CN115859825A - Multi-stage search-based constrained multi-target evolution method - Google Patents
Multi-stage search-based constrained multi-target evolution method Download PDFInfo
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Abstract
The invention relates to a multi-stage search-based constrained multi-target evolution method. Different search strategies are employed in the three stages of the method. In order to enable the population to rapidly pass through a large infeasible area and approach a Pareto frontier, constraint conditions are not considered in the first stage of the method, and a convergence index is used for guiding population search; in the second stage, a group of uniformly distributed weight vectors is adopted to maintain the diversity of the population, an improved epsilon constraint processing strategy is provided, and high-quality solutions in an infeasible area are reserved; and in the third stage, a constraint priority principle is adopted, and search preferences are concentrated in a feasible region to ensure the feasibility of a final solution set.
Description
Technical Field
The invention relates to a multi-stage search-based constrained multi-target evolution method.
Background
Constrained Multi-objective Optimization Problem (CMOP) is a common occurrence in everyday life and on engineering problems. Due to the existence of constraints, the target space of the CMOP is divided into a feasible region and an infeasible region, and the method needs to traverse the infeasible region in the searching process and find a group of well-distributed high-quality solution sets in the feasible region. To deal with CMOP, researchers have proposed a number of Constrained Multi-Objective evolutionary algorithms (CMOEA). The key to solving the CMOP is to design a proper constraint processing strategy, so that the final solution set can achieve ideal convergence and diversity while satisfying the constraint. However, when a problem with a large infeasible area is encountered, the population will gather at the edge of the feasible area due to the constraint always being the main individual selection index, thereby causing the stagnation of the method search. The epsilon constraint processing method utilizes one parameter to adjust the acceptance degree of an infeasible solution, so that the defect of a constraint priority principle is made up to a certain extent, but how to design a proper parameter value aiming at different problems is the greatest challenge faced by the method. The existing multi-objective evolutionary methods can be divided into three main categories according to the selected strategy: pareto-based dominant, decomposition-based, and index-based methods. Pareto dominated based methods decide to enter the solution set of the next generation according to non-dominated sorting and density estimation.
Disclosure of Invention
The invention aims to provide a multi-stage search-based constrained multi-objective evolution method, which is used for effectively solving the problem of multiple constrained objectives which are discontinuous and have multi-modal properties.
In order to achieve the purpose, the technical scheme of the invention is as follows: a multi-stage search based constrained multi-objective evolution method comprises the following steps:
the first stage is as follows: converting the constrained multi-objective optimization problem into a single-objective optimization problem without constraint, so that the method is free from constraint obstruction in the searching process and is fast close to the Pareto front edge;
and a second stage: adopting a decomposition-based idea, utilizing a weight vector to guide a population to uniformly search the whole target space, and simultaneously providing an improved epsilon constraint processing strategy to keep part of infeasible solutions;
and a third stage: and a constraint-based priority principle is adopted, so that a group of feasible solutions with high quality is obtained.
Compared with the prior art, the invention has the following beneficial effects: the method of the invention adopts different search strategies in three stages. In order to enable the population to rapidly pass through a large infeasible area and approach a Pareto frontier, constraint conditions are not considered in the first stage, and a convergence index is used for guiding population search; in the second stage, a group of uniformly distributed weight vectors is adopted to maintain the diversity of the population, an improved epsilon constraint processing strategy is provided, and high-quality solutions in an infeasible area are reserved; and in the third stage, a constraint priority principle is adopted, and search preferences are concentrated in a feasible region to ensure the feasibility of a final solution set.
Drawings
Figure 1 is a flow chart of the CMOEA-MSS of the present invention.
Detailed Description
The technical scheme of the invention is specifically explained in the following by combining the attached drawings.
The invention relates to a multi-stage search-based constrained multi-target evolution method (namely CMOEA-MSS), which specifically comprises the following steps as shown in figure 1:
the first stage is as follows: and the constrained multi-objective optimization problem is converted into a single-objective optimization problem without constraint, so that the method is free from constraint obstruction in the searching process and is fast close to the Pareto frontier.
And a second stage: and (3) adopting a decomposition-based idea, guiding the population to uniformly search the whole target space by using the weight vector, and simultaneously providing an improved epsilon constraint processing strategy to keep part of infeasible solutions.
And a third stage: and a constraint-based priority principle is adopted, so that a group of feasible solutions with high quality is obtained.
The method of the invention is concretely realized as follows:
CMOP problem:
a CMOP problem with m objective functions can be expressed as:
minF(x)＝(f 1 (x),f 2 (x),...,f m (x)) T
s.t.g i (x)≤0,i＝1,2,...,l
h j (x)＝0,j＝l+1,l+2,...,n
x＝(x 1 ,x 2 ,...,x d )∈S
where x is a d-dimensional decision vector, S is a decision space, F (x) is composed of m objective functions F (x), g i (x) Is the ith inequality constraint, h j (x) Is the jth equality constraint, and l and (n-l) are the number of inequality constraints and equality constraints, respectively.
In the wiring problem of the physical design of the very large scale integrated circuit, in order to reduce the length of the wiring and simultaneously satisfy the timing constraint, the length of the wiring and the longest distance from a source point to a certain sink point, namely the radius, need to be optimized simultaneously. It therefore has 2 objective functions f 1 (x) And f 2 (x) Respectively calculating the length and radius of the wire, and having inequality constraint g 1 ≧ 0 represents that the relaxation value of each pin in the wiring is non-negative to ensure the regularity of the timing, and its calculation formula is as follows:
f 1 (x)＝(∑ edge∈x WL edge )-WL rep
where edge represents the edge between two pins, WL edge Line length, WL, representing edge rep The length of the line representing the repeating edge.
Where v represents a pin, leaf (x) represents a set of leaf nodes for x, path v Representing the path from the source to pin v.
g 1 (x)＝rat(v)-aat(v),v∈x
Where, rat (v) is the required arrival time of the pin v signal, and aat (v) is the actual arrival time of the pin v signal.
2. Constraint violation degree:
the constraint violation degree is used for measuring the degree of constraint violation of an individual, and is specifically calculated as follows:
wherein, CV is i (x) For the degree of constraint violation of the individual x on the ith constraint, η is a very small positive number. If and only if CV (x) * ) If =0, then the individual x is said to be a feasible solution.
3. The convergence index is:
the first-stage search of the CMOEA-MSS converts the constraint multi-objective optimization problem into a constraint-free single-objective problem, and retains individuals with better convergence by using a convergence index conv (), so that the population rapidly passes through an infeasible area. The convergence index calculation formula is as follows:
wherein f is i (x) The objective function value of the individual x on the ith objective function is shown. After the search in the first stage, a group of solutions close to the front edge center of Pareto can be obtained, so that the population can be conveniently searched in other directions in the subsequent stage, and the population is widely distributed.
4. Selecting a first stage environment:
in the first stage, individual selection is performed mainly according to the convergence index. First, calculate the population P t The convergence index conv (x) of the medium unit is updated, and the maximum constraint violation degree epsilon is updated max . Then to the population P t In random drawing of two individuals x a And x b If conv (x) a )≤conv(x b ) Then the offspring individual p = x a (ii) a If conv (x) a )＞conv(x b ) Then the offspring individual p = x b . Finally, putting the individual P into a mating pool to generate an offspring population O, and combining the offspring population O with the parent population P t Merging, calculating the convergence index conv (x) of the individuals in the merged population) Selecting the best N individuals as the next generation population P t+1 。
5. The mechanism of diversity maintenance:
in order to expand the search range of the method, the invention adopts a Tchebycheff decomposition method, solves the multi-objective optimization problem into a group of single-objective optimization problems and simultaneously optimizes the single-objective optimization problems:
where ω is a set of weight vectors uniformly distributed in the target space, z * Is an ideal point.
Although the final goal of the method is to obtain a set of feasible solutions with high quality, if only the feasible solutions are retained in the searching process, the information carried by the infeasible solutions is lost. The invention provides an improved epsilon constraint processing strategy, which adaptively adjusts search preference according to a feasible solution ratio in a population, and comprises the following specific calculation formula:
wherein fr t Is the ratio of feasible solutions in the t generation population, lambda, delta and gamma are parameters for controlling and controlling the relaxation of constraint, and the value range of the lambda is [0,1 ]]. Alpha and beta are parameters that control the search preferences of the method in feasible and infeasible areas, epsilon max Is the maximum value of constraint violation in the population.
6. And second-stage environment selection:
alpha and beta divide the search preference of the method in feasible area and infeasible area into threeAnd (4) grading. When fr is present t When the value is less than alpha, a large number of infeasible solutions exist in the population, and the value of epsilon is exponentially reduced along with the increase of the iteration times, so that more feasible solutions are obtained. When alpha is less than or equal to fr t When the value is less than beta, the proportion of feasible solutions to infeasible solutions in the population is balanced, and epsilon is reduced at a certain speed along with the continuous searching of the method. When fr t When the solution is more than or equal to beta, most individuals in the population are feasible solutions, and epsilon is slowly reduced, so that a small part of infeasible solutions are reserved.
7. Constraint priority principle:
in the third stage, the individual is selected by adopting a constraint priority principle, namely when the individual x u And x v X when the following arbitrary conditions are satisfied u Is superior to x v 。
1)x u And x v Are all feasible solutions, and conv (x) u )＜conv(x v )；
2)x u Is a feasible solution, x v Is not feasible;
3)x u and x v Are all infeasible solutions, and CV (x) u )＜CV(x v )。
8. Third-stage environment selection:
after the CMOEA-MSS is searched in the first two stages, most candidate solutions in the population are feasible solutions meeting the constraint conditions, and good optimization effects on convergence and diversity are achieved, but due to the fact that the constraint conditions are relaxed in the second stage, after the second stage is finished, a small number of infeasible solutions which do not meet the constraint conditions still exist in the population, and the convergence and diversity are meaningful only on the basis of feasibility. Therefore, in the later stage of the method search, the population is already converged to the vicinity of the Pareto front edge, namely under the condition of better diversity and convergence, in the third stage, the CMOEA-MSS preferentially selects individuals with smaller constraint violation degree, guides the rest infeasible solutions to search in a feasible region, and ensures the feasibility of the final population. For the constraint violation degrees which are the same or the feasible solutions, the individual with better convergence index is preferentially selected, and the convergence of the population is further enhanced.
The above are preferred embodiments of the present invention, and all changes made according to the technical scheme of the present invention that produce functional effects do not exceed the scope of the technical scheme of the present invention belong to the protection scope of the present invention.
Claims (2)
1. A multi-stage search based constrained multi-target evolution method is characterized by comprising the following steps:
the first stage is as follows: converting the constrained multi-objective optimization problem into a single-objective optimization problem without constraint, so that the method is free from constraint obstruction in the searching process and is fast close to the Pareto front edge;
and a second stage: adopting a decomposition-based idea, guiding the population to uniformly search the whole target space by using the weight vector, and simultaneously providing an improved epsilon constraint processing strategy to keep part of infeasible solutions;
and a third stage: and a constraint-based priority principle is adopted, so that a group of feasible solutions with high quality is obtained.
2. The multi-stage search-based constrained multi-objective evolution method according to claim 1 is specifically realized as follows:
1) CMOP problem:
a constrained multi-objective optimization problem CMOP with m objective functions is expressed as:
minF(x)＝(f 1 (x),f 2 (x),...,f m (x)) T
s.t.g i (x)≤0,i＝1,2,...,l
h j (x)＝0,j＝l+1,l+2,...,n
x＝(x 1 ,x 2 ,...,x d )∈S
where x is a d-dimensional decision vector, S is a decision space, and F (x) is defined by m objective functions F 1 (x),f 2 (x),...,f m (x) Composition of (g) i (x) Is the ith inequality constraint, h j (x) Is the jth equality constraint, l and (n-l) are the number of inequality constraints and equality constraints, respectively;
in the wiring problem of physical design of very large scale integrated circuit, the wiring is performedThe time sequence constraint can be met while the line length is reduced, and the line length and the longest distance from a source point to a certain sink point, namely the radius, need to be optimized simultaneously; it therefore has 2 objective functions f 1 (x) And f 2 (x) Respectively calculating the length and radius of the wire, and having inequality constraint g 1 ≧ 0 represents that the relaxation value of each pin in the wiring is non-negative to ensure the regularity of the timing, and its calculation formula is as follows:
f 1 (x)＝(∑ edge∈x WL edge )-WL rep
where edge represents the edge between two pins, WL edge Line length, WL, representing edge rep A line length representing a repeating edge;
where v represents a pin, leaf (x) represents a set of leaf nodes for x, path v Represents the path from the source point to pin v;
g 1 (x)＝rat(v)-aat(v),v∈x
wherein, rat (v) is the required arrival time of the pin v signal, aat (v) is the actual arrival time of the pin v signal;
2) Constraint violation degree:
the constraint violation degree is used for measuring the degree of constraint violation of an individual, and is specifically calculated as follows:
wherein, CV is i (x) For individual x, the constraint violation on the ith constraint, η is a very small positive number; if and only if CV (x) * ) When =0, the individual x is called a feasible solution;
3) The convergence index is as follows:
the first stage of search converts a constraint multi-objective optimization problem CMOP into a single-objective problem without considering the constraint, and retains individuals with better convergence by using a convergence index conv () to realize that the population quickly passes through an infeasible area; the convergence index calculation formula is as follows:
wherein f is i (x) The objective function value of the individual x on the ith objective function is shown; after the search of the first stage, a group of solutions close to the front edge center of Pareto is obtained, so that the population can be conveniently searched to other directions in the subsequent stage, and the population is widely distributed;
4) Selecting a first stage environment:
in the first stage, individual selection is carried out according to the convergence index; first, calculate the population P t The convergence index conv (x) of the medium individual is updated, and the maximum constraint violation degree epsilon is updated max (ii) a Then to the population P t In the random drawing of two individuals x a And x b If conv (x) a )≤conv(x b ) Then the offspring individual p = x a (ii) a If conv (x) a )＞conv(x b ) Then the offspring individual p = x b (ii) a Finally, putting the individual P into a mating pool to generate an offspring population O, and combining the offspring population O with the parent population P t Merging, calculating the convergence index conv (x) of individuals in the merged population, and selecting the best N individuals as the next generation population P t+1 ；
5) The mechanism of diversity maintenance: in order to enlarge the search range of the method, a Tchebycheff decomposition method is adopted to transform a constraint multi-objective optimization problem CMOP into a group of single-objective optimization problems and simultaneously optimize:
where ω is a set of weight vectors uniformly distributed in the target space, z * Is an ideal point;
an improved epsilon constraint processing strategy is provided, the search preference is adaptively adjusted according to the feasible solution proportion in the population, and the specific calculation formula is as follows:
wherein fr t Is the ratio of feasible solutions in the t generation population, lambda, delta and gamma are parameters for controlling the relaxation of control constraints, and the value range of lambda is [0,1 ]](ii) a Alpha and beta are parameters that control the search preferences of the method in feasible and infeasible areas, epsilon max Is the maximum value of the violation degree of the constraint in the population;
6) And second-stage environment selection:
α and β divide the search preference between feasible and infeasible regions into three levels; when fr is present t When the value is less than alpha, a large number of infeasible solutions exist in the population, and the value of epsilon is exponentially reduced along with the increase of the iteration times, so that more feasible solutions are obtained; when alpha is less than or equal to fr t When the value is less than beta, the proportion of feasible solutions to infeasible solutions in the population is more balanced, and epsilon is reduced at a preset speed rate along with the continuous search of the method; when fr is present t When the solution is more than or equal to beta, most individuals in the population are feasible solutions, and epsilon is slowly reduced, so that a small part of infeasible solutions are reserved;
7) Constraint priority principle:
in the third stage, the individual is selected by adopting a constraint priority principle, namely when the individual x u And x v X when the following arbitrary conditions are satisfied u Is superior to x v ；
①x u And x v Are all feasible solutions, and conv (x) u )＜conv(x v )；
②x u Is a feasible solution, x v Is not feasible;
③x u and x v Are all infeasible solutions, and CV (x) u )＜CV(x v )；
8) Third-stage environment selection:
after the searching in the first two stages, the population is converged to the vicinity of the Pareto front edge, namely under the condition of better diversity and convergence, individuals with smaller constraint violation degree are preferentially selected in the third stage, and the remaining infeasible solution is guided to search towards a feasible region, so that the feasibility of the final population is ensured; for the same or feasible solution of constraint violation degree, the individual with better convergence index is selected preferentially.
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