stochastic integral as follows:

Definition 16 (stochastic Integral for elementary processes)

für alle s ∊ [t_i, t_i+1 [, i ∊ {0, ..., k –1} und alle ω ∊ Ω. Dann definiert man die C_b-B-messbare Zufallsvariable I_x mit

als stochastisches Integral von {X_s : s ∊ [a, b]}. Wir bezeichnen I_x mit

Let (Ω, S, P) be a probability space, {B _t : t ≥ 0} a one-dimensional Brownian motion on (Ω, S, P) with canonical filtration {C _t : t ≥ 0} and {X _s : s Ε [a, b]} is an elementary stochastic process with respect to {C _s : s [a, b]} with e, a ∈ R + / 0, a <b. Further, let {t ₀ , t ₁ , ..., t _k } be a decomposition of [a, b] such that

for all s ∈ [t _i , t _{i + 1} [, i ∈ {0, ..., k -1} and all ω ∈ Ω. Then one defines the C _b -B measurable random variable I _x

as a stochastic integral of {X _s : s ε [a, b]}. We denote I _x with

The stochastic integral for arbitrary stochastic processes {X _s : s ε [a, b]} with sup _a≤s≤b E (|| X _s || 2/2) <∞ is given by a kind of border crossing of the integrals of the elementary ones stochastic processes that approximate the stochastic process {X _s : s ε [a, b]}. This approximation is done in three steps: First, {X _s : s ε [a, b]} is approximated by stochastic processes with uniformly bounded paths, and these in turn by continuous stochastic processes. All stochastic integrals use this approach. The difference between the different stochastic integrals lies in the third step, the approximation of the continuous stochastic processes by elementary stochastic processes.

The Ito-Integral can for following stochastic processes are defined:

Definition 17 (Quantity of approximable stochastic processes)

• i) Y_s : Ω → R ist für jedes s ∊ [a, b] F_s-B-messbar.
• ii) Die Abbildung Y: [a, b] x Ω → R, (s,ω) ↦ Y_s (ω) ist B_{[a, b]} × S-B-messbar.
  

Let (Ω, S, P) be a probability space, [a, b] a closed interval with a, b ∈ R + / 0, a <b, and {F _s : s ∈ [a, b]} a filtration in S Then consider the set Π of all stochastic processes {Y _s : s ε [a, b]} with the following properties:

• i) Y _s : Ω → R is measurable for each s ε [a, b] F _s -B.
• ii) The mapping Y: [a, b] x Ω → R, (s, ω) ↦Y _s (ω) is B _{[a, b]} × SB-measurable.
  

The set Π is called the set of stochastic processes approximated with {F _s : s ε [a, b]}.

The integral for matrix-valued elementary stochastic processes can be represented in a similar way as just described. One calls the Ito integral I _t with

The For its part, Ito-Integral is again a steady stochastic one Process. It is used in the definition of stochastic differential equations.

Under With the aid of the principles set out above, it is now possible to define what is under a stochastic differential equation too understand is.

Again Let n, m ε N.

Definition 18

eine Itô-stochastische Differentialgleichung.Let a, b ∈ R with a <b and U ⊂ R ⁿ . Let X ~: Ω ⁿ → R ^{n be} a random variable with E (|| X ~ || 2/2) <∞. Let {B _t : t ε R + / 0} be an m-dimensional Brownian motion. Let the function h: [a, b] × R ⁿ → R ⁿ be B _{[a, b]} × B ⁿ -B ⁿ measurable, the function G: [a, b] × R ⁿ → R ^{(n, m)} let B _{[a, b]} × B ⁿ -B ^{n be} measurable. Then it means

an Itô-stochastic differential equation.

Instead of (1) one often uses the spelling dX t = h (t, X t ) dt + G (t, X t ) dB t , tε [a, b], X a = X ~.

H (t, X _t ) dt is called the drift coefficient, G (t, X _t ) dB n / t is called the diffusion coefficient.

Now It must be defined what is meant by a solution of (1). About that gives the following definition:

Definition 19 (Solution of stochastic differential equation)

• i) sup E(||X_s|| 2 / 2) < ∞
• ii) Für alle a ≤ t ≤ b ist (1) erfüllt, das heißt insbesondere die Integrale auf der rechten Seite von (1) existieren.

A stochastic process {X _s : s ε [a, b]} is called solution of (1) if

• i) sup E (|| X _s || 2/2) <∞
• (ii) For all a ≤ t ≤ b, (1) is satisfied, that is, in particular, the integrals on the right side of (1) exist.

A solution is called permissible if {X _s : s ε [a, b]} is continuous.

For the stochastic Differential equation according to the Itô calculus a condition for the existence of a solution can be given, where waived the specification of the exact condition at this point becomes.

In the method according to the invention is considered a stochastic process called a Markov chain. Therefore, below are definitions regarding Markov chains specified. The following definitions are as far as possible Taken from publication [6].

in the Let X be an arbitrary topological space and let B (X) be the corresponding Borel σ algebra X. First is the definition of a Markov process with the corresponding transition probabilities to clarify.

Definition 20 (transition probabilities)

• i) für jedes A ∊ B(X) ist P(·,A) eine nicht-negativ messbare Funktion auf X,
• ii) für jedes x ∊ X ist P(x,·) ein Wahrscheinlichkeitsmaß auf B(X).

Let P = {P (x, A): x ∈ X, A ∈ B (X)} given by

• i) for each A ε B (X), P (·, A) is a non-negatively measurable function on X,
• ii) for each x ε X, P (x, ·) is a probability measure on B (X).

Then is called P transition probability kernel.

With Help a transitional probability kernel Now a Markov chain can be defined.

Definition 21 (Markov chain)

für alle n ∊ N erfüllt.A stochastic process Φ: = {Φ _t : t ε N ₀ } defined on (X, B (X)) is called a time-homogeneous Markov process with initial distribution μ and transition probability kernel P, if the distribution of Φ

for all n ε N met.

In the following, the following concrete problem to be solved is considered. The technical system to be designed is described by a scalar objective function F: R ⁿ → R, F ε C ² , wherein the parameters x = (x1,..., Xn) ε R ^{n of} the technical system are optimally set when the Target function f (x) has a global minimum. The problem is thus to find a point x * ε R ⁿ , for which the following applies: f (x *) ≤ f (x) for all x ε R n ,

to solution This problem will be explained below on the procedure of Reference [1] based method is used, with objective functions which meet the following conditions A:

Condition A:

für alle x ∊ R \ {x ∊ Rⁿ : ||x||₂ ≤ r}, wobei ∇f (x) den Gradienten der Zielfunktion f : Rⁿ → R an der Stelle x bezeichne.There exists an ε> 0 such that there is an r> 0 with

for all x ∈ R \ {x ∈ R ⁿ : || x || ₂ ≤ r}, where ∇f (x) denotes the gradient of the objective function f: R ⁿ → R at x.

Condition A ensures that the objective function is located outside a sphere of radius r around the origin along all directions s with || s || ₂ = 1 rises sufficiently fast. This means that the function f has a global optimization. The inversion does not generally apply (see the function f: R → R, x ↦ sin (x)).

If the objective function f does not satisfy condition A, use the auxiliary function

This auxiliary function f has the following properties: f Ε C ² f (x) = f (x) for all x ε {x ∈ R ⁿ : || x || 2/2 ≤ c},
f (x)> f (x) for all x ε R ⁿ with || x || 2/2> c,
f meets requirement A.

Using the definitions given above, the following stochastic differential equation can be defined:

It can be proved (see document [1]) that every path of the solution process this differential equation is infinitely close in finite time the global optimum x * approaches f.

wobei ξ eine in einem vorangegangen Berechnungsschritt ermittelte Realisierung von

ist.In order to solve the stochastic differential equation (2), the semi-implicit Euler method is proposed in document [1], which determines a new point for a given step size σ

where ξ is a realization, determined in a preceding calculation step, of

is.

It can be proved that equation (3) for small σ represents an approximation of the solution of the stochastic differential equation (2). In the method according to the document [1], the step size σ is controlled as follows:
Select a start step size σ and determine from this step on the basis of ξ a new point Φ ~ _σ (ξ). Then you repeat from ξ an Euler step, but with increment σ / 2. From this newly generated point Φ ~ _{σ / 2} (ξ) one then performs another Euler step - also with step size σ / 2 - and obtains Φ ~ _{σ / 2} (Φ ~ _{σ / 2} (ξ)) Φ ~ _σ (ξ) and Φ ~ _{σ / 2} (Φ ~ _{σ / 2} (ξ)) maximally a given δ, one accepts Φ ~ _σ (ξ) and maintains the step size. If the difference exceeds δ, the step size is halved and the three specified Euler steps are executed again.

These Step size control is based on the goal, if possible, the exact solution of (2), which is not the case for global optimization of need is and therefore unnecessary means numerical effort.

on Interesse ist, sondern lediglich die Erzeugung von Zufallszahlen gemäß einer Verteilung, bei der ein relativ großes Gewicht auf den globalen Minimierer der Zielfunktion f liegt. Deshalb wurde von den Erfindern untersucht, mit welcher Wahl der Schrittweite σ die Stabilitätseigenschaften erhalten bleiben, und zwar in dem Sinn, dass die vom semi-implizierten Euler-Verfahren in duzierte Markov-Kette eine stationäre Verteilung n besitzt. Hierzu wird die Vorgehensweise des semi-impliziten Euler-Verfahren nicht mehr länger als numerische Lösung der Stochastischen Differentialgleichung (2) aufgefasst, sondern als ein eigenes theoretisches Verfahren.In contrast to the method of document [1], another step size control is used in the method according to the invention. This step size control is based on the insight that for the purpose of global optimization is not the exact solution

is interested, but only the generation of random numbers according to a distribution in which a relatively large weight is on the global minimizer of the objective function f. Therefore, the inventors investigated which choice of step size σ preserves the stability properties, in the sense that the Markov chain induced by the semi-implied Euler method has a stationary distribution n. For this purpose, the procedure of the semi-implicit Euler method is no longer regarded as a numerical solution of the stochastic differential equation (2), but as a separate theoretical method.

Thus one uses a step-size function σ: R ⁿ → R, σ (ξ)> 0 for all ξ ε R ⁿ , and generates a Markov chain Φ ^σ : = {Φ σ / t, starting from the starting point or from the realization ξin the previous step. t ε N} according to the Education Act

The goal is to determine the function σ: R ⁿ → R, σ (ξ)> 0 for all ξ ε R ⁿ , depending on the starting point ξ, so that the Markov chain resulting from the rule (4) already has its own Property has an invariant probability measure and to converge in distribution against a random variable, which is distributed according to this invariant probability measure. If such a function σ can be found, a step size control can be used, which is determined a priori and not determined from the method itself.

The inventors were able to prove that the function σ must be substantially continuous and to the objective function f, in addition to condition A, the following requirement B must be provided:

Condition B:

There exists a r 'ε R, r'> 0, and a continuous function σ R ⁿ → R, σ (x)> 0 for all x ε R ⁿ , so that for all x ε R ⁿ the matrix A _{σ, x} : = E _n + σ (x) ∇ ² f (x) is invertible and for all x ∈ R ⁿ with || x || ₂ > r 'applies

Especially the inventors could prove that by such a choice of the Function σ that Method is stochastically stable, that is that the Markov chain against a boundary distribution converges. Furthermore, numerical simulations have approved, that by the step size control according to the invention whatever the global optimum of the function f is found.

wobei ε > 0 und Φ σ / i die Entwicklung der stochastischen Größe im i-ten Berechnungsschritt und k die Anzahl der bereits durchgeführten Berechnungsschritte ist.In addition, the inventors could define a suitable termination criterion in which it can be assumed that at least one of the functional values determined using the method according to the invention was close to the global optimum of the function. The abort criterion here is the convergence of the following size:

where ε> 0 and Φ σ / i is the evolution of the stochastic quantity in the ith calculation step and k is the number of calculation steps already performed.

The following is based on 1 a concrete implementation of the method according to the invention described.

First, in an initialization step 101 the following values are set:
k: = 1
f _max : = f (Φ σ / 0)
f _min : = f (Φ σ / 0)
σ: = σ _s
Φ σ / 0: = X _s

bestimmt.In this case, arbitrary starting points x _s ∁ R ⁿ and an arbitrary starting step size σ _s ε R are determined. In addition, a block size 1 for investigating the variation of the function values and a minimum acceptable variable size δ and a weighting factor ε are set. In the next step 102 become the derivatives

certainly.

ermittelt. Ist diese Größe positiv-definit, ist die Voraussetzung B erfüllt und es wird zu Schritt 104 übergegangen. Anderenfalls wird σ halbiert und nachmals zu Schritt 103 zurückgegangen.Subsequently, in step 103 the size

determined. If this size is positive-definite, condition B is fulfilled and it becomes step 104 passed. Otherwise σ is halved and then step by step 103 declined.

In step 104 Then a normally distributed random number is determined by calculating a realization N * of an N (0, σ E _n ) Gaussian distributed random vector. This corresponds to the simulation of a Brownian motion as used in equation (4). Subsequently, in step 105 the random number Φ σ / k is calculated by means of the solution of the following linear system of equations, which represents a transformation of equation (4):

Subsequently, Φ σ / k is defined as follows:

ist. Falls ja, wird die Schrittweite verdoppelt (Schritt 107), das heißt σ := 2σ, und es wird überprüft, ob f(Φ σ / k) < f_min ist. Ist dies der Fall, wird f_min auf f(Φ σ / k) gesetzt. Ist hingegen

wird die Schrittweite halbiert, das heißt σ := 1/2 σ (Schritt 108). Ferner wird f_max auf f(Φ σ / k) gesetzt, sofern f(Φ σ / k) > f_max ist.In practice it has been shown that the running time of the method can be shortened if the step size is increased, provided that the function value of the newly determined point is less than or equal to the function value of the current starting point. Therefore, in the next step 106 checked if

is. If so, the step size is doubled (step 107 ), ie σ: = 2σ, and it is checked whether f (Φ σ / k) <f _min . If this is the case, f _{min is set} to f (Φ σ / k). Is on the other hand

the step size is halved, that is σ: = 1/2 σ (step 108 ). Further, f _{max is set} to f (Φ σ / k) if f (Φ σ / k)> f _max .

Then it becomes step 109 in which it is checked whether 1 calculation steps of the method have already been passed, that is, whether k = 0 mod l. If this is the case, it is checked whether the difference between f _max and f _{min is} smaller or equal to δ. If this is the case, σ is set to 1/5 σ and f _max and f _{min are set} to f (Φ σ / k). The step 109 is performed because it has been shown in practice that decreasing the step size can more quickly override stagnation of the algorithm in a local minimizer.

Finally, in step 110k set to k + 1 and go to step 102 declined.

The just described implementation of the method is repeated until a predetermined termination criterion is met, wherein the termination criterion is in particular the criterion according to equation (5). After the method aborts, the function value f _{min is} subjected to a local minimization procedure using known prior art minimization techniques. The objective function value obtained after the minimization procedure is then the global minimum of the function f. The implementation of the method was simulated for different objective functions and it has actually been shown that the global minimum of the objective function is always found.

The inventive method has opposite the method of document [1] the advantage that the step size control Much less numerical effort means fewer calculation steps to determine a new increment must be performed.

In 2 a processor unit PRZE for carrying out the method according to the invention is shown. The processor unit PRZE comprises a processor CPU, a memory MEM and an input / output interface IOS, which is used in different ways via an interface IFC. Via a graphic interface, an output is visible on a monitor MON and / or on a printer PRT issued. An entry is made via a mouse MAS or a keyboard TAST. Also, the processor unit PRZE has a data bus BUS, which ensures the connection of a memory MEM, the processor CPU and the input / output interface IOS. Furthermore, additional components can be connected to the data bus BUS, eg additional memory, data memory (hard disk) or scanner.

Bibliography:

• [1] S. Schäffler: Global Optimizing Using Stochastic Integration, S. Roderer Verlag, Regensburg 1995
• Schäffler, T. F. Sturm: Probability Theory and Statistics I. Series of the institute for Applied Mathematics and Statistics of the Technische Universität München Nr. 5, October 1994
• S. Schäffler, T. F. Sturm: Probability Theory and Statistics II. Series of the institute for Applied Mathematics and Statistics of the Technische Universität München Nr. 6th March 1995
• [4] B. Øksendal: Stochastic Differential Equations - An Introduction with Applications, Springer, Berlin, Heidelberg 1998
• [5] P. Protter: Stochastic Integration and Differential Equations, Berlin, New York 1990
• [6] S.P. Meyn, R.L. Tweedie: Markov-Chains and Stochastic Stability. Springer-Verlag Berlin Heidelberg New York, 1996

Claims (10)

Method of designing a technical system - in which the technical system comprises an objective function (f (x)), which is derived from depends on at least one parameter (x) in a variable space; - in which Objective function values determined iteratively by several calculation steps with one in each calculation step of the objective function (f (x)) dependent and development based on a stochastic differential equation a stochastic size, in particular a Markov chain, is calculated; - in which the development of the stochastic size of one Step size (σ (x)) depends which is chosen that development with increasing number of calculation steps against a stationary one stochastic distribution converges; - in the case of a number carried out by Calculation steps a local optimization method for determining an optimal functional value in terms of an optimality criterion carried out is, where as the starting value of the local optimization method a the number of performed Calculation steps determined target function values is used; - in which the optimal objective function value determined by the local optimization method and the associated ones Parameters are used for the design of the technical system. The method of claim 1, wherein the parameters are continuous parameters and the objective function f (x) is a scalar-valued function f: R ⁿ -> R. Method according to Claim 2, in which the objective function (f (x)) satisfies the following requirements: There exists an ε> 0 such that there is an r> 0 with

for all x ∈ R \ {x ∈ R ⁿ : || x || ₂ ≤ r}, where ∇f (x) denotes the gradient of the objective function f: R ⁿ → R at x; and there exists a r 'ε R, r'> 0, and a continuous function σ: R ⁿ → R, σ (x)> 0 for all x ε R ⁿ , so that for all x ε R ⁿ the matrix A _{σ , x} : = E _n + σ (x) ∇ ² f (x) is invertible and for all x ∈ R ⁿ with || x || ₂ > r 'applies

where σ (x) is the step size. Method according to one of the preceding claims, wherein the stochastic differential equation is as follows:

where f (x) is the objective function and σ: R ⁿ -> R is a step-size function, {Φ _t ^{ε, x} 0 (ω)} is a stochastic process and {B _t : t ≥ 0} is a Brownian motion. The method of claim 4, wherein the evolution of the stochastic size is as follows:

where ξ is a realization, determined in a preceding calculation step, of

is. Method according to one of the preceding claims, in the function value determined after each calculation step (f (x)) with the function value determined in the preceding calculation step is compared and the step size (σ (x)) is increased when the determined Function value less than or equal to that in the previous calculation step determined function value, and otherwise the step size (σ (x)) reduced becomes. Method according to one of the preceding claims, wherein after a predetermined number of Calculation steps, the difference of the maximum and the minimum function value, which were calculated in the previous calculation steps, is determined and the step size (σ (x)) is reduced, if the difference is smaller than a predetermined value. Method according to one of the preceding claims, in which the number of calculation steps carried out is determined by the following termination criterion: Convergence of the expression:

where ε> 0 and Φ σ / i is the evolution of the stochastic quantity in the ith calculation step and k is the number of calculation steps already performed. Arrangement for the design of a technical system with a processor unit configured such that a method according to one of the preceding claims is feasible. Computer program product, which is a storage medium on which a computer program is stored with A method according to any one of claims 1 to 8 is carried out, when the computer program runs on a computer.