A method for routing optimization of very large scale integrated circuit physical design using a constraint multi-objective evolutionary method based on multi-stage search

Abstract

The present application relates to a constrained multi-objective evolutionary method based on multi-stage search. Different search strategies are adopted in three stages of the method. In order to make the population quickly cross the large infeasible region and approach the Pareto front, the method does not consider the constraint condition in the first stage, and a convergence index is used to guide the population search; a set of uniformly distributed weight vectors is used to maintain the diversity of the population in the second stage, and an improved epsilon constraint processing strategy is proposed to retain high-quality solutions in the infeasible region; and the constraint priority principle is adopted in the third stage to concentrate the search preference in the feasible region to ensure the feasibility of the final solution set.

Description

TECHNICAL FIELD

[0001] The present application relates to a routing optimization method for ultra large scale integrated circuit physical design using a constrained multi-objective evolutionary method based on multi-stage search. BACKGROUND

[0002] Constrained multi-objective optimization problem (CMOP) often appears in daily life and engineering problems. Due to the existence of constraints, the objective space of CMOP is divided into feasible region and infeasible region, and the method needs to cross the infeasible region in the search process and find a set of well-distributed high-quality solutions in the feasible region. In order to deal with CMOP, scholars have proposed many constrained multi-objective evolutionary algorithms (CMOEA). The key to solving CMOP is to design a suitable constraint handling strategy, so that the final solution set meets the constraints while achieving ideal convergence and diversity. However, when facing problems with large infeasible regions, the population will gather at the edge of the feasible region because the constraint is always used as the main individual selection indicator, resulting in stagnation of the method search. The epsilon constraint handling method uses a parameter to adjust the acceptance degree of infeasible solutions, which to some extent makes up for the defects of the constraint priority principle, but how to design a suitable parameter value for different problems is the biggest challenge faced by this method. Existing multi-objective evolutionary methods can be divided into three categories according to the selection strategy: Pareto-based, decomposition-based and indicator-based methods. The Pareto-based method determines the solution set entering the next generation according to non-dominated sorting and density estimation. SUMMARY

[0003] The purpose of the present application is to provide a routing optimization method for ultra large scale integrated circuit physical design using a constrained multi-objective evolutionary method based on multi-stage search, to effectively solve the discontinuous and multi-modal constrained multi-objective problem.

[0004] To achieve the above purpose, the technical scheme of the present application is as follows: a routing optimization method for ultra large scale integrated circuit physical design using a constrained multi-objective evolutionary method based on multi-stage search, comprising the following steps:

[0005] First stage: convert the constrained multi-objective optimization problem into a single-objective optimization problem without considering constraints, so that the method is not hindered by constraints in the search process and quickly approaches the Pareto front;

[0006] Second stage: Based on the decomposition, the weight vector is used to guide the population to search the whole target space uniformly, and an improved epsilon constraint processing strategy is proposed to reserve some infeasible solutions.

[0007] Third stage: Based on the constraint priority principle, a group of high-quality feasible solutions is obtained.

[0008] Compared with the prior art, the method has the following beneficial effects: the method adopts different search strategies in three stages. In order to make the population quickly cross the large infeasible region and approach the Pareto front, the method does not consider the constraint condition in the first stage, and a convergence index is used to guide the population search; a group of uniformly distributed weight vectors is used to maintain the diversity of the population in the second stage, and an improved epsilon constraint processing strategy is proposed to reserve high-quality solutions in the infeasible region; the constraint priority principle is adopted in the third stage, and the search preference is concentrated in the feasible region to ensure the feasibility of the final solution set. BRIEF DESCRIPTION OF DRAWINGS

[0009] Figure 1 The flowchart of the CMOEA-MSS of the application. DETAILED DESCRIPTION

[0010] The technical solutions of the application will be specifically described below with reference to the drawings.

[0011] The application is a wiring optimization method for ultra-large scale integrated circuit physical design based on the constraint multi-objective evolutionary method of multi-stage search (CMOEA-MSS), which specifically comprises the following steps, as shown in the figure. Figure 1

[0012] First stage: The constraint multi-objective optimization problem is converted into a single-objective optimization problem without considering the constraint, so that the method is not hindered by the constraint in the search process and quickly approaches the Pareto front.

[0013] Second stage: Based on the decomposition, the weight vector is used to guide the population to search the whole target space uniformly, and an improved epsilon constraint processing strategy is proposed to reserve some infeasible solutions.

[0014] Third stage: Based on the constraint priority principle, a group of high-quality feasible solutions is obtained.

[0015] The method of the application is specifically implemented as follows:

[0016] 1. CMOP problem:

[0017] A CMOP problem with m objective functions can be expressed as:

[0018] ​min F(x) = (f1(x), f2(x),..., f m (x)) T

[0019] s.t.g i (x)≤0,i = 1,2,...,l

[0020] h j (x) = 0,j = l + 1,l + 2,...,n

[0021] x = (x1,x2,...,x d ) ∈ S

[0022] where x is a d-dimensional decision vector, S is the decision space, F(x) consists 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 numbers of inequality constraints and equality constraints, respectively.

[0023] In the routing problem of very large scale integrated circuit physical design, in order to make the routing reduce the wire length while also being able to meet the timing constraints, it is necessary to optimize the wire length and the longest distance from a source point to a sink point, i.e. the radius, at the same time. Therefore, it has two objective functions f1(x) and f2(x) to calculate the wire length and the radius, respectively, and the inequality constraint g1≥0 to represent the non-negative relaxation value of each pin in the routing to ensure the normality of the timing, and its calculation formula is as follows:

[0024] f1(x) = (∑ edge∈x WL edge )- WL rep

[0025] where edge represents an edge between two pins, WL edge represents the wire length of edge, and WL rep represents the wire length of the repeated edge.

[0026]

[0027] where v represents a pin, leaf(x) represents the set of leaf nodes of x, and path v represents the path from the source point to the pin v.

[0028] g1(x) = rat(v) - aat(v), v ∈ x

[0029] where rat(v) is the required arrival time of the signal of pin v, and aat(v) is the actual arrival time of the signal of pin v.

[0030] 2. Constraint violation degree:

[0031] The constraint violation degree is used to measure the degree of constraint violation of individuals, and is calculated as follows:

[0032]

[0033] where CV i (x) is the constraint violation degree of individual x on the ith constraint, and η is a very small positive number. If and only if CV * (x) = 0, then individual x is called a feasible solution.

[0034] 3. Convergence index:

[0035] The first stage search of CMOEA-MSS transforms the constrained multi-objective optimization problem into a single-objective problem without considering constraints, and uses the convergence index conv() to retain individuals with better convergence, so as to realize the rapid traversal of the infeasible region by the population. The calculation formula of the convergence index is as follows:

[0036]

[0037] where f i (x) represents the objective function value of individual x on the ith objective function. After the first stage search, a set of solutions close to the center of the Pareto front can be obtained, which facilitates the population to search in other directions in the subsequent stage, and realizes the extensive distribution of the population.

[0038] 4. First stage environment selection:

[0039] The first stage mainly selects individuals according to the convergence index. First, the convergence index conv(x) of individuals in the population P t is calculated, and the maximum constraint violation degree ε max is updated. Then two individuals x a and x b are randomly selected from the population P a . If conv(x b )≤conv(x a ), then the offspring individual p=x a ; if conv(x b )>conv(x b ), then the offspring individual p=x t . Finally, the individual p is put into the mating pool to generate the offspring population O, and the offspring population O and the parent population P t+1 are combined to calculate the convergence index conv(x) of individuals in the combined population, and the best N individuals are selected as the next generation population P * .

[0040] 5. Diversity maintenance mechanism:

[0041] To enlarge the search range of the method, the Tchebycheff decomposition method is used to transform the multi-objective optimization problem into a set of single-objective optimization problems and optimize them simultaneously:

[0042]

[0043] where ω is a set of weight vectors uniformly distributed in the objective space, z * is the ideal point.

[0044] Although the ultimate goal of the method is to obtain a set of high-quality feasible solutions, if only feasible solutions are retained during the search process, the information carried by infeasible solutions will be lost. An improved epsilon constraint handling strategy is proposed, which adaptively adjusts the search preference according to the proportion of feasible solutions in the population. The specific calculation formula is as follows:

[0045]

[0046] where fr t is the proportion of feasible solutions in the tth generation population, λ, δ and γ are parameters that control the constraint relaxation, λ takes the value range [0, 1]. α and β are parameters that control the search preference of the method in the feasible region and the infeasible region, ε max is the maximum constraint violation in the population.

[0047] 6. Second stage environment selection:

[0048] α and β divide the search preference of the method in the feasible region and the infeasible region into three levels. When fr t < α, there are a large number of infeasible solutions in the population, at this time the value of ε will decrease exponentially with the increase of the number of iterations, so as to obtain more feasible solutions. When α ≤ fr t < β, the proportion of feasible solutions and infeasible solutions in the population is balanced, with the continuous search of the method, ε decreases at a certain rate. When fr t ≥ β, most individuals in the population are feasible solutions, ε decreases slowly, so as to retain a small number of infeasible solutions.

[0049] 7. Constraint priority principle:

[0050] The third stage adopts the constraint priority principle to select individuals, that is, when individuals x u and x v satisfy any of the following conditions, then x u is superior to x v .

[0051] 1) x u and x v are both feasible solutions, and conv(xu ) < conv(x v ) ;

[0052] 2) x u is a feasible solution, x v is an infeasible solution;

[0053] 3) x u and x v are both infeasible solutions, and CV(x u ) < CV(x v ).

[0054] 8. The third stage of environment selection:

[0055] After the search of the previous two stages, most of the candidate solutions in the population are feasible solutions that satisfy the constraint conditions, and good optimization results have been achieved in convergence and diversity. However, due to the relaxation of the constraint conditions in the second stage, there are still a small number of infeasible solutions in the population that do not satisfy the constraint conditions after the end of the second stage. Convergence and diversity only have meaning on the basis of feasibility. Therefore, in the later stage of the method search, the population has converged to the vicinity of the Pareto front, that is, under the condition of good diversity and convergence, the CMOEA-MSS in the third stage preferentially selects individuals with smaller constraint violation degrees, guiding the remaining infeasible solutions to search in the feasible region, and ensuring the feasibility of the final population. For individuals with the same constraint violation degree or feasible solutions, individuals with better convergence indicators are preferentially selected, further strengthening the convergence of the population.

[0056] The above is the preferred embodiment of the present application, any changes made according to the technical solutions of the present application, as long as the resulting functional effects do not exceed the scope of the technical solutions of the present application, are within the scope of protection of the present application.

Claims

1. A routing optimization method for physical design of very large scale integrated circuits using a constrained multi-objective evolutionary method based on a multi-stage search, characterized by, Comprises: 1) CMOP problem: A constrained multi-objective optimization problem CMOP with m objective functions is formulated as: minF(x) = (f1(x), f2(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 = (x1, x2,..., x d ) ∈ S x is a d-dimensional decision vector, S is the decision space, F(x) is composed of m objective functions f1(x), f2(x),..., f m i gi(x) is the ith inequality constraint, hi j hj(x) is the jth equality constraint, l and (n-l) are the number of inequality constraints and equality constraints, respectively;​ In the routing problem of ultra large scale integrated circuit physical design, in order to make the routing reduce the line length while meeting the timing constraints, it is necessary to optimize the line length and the longest distance from a source point to a sink point, i.e. the radius, simultaneously; therefore, it has two objective functions f1(x) and f2(x) to calculate the line length and the radius respectively, and an inequality constraint g1 >= 0 to represent the non-negative relaxation value of each pin in the routing to ensure the normality of the timing, whose calculation formula is as follows: f1(x) = (∑ edge∈x WL edge )-WL rep edge represents an edge between two pins, WL edge line length of edge, WL rep line length of a repeating edge; v represents a pin, leaf(x) represents a set of leaf nodes of x, path v represents a path from the source to the pin v; g1(x) = rat(v) - aat(v), v e x rat(v) is the required arrival time of the signal of pin v, and aat(v) is the actual arrival time of the signal of pin v; 2) Constraint violation degree: The constraint violation degree is used to measure the degree of constraint violation of an individual; 3) Convergence index: The first stage search converts the constrained multi-objective optimization problem CMOP into a single objective problem without considering constraints, and uses the convergence index conv() to retain individuals with better convergence to realize the rapid traversal of the infeasible region by the population; 4) First stage environment selection: The first stage selects individuals according to the convergence index; calculates the population P t The convergence index conv(x) of the middle individual is updated, and the maximum constraint violation degree ε is updated max ; two individuals x t and x a are randomly selected from the population P b ; if conv(x a )≤conv(x b ), then the offspring individual p=x a ; if conv(x a )>conv(x b ), then the offspring individual p=x b ; the individual p is put into the mating pool, the offspring population O is generated, the offspring population O is combined with the parent population P t , the convergence index conv(x) of the individual in the combined population is calculated, and the best N individuals are selected as the next generation population P t+1 ; 5) Diversity maintenance mechanism: to expand the search range of the method, the Tchebycheff decomposition method is used to convert the constrained multi-objective optimization problem CMOP into a group of single objective optimization problems and optimize them simultaneously; An improved epsilon constraint processing strategy is proposed, which adaptively adjusts the search preference according to the proportion of feasible solutions in the population, and the specific calculation formula is as follows: where fr t is the proportion of feasible solutions in the tth generation population, λ, δ and γ are parameters to control the relaxation of constraints, λ is in the range of [0, 1]; α and β are parameters to control the search preference of the method in the feasible region and the infeasible region, ε max is the maximum constraint violation in the population; 6) Second stage environment selection: α and β divide the search preference of feasible and infeasible regions into three levels; when fr t < α, there are a large number of infeasible solutions in the population, at this time the value of ε will decrease exponentially with the increase of the number of iterations, so as to get more feasible solutions; when α ≤ fr t < β, the proportion of feasible and infeasible solutions in the population is balanced, with the continuous search of the method, ε decreases at a predetermined rate; when fr t ≥ β, most individuals in the population are feasible solutions, ε decreases slowly, so as to retain a small number of infeasible solutions; 7) Constraint priority principle: The third stage uses the constraint priority principle to select individuals; 8) Third stage environment selection: After the search of the previous two stages, the population has converged to the vicinity of the Pareto front, i.e. under the condition of good diversity and convergence, the third stage preferentially selects individuals with smaller constraint violation degrees, guiding the remaining infeasible solutions to search in the feasible region, and ensuring the feasibility of the final population; for individuals with the same constraint violation degree or feasible solutions, the individual with better convergence index is preferentially selected.

2. The routing optimization method for physical design of very large scale integrated circuits using a constrained multi-objective evolutionary method based on a multi-stage search according to claim 1, wherein, 2) in, constraint violation degree, the specific calculation is as follows: where CV(x) = 0 if and only if x is a feasible solution. i (x) is the constraint violation of individual x on the ith constraint, and η is a very small positive number; x* is called a feasible solution if and only if CV(x * ) = 0. 3) in, the convergence index calculation formula is as follows: where f i (x) represents the objective function value of individual x on the ith objective function; after the first stage of search, a set of solutions close to the center of the Pareto front is obtained, which facilitates the population to search in other directions in the subsequent stage, achieving the extensive distribution of the population; 5) in, the constrained multi-objective optimization problem CMOP is converted into a group of single objective optimization problems and optimized simultaneously, and the specific formula is as follows: where ω is a set of weight vectors uniformly distributed in the target space, z * is the ideal point; 7) In the middle, the selection of individuals is done using the principle of constraint precedence, i.e. when an individual x u and x v satisfies any of the following conditions, then x u is preferred to x v ; u and x v are feasible solutions, and conv(x u ) < conv(x v ).​ x u is a feasible solution, x v is an infeasible solution; u and x v are infeasible solutions, and CV(x u ) < CV(x v ).​

Images