CN117669705A - Constrained multi-objective evolution method based on three-stage optimization - Google Patents
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Abstract
The invention relates to a constrained multi-objective evolution method based on three-stage optimization. The constraint multi-objective optimization problem is converted into a single-objective optimization problem without constraint, so that the population passes through a large-scale infeasible area and approaches the Pareto front rapidly. Secondly, in the diversity optimization stage, the population is divided into a plurality of sub-populations, and better candidate solutions are further selected from each sub-population, so that the convergence is considered while the diversity is maintained. Finally, in the refining optimization stage, a steady-state-based evolution mode is adopted, and the feasibility, convergence and diversity of candidate solutions are considered, so that the overall quality of the population is further improved.
Description
Technical Field
The invention relates to the technical field of multi-objective evolutionary optimization algorithms, in particular to a constraint multi-objective evolutionary method based on three-stage optimization.
Background
Existing multi-stage-based methods consider objective functions and constraints separately, alleviating the contradiction between objective function optimization and constraint satisfaction to some extent, but they still have some drawbacks. The first stage of ToP uses constraint rules, i.e. the feasible solutions always have higher priority than the infeasible solutions, which, according to the above analysis, is unfavorable for the population to traverse large infeasible areas. The PPS, although ignoring constraints during the push search phase, maintains diversity while optimizing convergence, resulting in very limited computational resources left to the pull search phase and thus affecting the proportion of feasible solutions in the final solution set.
In recent years, related scholars have proposed a number of Multi-objective evolutionary methods (Multi-Objective Evolutionary Algorithm, MOEA) that achieve satisfactory results when solving the unconstrained Multi-objective optimization problem (unconstrained Multi-objective Optimization Problem, MOP), but do not yield the ideal solution set when processing CMOP due to the lack of constrained processing techniques (Constraint Handling Technique, CHT) for MOEA. To enable the method to solve for CMOP, scholars designed a wide variety of CHTs and combined with MOEAs to form a constrained multi-objective evolutionary method (Constrained Multi-Objective Evolutionary Algorithm, CMOEA). The introduction of the constraint can divide the search space into narrower discontinuous areas, so that the proportion of the feasible region in the search space is greatly reduced; it is also possible to leave part or the whole PF in the original MOP out of constraint. The effectiveness of CHT can therefore greatly affect the performance of CMOEA.
However, the common point of the present method considering CHT is that the searching strategy of the method is unchanged in the whole searching process, namely, the feasibility, the convergence and the diversity are optimized at the same time, however, the conflict exists among the three methods, and the problem of solving the CMOP is also solved.
Disclosure of Invention
The invention aims to overcome the defect that most of the existing methods are difficult to balance the targets of feasibility, convergence and diversity when solving the constraint multi-target optimization problem, and therefore provides a constraint multi-target evolution method (Three-Stages Evolutionary Algorithm for Constrained Multi-objective Optimization, CMOEA-TS) based on Three-stage optimization, so that the convergence, the diversity and the feasibility are optimized simultaneously under the condition of meeting the constraint.
In order to achieve the above purpose, the technical scheme of the invention is as follows: a constrained multi-objective evolutionary method based on three-phase optimization, comprising:
(1) Convergence optimization stage: converting the original constraint multi-objective optimization problem into an unconstrained single-objective optimization problem; the population is not hindered by an infeasible area in the convergence optimization stage, and convergence information is only used as an optimization target, so that the population approaches PF rapidly;
(2) Diversity optimization stage: after the population is close to PF, the population is divided into a plurality of sub-populations by using weight vectors which are uniformly distributed, so that the searching range of the population is enlarged;
(3) Refining: and finally, locally adjusting the population, eliminating a small part of possibly existing infeasible solutions, and further optimizing the convergence and diversity of the population.
In an embodiment of the present invention, the step (1) is specifically implemented as follows:
using the objective function of the candidate solution and as its convergence information CI:
wherein f i (x) For the ith objective function value of the candidate solution x, m is the number of objective functions, and the smaller CI (x) is, the better the convergence of the candidate solution x is;
to measure the convergence of the whole population, a low valley point change rate cr is introduced max And the ideal point change rate cr min The specific calculation mode is shown in the following formula:
where T is an iteration constant, r i k The i-th objective function value, l, of the low valley point at the kth iteration i k For the ith objective function value of the ideal point in the kth iteration, n is the size of the population, j represents the jth individual of the population, and the specific calculation mode is as follows:
if cr max And cr min The value of (2) is larger, which indicates that in the latest T iterations, the convergence optimization of the whole population is obvious; conversely, if cr max And cr min The smaller value of (2) indicates that the population as a whole is already closer to the PF; thus according to cr max And cr min As the basis for judging whether the population is converged, when cr max And cr min And when the convergence thresholds epsilon are smaller than the convergence threshold epsilon, switching from the convergence optimization stage to the diversity optimization stage.
In an embodiment of the present invention, the step (2) is specifically implemented as follows:
after the convergence optimization stage is finished, the objective function and the convergence information are used as convergence information in the convergence optimization stage, and the population at the moment is close to the PF but is converged at a local position in the middle of the PF; the diversity optimization stage aims at optimizing the diversity of the population, namely the universality and uniformity of candidate solution distribution, and a method for dividing the population by using weight vectors is provided for the purpose; in the target space, weight vectors consistent with the population scale are uniformly generated, and each weight vector satisfies the following formula:
where H is the number of divisions on each objective function; generating a reference vector by adopting a Deb and Jain method; next, each candidate solution in the population will be associated with its nearest reference vector, forming a sub-population; kth sub-population SP k Expressed as:
wherein P is the current population, x is a candidate solution in the population P, f (x) is an objective function vector, angle () is the angle between the two vectors, and N is the size of the population scale;
in the diversity optimization stage, selecting a feasible solution is prioritized; specifically, all the objective function values of the infeasible solutions are punished according to the constraint violation degree, and the punished infeasible solutions are placed in the dominant region of the low valley point; the punished objective function value is calculated as follows:
wherein x is nadir CV (x) is the constraint violation degree, ω (x), of the candidate solution in the population P for the low valley point ndir ) Is a weight vector;
in order to evaluate the candidate solutions in each sub-population, the weighted sum WS of the objective function value and the weight vector of the candidate solution is used as the basis for measuring the quality of the candidate solution, and the WS calculation mode is as follows:
wherein ω (x) k ) Is a weight vector;
if the population just forms N sub-populations in the current iteration round, only the candidate solution with the minimum WS from each sub-population is needed to be selected and reserved to the next generation; when the number of sub-populations is less than N, it means that a plurality of candidate solutions need to be selected in at least one sub-population, in which case the diversity information DI of the candidate solutions will be further compared, the DI is calculated as follows:
DI(x)＝||F(x)-F(x ref )||
wherein x is ref To the x th distance candidate solutionNear individuals, |·| is the euclidean distance of the corresponding vector;
the fitness function Fit of the individual in the diversity optimization stage is designed, and the calculation mode is as follows:
wherein WS is no For each sub-population SP k Is ranked according to ascending order of WS values.
In an embodiment of the present invention, the specific implementation manner of the step (3) is: optimizing the original population by adopting a steady-state-based evolution mode and adopting a local exploration mode; firstly, calculating constraint violation degrees, convergence information and diversity information of all candidate solutions in a population; selecting N parent candidate solutions by adopting a binary tournament mode according to the violation degree of the candidate solution constraint to construct a mating pool, and generating a child candidate solution y by adopting simulated binary intersection and polynomial variation; then, screening all candidate solutions with smaller violation degree than y constraint from the population to form a set Q 1 Randomly replacing Q with y 1 Is a single individual; finally, screening all candidate solutions with worse convergence and diversity than y from the population to form a set Q 2 Randomly replacing Q with y 2 Is a single individual.
Compared with the prior art, the invention has the following beneficial effects: the method converts the constraint multi-objective optimization problem into a single-objective optimization problem without considering constraint, so that the population passes through a large-scale infeasible area and approaches the Pareto front rapidly. Secondly, in the diversity optimization stage, the population is divided into a plurality of sub-populations, and better candidate solutions are further selected from each sub-population, so that the convergence is considered while the diversity is maintained. Finally, in the refining optimization stage, a steady-state-based evolution mode is adopted, and the feasibility, convergence and diversity of candidate solutions are considered, so that the overall quality of the population is further improved.
Drawings
FIG. 1 CMOEA-TS flow chart.
FIG. 2 is a flow chart of the diversity optimization phase environment selection.
Detailed Description
The technical scheme of the invention is specifically described below with reference to the accompanying drawings.
As shown in FIG. 1, the invention provides a constrained multi-objective evolution method (Three-Stages Evolutionary Algorithm for Constrained Multi-objective Optimization, CMOEA-TS) for Three-stage search, which specifically comprises the following steps:
convergence optimization stage: the original constraint multi-objective optimization problem is converted into an unconstrained single-objective optimization problem. The population is not hindered by an infeasible area in the convergence optimization stage, and only convergence information is used as an optimization target, so that the PF is approximated rapidly.
Diversity optimization stage: after the population is close to PF, the uniformly distributed weight vector is used to divide the population into a plurality of sub-populations, so as to enlarge the searching range of the population.
Refining: and finally, locally adjusting the population, eliminating a small part of possibly existing infeasible solutions, and further optimizing the convergence and diversity of the population.
The following is a specific implementation procedure of the present invention.
1. Convergence optimization stage
The invention uses the objective function of the candidate solution and its convergence information CI.
Wherein f i (x) The ith objective function for candidate solution xAnd the value m is the number of objective functions. A smaller CI (x) indicates a better convergence of the candidate solution x.
In order to measure the convergence degree of the whole population, the invention introduces a low valley point change rate cr max And the ideal point change rate cr min The specific calculation modes are shown in the formula (1) and the formula (2) respectively:
where T is an iteration constant, r i k The i-th objective function value, l, of the low valley point at the kth iteration i k For the ith objective function value of the ideal point in the kth iteration, the specific calculation modes are shown in the formula (3) and the formula (4):
if cr max And cr min The larger the value of (c), the more obvious the convergence optimization of the population as a whole in the last T iterations. Conversely, if cr max And cr min The smaller value of (2) indicates that the population as a whole is already closer to the PF. Thus according to cr max And cr min As the basis for judging whether the population is converged, when cr max And cr min When both are smaller than the convergence threshold epsilon, the CMOEA-TS will switch from the convergence optimization phase to the diversity optimization phase.
2. Diversity optimization stage
After the convergence optimization phase is completed, the objective function and the convergence information are used as the convergence optimization phase, and the population at the moment is close enough to the PF, but is converged at a local position in the middle of the PF. The diversity optimization stage aims at optimizing the diversity of the population, namely the universality and uniformity of candidate solution distribution, and the invention provides a method for dividing the population by using weight vectors. Uniformly generating weight vectors consistent with the population size in a target space, wherein each weight vector satisfies a formula (5) and a formula (6):
where H is the number of divisions on each objective function. The invention adopts the Deb and Jain method to generate the reference vector. Each candidate solution in the population will then be associated with its nearest reference vector, forming a sub-population. Kth sub-population SP k Can be expressed as:
where P is the current population and angle () is the angle between the two vectors.
In the convergence phase, CMOEA-TS can fully explore the infeasible area and the feasible area regardless of the existence of constraints. Since convergence and diversity must be built on top of feasibility, the present invention prioritizes the selection of feasible solutions in the diversity optimization stage. Specifically, the invention penalizes the objective function values of all the infeasible solutions according to the size of constraint violation, and the infeasible solutions after penalty are placed in the dominant region of the low valley point. The punished objective function value is calculated as follows:
wherein x is nadir Is a low valley point.
In order to evaluate the candidate solutions in each sub-population, the invention takes the weighted sum WS of the candidate solution objective function value and the weight vector as the basis for measuring the quality of the candidate solutions, and the WS calculation mode is as follows:
if the population forms exactly N sub-populations in the current iteration round, only the candidate solution with the minimum WS from each sub-population needs to be selected and reserved to the next generation. When the number of sub-populations is less than N, which means that a plurality of candidate solutions need to be selected in at least one sub-population, the invention will in this case further compare the diversity information DI of the candidate solutions. The DI calculation is as follows:
DI(x)＝||F(x)-F(x ref )|| (10)
wherein x is ref To the x th distance candidate solutionNear individuals, |·| is the euclidean distance of the vector.
According to the method thought, the fitness function Fit of the individual in the diversity optimization stage is designed. The calculation method is as follows:
wherein WS is no For each sub-population SP k Is ranked according to ascending order of WS values.
FIG. 2 is an example of diversity optimization stage environment selection. As shown in fig. 2 (a), 5 parent individuals were pooled with 5 child individuals, for a total of 10 individuals (a-J), from which the environmental selection required 5 individuals to be selected for retention to the next generation. As shown in fig. 2 (B), each individual is associated to the nearest reference vector, forming 4 sub-populations (SP 1-SP 4), where SP1 = { a }, SP2 = { B, G, H, I }, SP3 = { C, F, J }, SP4 = { D, E }. As shown in fig. 2 (c), WS of all individuals in each sub-population are calculated separately, and individuals in each sub-population are sorted in ascending order according to the value of WS, and the values of WS in SP1, SP2, SP3 and SP4 are minimized for individuals A, B, C and D, respectively, thus keeping these individuals to the next generation. Next, DI for WS-sub-individuals (H, G, F and E) in each sub-population are calculated and compared, where individual E possesses the greatest DI, thus retaining individual E to the next generation.
3. Refining optimization stage
After the first two stages of optimization, the CMOEA-TS has been optimized such that most of the candidate solutions in the population are already relatively close to the PF, while being widely distributed across the PF, but a small portion of the infeasible solutions may still exist in the population. Therefore, in the final refining stage, the invention adopts an evolution mode based on a steady state, and optimizes the original population by a local exploration mode. First, constraint violation degrees, convergence information and diversity information of all candidate solutions in the population are calculated. And then selecting N parent candidate solutions by adopting a binary tournament mode according to the size of the constraint violation degree of the candidate solutions to construct a mating pool, and generating a child candidate solution y by adopting simulated binary intersection and polynomial variation. Then, screening all candidate solutions with smaller violation degree than y constraint from the population to form a set Q 1 Randomly replacing Q with y 1 Is a single individual. Finally, screening all candidate solutions with worse convergence and diversity than y from the population to form a set Q 2 Randomly replacing Q with y 2 Is a single individual.
The invention can be applied to complex examples where multiple objective optimization problems need to be solved and constraints exist. For example, in engineering, it is often necessary to consider a number of conflicting objectives while satisfying various constraints, such as material cost, structural strength, weight, etc., where these criteria and constraints are converted into the form of objective functions and constraint functions and then solved as described above.
The above is a preferred embodiment of the present invention, and all changes made according to the technical solution of the present invention belong to the protection scope of the present invention when the generated functional effects do not exceed the scope of the technical solution of the present invention.
Claims (4)
1. A constrained multi-objective evolutionary method based on three-phase optimization, comprising:
(1) Convergence optimization stage: converting the original constraint multi-objective optimization problem into an unconstrained single-objective optimization problem; the population is not hindered by an infeasible area in the convergence optimization stage, and convergence information is only used as an optimization target, so that the population approaches PF rapidly;
(2) Diversity optimization stage: after the population is close to PF, the population is divided into a plurality of sub-populations by using weight vectors which are uniformly distributed, so that the searching range of the population is enlarged;
(3) Refining: and finally, locally adjusting the population, eliminating a small part of possibly existing infeasible solutions, and further optimizing the convergence and diversity of the population.
2. The constrained multi-objective evolutionary method based on three-phase optimization of claim 1, wherein step (1) is specifically implemented as follows:
using the objective function of the candidate solution and as its convergence information CI:
wherein f i (x) For the ith objective function value of the candidate solution x, m is the number of objective functions, and the smaller CI (x) is, the better the convergence of the candidate solution x is;
to measure the convergence of the whole population, a low valley point change rate cr is introduced max And the ideal point change rate cr min The specific calculation mode is shown in the following formula:
where T is an iteration constant, r i k The i-th objective function value, l, of the low valley point at the kth iteration i k For the ith objective function value of the ideal point in the kth iteration, n is the size of the population, j represents the jth individual of the population, and the specific calculation mode is as follows:
if cr max And cr min The value of (2) is larger, which indicates that in the latest T iterations, the convergence optimization of the whole population is obvious; conversely, if cr max And cr min The smaller value of (2) indicates that the population as a whole is already closer to the PF; thus according to cr max And cr min As the basis for judging whether the population is converged, when cr max And cr min And when the convergence thresholds epsilon are smaller than the convergence threshold epsilon, switching from the convergence optimization stage to the diversity optimization stage.
3. The constrained multi-objective evolutionary method based on three-phase optimization of claim 1, wherein step (2) is specifically implemented as follows:
after the convergence optimization stage is finished, the objective function and the convergence information are used as convergence information in the convergence optimization stage, and the population at the moment is close to the PF but is converged at a local position in the middle of the PF; the diversity optimization stage aims at optimizing the diversity of the population, namely the universality and uniformity of candidate solution distribution, and a method for dividing the population by using weight vectors is provided for the purpose; in the target space, weight vectors consistent with the population scale are uniformly generated, and each weight vector satisfies the following formula:
where H is the number of divisions on each objective function; generating a reference vector by adopting a Deb and Jain method; next, each candidate solution in the population will be associated with its nearest reference vector, forming a sub-population; kth sub-population SP k Expressed as:
wherein P is the current population, x is a candidate solution in the population P, f (x) is an objective function vector, angle () is the angle between the two vectors, and N is the size of the population scale;
in the diversity optimization stage, selecting a feasible solution is prioritized; specifically, all the objective function values of the infeasible solutions are punished according to the constraint violation degree, and the punished infeasible solutions are placed in the dominant region of the low valley point; the punished objective function value is calculated as follows:
wherein x is nadir CV (x) is the constraint violation degree, ω (x), of the candidate solution in the population P for the low valley point ndir ) Is a weight vector;
in order to evaluate the candidate solutions in each sub-population, the weighted sum WS of the objective function value and the weight vector of the candidate solution is used as the basis for measuring the quality of the candidate solution, and the WS calculation mode is as follows:
wherein ω (x) k ) Is a weight vector;
if the population just forms N sub-populations in the current iteration round, only the candidate solution with the minimum WS from each sub-population is needed to be selected and reserved to the next generation; when the number of sub-populations is less than N, it means that a plurality of candidate solutions need to be selected in at least one sub-population, in which case the diversity information DI of the candidate solutions will be further compared, the DI is calculated as follows:
DI(x)＝||F(x)-F(x ref )||
wherein x is ref To the x th distance candidate solutionNear individuals, |·| is the euclidean distance of the corresponding vector;
the fitness function Fit of the individual in the diversity optimization stage is designed, and the calculation mode is as follows:
wherein WS is no For each sub-population SP k Is ranked according to ascending order of WS values.
4. The constrained multi-objective evolutionary method based on three-phase optimization of claim 1, wherein the step (3) is specifically implemented as follows: optimizing the original population by adopting a steady-state-based evolution mode and adopting a local exploration mode; firstly, calculating constraint violation degrees, convergence information and diversity information of all candidate solutions in a population; selecting N parent candidate solutions by adopting a binary tournament mode according to the violation degree of the candidate solution constraint to construct a mating pool, and generating a child candidate solution y by adopting simulated binary intersection and polynomial variation; then screening out all candidates with smaller violation degree than y constraint from the populationSolve and form set Q 1 Randomly replacing Q with y 1 Is a single individual; finally, screening all candidate solutions with worse convergence and diversity than y from the population to form a set Q 2 Randomly replacing Q with y 2 Is a single individual.
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