CN112199849B - High-dimensional problem optimization method based on information entropy online decomposition - Google Patents
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Abstract
The invention provides a high-dimensional problem optimization method based on information entropy online decomposition, which is characterized in that decision variables of an optimization problem with n decision variables are randomly arranged to form m sub-problems, a covariance adaptive evolution strategy is used for performing one-time solution space sampling on each sub-problem, an edge probability model is established, evaluation of the model is calculated, iteration and optimization are carried out, the decision variables are regrouped by using an information entropy based method, and the optimization effect can be improved. Simulation results show that the method has a good optimization effect on high-dimensional problems and has a good engineering application prospect.
Description
Technical Field
The invention relates to the field of optimization of high-dimensional problems, in particular to a high-dimensional problem optimization method based on information entropy.
Background
In the high-dimensional problem, the problem search space is large due to the fact that many decision variables exist, and the complex interaction relation among the decision variables further improves the difficulty in solving the high-dimensional optimization problem. Grouping the decision variables, i.e., decomposing the high-dimensional problem into sub-problems with smaller dimensions, is an effective method for solving the high-dimensional problem. The grouping of variables has an important effect on the optimization. Generally, variable grouping randomly divides decision variables into a certain number of subgroups to form a sub-problem with lower dimensionality.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a high-dimensional problem optimization method based on information entropy online decomposition. In order to improve the optimization effect of the high-dimensional problem, the method is executed when the optimization algorithm is trapped in stagnation, and an edge distribution probability distribution model is established for the sampling individuals of the optimization algorithm. Firstly, regarding all variables as a subproblem, and calculating model evaluation of the subproblem; then, combining all the sub-problems step by step to generate a new grouping model, and calculating model evaluation; and selecting the grouping model with the minimum model evaluation value, and repeating the previous steps of combining, evaluating and selecting the minimum until the model evaluation value is not reduced any more. And finally, the grouping model with the minimum model evaluation is the result of variable grouping. The invention realizes the online grouping, namely grouping while optimizing, and the simulation result shows, and the method for grouping the decision variables according to the correlation among the decision variables is a method capable of improving the optimization effect of the high-dimensional problem.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1: the optimization problem miny ═ f (x) with n decision variables 1 ,x 2 ,…,x n ) The decision variables are randomly arranged, starting from the first randomly arranged decision variable sequence, every k decision variables form a subproblem, and M subproblems are formed, wherein n is k M, and the random grouping model is marked as M 0 (ii) a Randomly generating a solution
Calculating y * ＝f(X * ) (ii) a The initial evolution iter is 0,the evolution stagnation number stg is 0;
step 2: grouping model M 0 Using a covariance adaptive evolution strategy to perform one-time solution space sampling in each subproblem, combining solutions obtained by sampling of each subproblem according to a sampling sequence to obtain lambda complete solutions X 1 ,X 2 ,…,X λ And evaluating lambda complete solutions to obtain y 1 ,y 2 ,…,y λ Marking the solution with the minimum y value in the solutions obtained by current sampling as X b And an evaluation value y b (ii) a Evolution algebra is accumulated for one generation, and iter is added with 1; if y b <y * If stg is 0, X * ←X b ，y * ←y b (ii) a If y b ≥y * If stg is stg +1, X * 、y * Keeping the original shape;
and 3, step 3: if the evolution stagnation times stg are larger than the threshold value, performing a step 4; otherwise, returning to the step 2;
and 4, step 4: regarding n decision variables as n subproblems to form a grouping model M 0 ←([x 1 ][x 2 ]…[x n ]) (ii) a With X 1 ,X 2 ,…,X λ For a sample point, establish M 0 The edge probability model of (1);
and 5: calculation model M 0 Evaluation (5) C 0 The formula is as follows:
wherein, N is the number of individuals in the population, S [ I ] is the number of decision variables of the I-th sub-problem, E (I) is the information entropy of the edge distribution of the sub-problem I;
step 6: the current model M 0 Any two sub-problems are combined to generate t new grouping models M 1 ～M t (ii) a With X 1 ,X 2 ,…,X λ For a sample point, establish M 1 ～M t The edge probability model of (2); then, the model M is calculated using the formula (1) 1 ～M n Evaluation value C of 1 ～C n ；
And 7: will evaluate the value C 1 ～C n The minimum value in (1) is marked as C * Model with minimal evaluation labeled M * If C is * <C 0 Then C is added 0 Assigned a value of C * ，M 0 Assigned a value of M * I.e. C 0 ←C * 、M 0 ←M * And returning to the step 6; if C * ≥C 0 Then C is 0 、M 0 Keeping the state unchanged, and performing step 8;
step 8; outputting the current model M 0 Grouping the structure for the final variable;
and step 9: if iter is greater than or equal to T max ，T max Is the maximum number of times, output X * For the optimization problem to be solved min y ═ f (x) 1 ,x 2 ,…,x n ) And ending the optimal solution; if iter<T max Returning to the step 2; therefore, high-dimensional problems are randomly grouped, covariance self-adaptive evolution is utilized, decision variables are grouped again by utilizing a method based on information entropy, and optimization effect is improved.
The threshold value in step 3 is 1000.
The method has the advantages that firstly, high-dimensional problems are randomly grouped and optimized by using a covariance adaptive evolution strategy, when the optimal value is not updated for 1000 continuous generations, decision variables are regrouped by using an information entropy-based method, the interaction among the variables can be found by using an information entropy-based variable decomposition method, and the optimization effect can be improved by using the found variable grouping structure. Simulation results show that the method has a good optimization effect on high-dimensional problems. Therefore, the method has better engineering application prospect.
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FIG. 1 is a schematic flow diagram of the optimization method of the present invention.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
In order to verify the high-dimensional problem optimization method based on information entropy online decomposition proposed by the invention, two embodiments are listed here. The method provided by the invention is used in both embodiments, the embodiment 1 is standard test function optimization, and the embodiment 2 is unmanned underwater vehicle route planning.
Example 1: in this example, a 1000-dimensional Ackley test function is taken as an example to illustrate the execution steps of the present invention, and compared with a high-dimensional optimization method based on random grouping, the present invention illustrates the effect of the method provided by the present invention. The 1000 dimensional Ackley function is formulated as:
step 1: the decision variables of the 1000-dimensional Ackley function are labeled as [ x1, x2, x3, …, x1000], and the 1000 decision variables are randomly decomposed into 20 50-dimensional subproblems. The decomposed decision variables are grouped as shown in table 1.
TABLE 1 variable grouping results from random grouping
Step 2: and (3) performing cyclic optimization calculation on all the sub-problems by using a covariance adaptive evolution strategy as an optimization operator, and outputting the currently obtained optimal individual x and an optimal value f. The evolution algebra is accumulated to iter + 1.
And 3, step 3: if the found optimum value f is continuous G max If the generation is not changed, the step 4 is carried out; otherwise, returning to the step 2, and continuing to use the optimization operator to perform optimization calculation.
And 4, step 4: first, each variable is regarded as a subproblem, and the grouping structure is made to be an optimal grouping model M. Calculating the complexity C of the optimal model M 1 Entropy of information C 2 Model evaluation C 0 . Table 2 shows the grouping model and model evaluation when each variable is treated as a subproblem.
TABLE 2 Primary grouping model and model evaluation
| Grouping model | Model evaluation |
| [x1][x2][x3]……[x998][x999][x1000] | 3.79×10 3 |
And 5: and combining any two sub-problems of the current optimal model to generate a plurality of new grouping models, and calculating the evaluation of the new models. Table 3 shows the newly generated grouping models and model evaluations after some subproblems are combined.
TABLE 3 second generation grouping model and model evaluation
| Grouping model | Model evaluation |
| [x1,x2][x3]……[x998][x999][x1000] | 3.6×10 3 |
| [x1][x2,x3]……[x998][x999][x1000] | 3.5×10 3 |
| …… | …… |
| [x1][x2][x3]……[x998][x999,x1000] | 3.45×10 3 |
And 6: since the model [ x1] [ x2] [ x3] … … [ x998] [ x999, x1000] has the lowest model evaluation, the new optimal current grouping model M is selected, and step 4 is executed again on the optimal current grouping model M until the model evaluation value is not reduced any more. Outputting the final optimal current model [ x1, x2, x3]
[ x4, x5, x6] … … [ x998, x999, x1000] is the final variable grouping structure.
And 7: if iter<T max Using the sub-problem formed by the current grouping model to return to the step 2; if ter>T max And outputting a final optimization result: an optimal solution x, and an optimal function value f.
The specific parameters are set as follows: t is a unit of max ＝1000，G max ＝1000。
The optimization results of the 1000-dimensional Ackley function by using and not using the variable grouping high-dimensional optimization method based on the information entropy are shown in the table 4, the optimal value is 0, and the closer the optimization result is to 0, the better the optimization result is. It can be seen from the table that the proposed method can significantly improve the performance.
TABLE 4 comparison of the results of the Ackley function optimization
| Method | Based on random grouping | Information entropy variable decomposition |
| Optimizing result (function value) | 5.19E-13 | 12.2895 |
The method comprises the steps of firstly randomly grouping high-dimensional problems and optimizing by using a covariance adaptive evolution strategy, and when an optimal value is not updated for 1000 continuous generations, regrouping decision variables by using an information entropy-based method. The interaction between the variables can be found by using the variable decomposition method of the information entropy, and the optimization effect is improved. Simulation results show that the method has good optimization effect on high-dimensional problems. Therefore, the method has better engineering application prospect.
Example 2: in the embodiment, different effects of two methods on the route planning of the unmanned underwater vehicle under the influence of ocean currents are compared, wherein one method is an optimization algorithm based on random grouping, and the other method is an information entropy online decomposition-based method provided by the invention. The case compares which scheme optimizes the route to reach the target point in a shorter time.
The farther the underwater vehicle travels, the more waypoints need to be planned, and the problem of path optimization will be a high-dimensional problem. However, since the motion of the underwater vehicle is greatly influenced by ocean currents, if a better route is to be obtained, route points need to be obtained more densely, and the dimension of the route planning problem is also increased.
In this embodiment, the number of waypoints is 200, the longitudinal coordinate of each waypoint is a decision variable, that is, the optimized problem is 200 dimensions, and the ocean current change from the starting point to the terminal point is known.
TABLE 5 required time for flight path
| Method | Based on random grouping | Information entropy variable decomposition |
| Time required for flight path | 734.7 hours | 622 hours |
Table 5 gives the results of the path optimization for both methods. The time consumption of the route planned by the method is shorter, because the interaction between some route points can be identified by the information entropy online decomposition-based method, and some route points are combined into a sub-problem to be optimized so as to conform to the ocean current, so that a better optimization effect can be obtained.
Claims (2)
1. A high-dimensional problem optimization method based on information entropy online decomposition is characterized by comprising the following steps:
step 1: in the route planning of the unmanned underwater vehicle, the longitudinal coordinate of each route point is a decision variable, the number of the route points is n, the route points are n-dimensional, and the optimization problem miny with n decision variables is f (x) and 1 ,x 2 ,…,x n ) Is randomly arranged, function f (x) 1 ,x 2 ,…,x n ) Representing the time required by the route, starting from the first of a randomly arranged decision variable sequence, forming a subproblem by every k decision variables, forming M subproblems in total, wherein n is k M, and recording the random grouping model as M 0 (ii) a Randomly generating a solution
Calculating y * ＝f(X * ) (ii) a Initializing evolution algebra iter to be 0 and evolution stagnation times stg to be 0;
and 2, step: for packet model M 0 Using a covariance adaptive evolution strategy to perform one-time solution space sampling in each subproblem, combining solutions obtained by sampling of each subproblem according to a sampling sequence to obtain lambda complete solutions X 1 ,X 2 ,…,X λ And evaluating lambda complete solutions to obtainy 1 ,y 2 ,…,y λ Marking the solution with the minimum y value in the solutions obtained by current sampling as X b And an evaluation value y b (ii) a Evolution algebra is accumulated for one generation, and iter is added with 1; if y b <y * If stg is equal to 0, X * ←X b ，y * ←y b (ii) a If y b ≥y * If stg is stg +1, X * 、y * Keeping the original shape;
and step 3: if the evolution stagnation time stg is larger than the threshold value, performing the step 4; otherwise, returning to the step 2;
and 4, step 4: regarding n decision variables as n subproblems to form a grouping model M 0 ←([x 1 ][x 2 ]…[x n ]) (ii) a With X 1 ,X 2 ,…,X λ For a sample point, establish M 0 The edge probability model of (2);
and 5: calculation model M 0 Evaluation (C) of 0 The formula is as follows:
wherein, N is the number of individuals in the population, S [ I ] is the number of decision variables of the I-th sub-problem, E (I) is the information entropy of the edge distribution of the sub-problem I;
step 6: the current model M 0 Any two sub-problems are combined to generate t new grouping models M 1 ～M t (ii) a With X 1 ,X 2 ,…,X λ For a sample point, establish M 1 ～M t The edge probability model of (1); then, the model M is calculated using the formula (1) 1 ～M n Evaluation value C of 1 ～C n ；
And 7: will evaluate the value C 1 ～C n The minimum value in (1) is marked as C * Model with minimal evaluation labeled M * If C is * <C 0 Then C will be 0 Assigned a value of C * ，M 0 Assigned a value of M * I.e. C 0 ←C * 、M 0 ←M * And go back toReturning to the step 6; if C * ≥C 0 Then C is 0 、M 0 Keeping the state unchanged, and performing step 8;
step 8; outputting the current model M 0 Grouping the structure for the final variable;
and step 9: if iter is greater than or equal to T max ，T max Is the maximum number of times, output X * For the optimization problem to be solved, miny ═ f (x) 1 ,x 2 ,…,x n ) And ending the optimal solution; if iter<T max Returning to the step 2; therefore, high-dimensional problems are randomly grouped, covariance self-adaptive evolution is utilized, decision variables are grouped again by utilizing a method based on information entropy, and optimization effect is improved.
2. The high-dimensional problem optimization method based on information entropy online decomposition as claimed in claim 1, wherein:
the threshold value in step 3 is 1000.
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