High-dimensional many-objective evolutionary method based on improved dominance criteria

Abstract

The application provides a high-dimensional multi-objective evolutionary method based on an improved dominance criterion, which is used for optimizing the wiring design of a very large scale integrated circuit physical design and comprises the following steps: step one, convergence of a non-dominated solution set is ensured according to a defined convergence index, and an adaptive parameter based on a genetic algorithm niche is combined to control the diversity of the solution set, the radius target and the line length target of the wiring design are optimized by minimizing MOP, and the dominance criterion is improved; step two, a convergence index and a diversity index are designed, the two indexes jointly constitute a dynamic fitness function, and individuals with better convergence and diversity are adaptively reserved to perform MaOEA-IDR environment selection; and step three, an adaptive t distribution crossover operator is provided, which can balance the global search capability of a Cauchy operator and the local exploration capability of a Gaussian operator in a high-dimensional space, and the ASDX adaptive distribution crossover operator is adaptively adjusted. t The application can effectively solve the wiring problem of the very large scale integrated circuit physical design.

Description

Technical Field

[0001] This invention relates to the field of integrated circuit design technology, and in particular to a high-dimensional multi-objective evolutionary method based on improved dominance criteria. Background Technology

[0002] With the development of electronic technology, the routing stage of VLSI physical design needs to consider more and more constraints. Each constraint corresponds to an objective function, making the routing problem a multi-objective optimization problem (MOP).

[0003] like Figure 1 As shown, during the routing phase, the two optimization objectives of line length and radius usually cannot be optimized simultaneously. Figure 1 (a) in the diagram represents the minimum line length spanning tree; Figure 1 (b) in the diagram represents the minimum radius spanning tree; Figure 1 (c) in the diagram represents the spanning tree designed in this invention that comprehensively considers the relationship between the two. This type of problem requires the simultaneous optimization of multiple objective functions, and there are often conflicts between the objective functions, meaning that there is no single optimal solution that makes all objective function values ​​optimal. Therefore, the ultimate goal of solving the MOP is to find a set of Pareto optimal solutions.

[0004] The Multi-objective Optimization Evolutionary Algorithm (MOEA), as a population-based heuristic algorithm, can optimize a set of solutions simultaneously in each iteration, and has attracted widespread attention from researchers. However, MOEA often struggles to obtain a satisfactory solution set when solving high-dimensional multi-objective optimization problems (MaOP) with an objective dimension greater than or equal to 4. In recent years, researchers have proposed many strategies to address the challenges of MaOP. These strategies can be categorized into two aspects based on their focus on different problem-solving approaches. One aspect considers increasing selection pressure to improve population convergence, thereby reducing the impact of domination resistance. However, methods that improve domination relationships may not necessarily map uniformly distributed non-dominated solution sets to uniformly distributed projection points on the hyperplane. Therefore, simply increasing selection pressure can improve convergence but may negatively impact diversity. The other aspect considers maintaining diversity and distribution. The main method is diversity density assessment, which can be further divided into two systems: the first is distribution diversity assessment based on distance information, but this method often has a significant negative impact on population convergence; the second is distribution diversity assessment based on angle information, but these methods suffer from difficulties in setting angle thresholds and are susceptible to the influence of frontier shape characteristics.

[0005] Therefore, it is necessary to propose a method that can adaptively balance convergence and diversity and reduce the adverse effects of dominance resistance. Summary of the Invention

[0006] This invention proposes a high-dimensional multi-objective evolutionary method based on an improved dominance criterion, which can effectively solve the wiring problem in the physical design of very large-scale integrated circuits.

[0007] The present invention adopts the following technical solution.

[0008] A high-dimensional multi-objective evolutionary method based on improved dominance criteria is used to optimize the wiring design of VLSI physical design, including the following steps;

[0009] Step 1: Based on the defined convergence index, the convergence of the non-dominated solution set is guaranteed. Adaptive parameters based on the niche of the genetic algorithm are combined to control the diversity of the solution set. The radius target and line length target of the wiring design are optimized by minimizing the MOP, and the dominance criterion is improved.

[0010] Step 2: Design convergence and diversity indices to form a dynamic fitness function that adaptively retains individuals with good convergence and diversity for MaOEA-IDR environment selection.

[0011] Step 3: Propose an adaptive t-distribution crossover operator that can take into account both the global search capability of the Cauchy operator and the local exploration capability of the Gaussian operator in high-dimensional space, so that the t-distribution crossover operator ASDX is adaptive.

[0012] Step one adopts the following method.

[0013] Method A1, minimizing MOP, specifically:

[0014] In the routing problem of very large-scale integrated circuits, considering both the radius and the line length, the minimization of the mean squared error (MOP) problem can be expressed as follows:

[0015]

[0016] Where x = (x1, x2, ..., x n F represents taking n feasible routing solutions, where Ω is the solution space; F: Ω → R 2 Composed of two objective functions, R 2 It is the target space;

[0017] For two candidate solutions x1 and x2 in two decision spaces, if and only if f i (x1)≤f i (x2) and f i (x1)<f i When x1 dominates x2, the candidate solution x1 is said to dominate x2. If the candidate solution x1 is not dominated by any other candidate solution, then the candidate solution x1 is called a Pareto optimal solution. The set of all Pareto optimal solutions is called the Pareto optimal set (PS), and its mapping on the target space is called the Pareto front (PF).

[0018] The function f1(x) is the formula for calculating the wire length of the wiring, and its calculation process is shown below:

[0019]

[0020] Where 'e' represents the edge between the two pins, WL e WL represents the length of line e. r The length of the repeating edge is represented by f2(x); the function f2(x) is the formula for calculating the radius of the wiring, and its calculation process is shown below.

[0021]

[0022] Where p represents the pin, leaf(x) represents the set of leaf nodes in the spanning tree of the wiring solution x, and l p This represents the path from the source point to pin p;

[0023] Method A2, Improved Domination Criterion IDR:

[0024] The IDR (Independent Dependent Relationship) uses convergence indices as the primary criterion for determining the dominance relationship between solutions, aiming to enhance the convergence of the population. To improve population diversity, solutions are assigned to their respective niches based on cosine similarity. IDR utilizes a customized niche technique to balance the convergence and diversity of the non-dominated solution set. Specifically, in IDR, candidate solution x dominates candidate solution y if and only if:

[0025] Con(x)·λ<Con(y) Formula 4;

[0026] Where Con(x) is the convergence index, and λ is the adaptive parameter, calculated as follows:

[0027]

[0028]

[0029] in, The value is the size of the individual's microhabitat, and θ(x,y) is the angle between the solutions. The calculation formula is as follows:

[0030]

[0031] microhabitat size Different values ​​of will affect the convergence and diversity of the population; if If the value is too small, the convergence of the non-dominated solution set dominated by IDR will be poor; if If the value is too large, the diversity of non-dominated solution sets dominated by the IDR will be poor; therefore, this parameter... The following non-repeating angles are set to be the |P| / 2 smallest in the set P:

[0032] Step two is performed using the following method;

[0033] Method A3, Fitness Function Calculation: A fitness function that simultaneously considers both convergence and diversity metrics is proposed, specifically as follows:

[0034] The dynamic fitness function Fitness(x) and adaptive parameters of individual x in the population are calculated using the following formulas IX and X:

[0035]

[0036]

[0037] Wherein, Con(x) is the convergence index, calculated as shown in Formula 5, R Con(x) R is the ascending ranking of all individuals Con(x) in the population; FE is the current evaluation count, and maxFE is the maximum evaluation count; Div(x) is the diversity index, calculated as shown in Formula 11 below. Div(x) It is a descending ranking of the Div(x) values ​​of all individuals in the population;

[0038]

[0039] Among them, ManhattanDist 1st and ManhattanDist 2nd These refer to the Manhattan distances to the nearest and second-nearest individuals in the evaluation of individual x, respectively.

[0040] Method A4, MaOEA-IDR environment selection, specifically:

[0041] First, merge the parent population with the offspring population, and use IDR to perform non-dominated sorting on the merged population to obtain several levels of non-dominated solution sets.

[0042] Secondly, according to the hierarchy of non-dominated solutions, non-dominated solutions are retained to the next generation from smallest to largest. If the number of retained individuals exceeds the population size n, then the last set of non-dominated solutions to be retained is taken as the critical layer F. l ;

[0043] Then, the convergence index Con(x) of the critical layer individuals is calculated according to Formulas 5 and 11. i ) and diversity index Div(x i ), calculate the dynamic fitness value of all critical layer individuals according to Formula 9; finally, retain the critical layer individuals to the next generation in order of increasing dynamic fitness value.

[0044] In step three, during the early search phase of the multi-objective evolutionary method, the population is far from the Pareto front, R Con(x) It has a larger weight in the dynamic fitness function Fitness(x), and preferentially selects individuals with better convergence to enter the next generation, so that the population can quickly approach the Pareto front;

[0045] In the later stages of the multi-objective evolutionary search, most individuals in the population are sufficiently close to the Pareto front. To ensure that the population can evenly and broadly cover the entire Pareto front, R at this point... Div(x) The method prioritizes individuals with higher diversity in the dynamic adaptive function Fitness(x), thus ensuring the diversity of the population.

[0046] In step three, the following method is used;

[0047] Method A5: The analog binary SBX operator is used as the crossover operator, and its definition is expressed by the formula:

[0048]

[0049] Where x1 and x2 represent two parent individuals, c1 and c2 represent offspring individuals, and β represents a random variable. The calculation formula is as follows:

[0050]

[0051] Where μ is a random number, uniformly distributed in the interval (0,1); η is a constant, representing the crossover parameter;

[0052] Method A6: To address the limited search range of β in the simulated binary crossover operator, which easily leads to unstable evolutionary processes and local optima, the t-distribution is introduced into the crossover operator, and its degrees of freedom are set as a dynamic parameter δ related to the number of evaluations, replacing the random variable β in the original formula 12, thus forming the adaptive t-distribution crossover operator ASDX; the formula is as follows:

[0053]

[0054] Among them, (x 1,i +x 2,i ) / 2 represents the midpoint of the two parent individuals x1 and x2 in the i-th dimension, while (x1,i -x 2,i ) / 2 represents the search step size in this dimension, and the formula for calculating the degrees of freedom δ is as follows:

[0055]

[0056] Among them, when the degrees of freedom are 50, the t-distribution tends to a Gaussian distribution.

[0057] In step three, the adaptive t-distribution crossover operator ASDX is combined with the advantages of Cauchy and Gaussian operators. In the early stage of evolution, the strong local exploitation ability of Gaussian distribution is used to ensure that the population can converge quickly. In the later stage of the method, the global search ability of Cauchy operator is used to maintain population diversity.

[0058] The present invention integrates the t-distribution and dynamic degrees of freedom δ into the traditional binary crossover to obtain the ASDX operator, which can better balance the space search capability and local exploitation capability compared with the SBX operator, and prevents it from getting trapped in local optima in high-dimensional space.

[0059] This invention can adaptively balance convergence and diversity, and reduce the adverse effects of dominance resistance, thereby effectively solving the wiring problem in the physical design of very large-scale integrated circuits. Attached Figure Description

[0060] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0061] Appendix Figure 1 This is a diagram of a spanning tree considering line length and radius in wiring design;

[0062] Appendix Figure 2 This is a schematic diagram of the dominance region of IDR in this invention; Detailed Implementation

[0063] As shown in the figure, a high-dimensional multi-objective evolutionary method based on improved dominance criteria is used to optimize the wiring design of VLSI physical design, including the following steps;

[0064] Step 1: Based on the defined convergence index, the convergence of the non-dominated solution set is guaranteed. Adaptive parameters based on the niche of the genetic algorithm are combined to control the diversity of the solution set. The radius target and line length target of the wiring design are optimized by minimizing the MOP, and the dominance criterion is improved.

[0065] Step 2: Design convergence and diversity indices to form a dynamic fitness function that adaptively retains individuals with good convergence and diversity for MaOEA-IDR environment selection.

[0066] Step 3: Propose an adaptive t-distribution crossover operator that can take into account both the global search capability of the Cauchy operator and the local exploration capability of the Gaussian operator in high-dimensional space, so that the t-distribution crossover operator ASDX is adaptive.

[0067] Step one adopts the following method.

[0068] Method A1, minimizing MOP, specifically:

[0069] In the routing problem of very large-scale integrated circuits, considering both the radius and the line length, the minimization of the mean squared error (MOP) problem can be expressed as follows:

[0070]

[0071] Where x = (x1, x2, ..., x n F represents taking n feasible routing solutions, where Ω is the solution space; F: Ω → R 2 Composed of two objective functions, R 2 It is the target space;

[0072] For two candidate solutions x1 and x2 in two decision spaces, if and only if f i (x1)≤f i (x2) and f i (x1)<f i When x1 dominates x2, the candidate solution x1 is said to dominate x2. If the candidate solution x1 is not dominated by any other candidate solution, then the candidate solution x1 is called a Pareto optimal solution. The set of all Pareto optimal solutions is called the Pareto optimal set (PS), and its mapping on the target space is called the Pareto front (PF).

[0073] The function f1(x) is the formula for calculating the wire length of the wiring, and its calculation process is shown below:

[0074]

[0075] Where 'e' represents the edge between the two pins, WL e WL represents the length of line e. r The length of the repeating edge is represented by f2(x); the function f2(x) is the formula for calculating the radius of the wiring, and its calculation process is shown below.

[0076]

[0077] Where p represents the pin, leaf(x) represents the set of leaf nodes in the spanning tree of the wiring solution x, and l p This represents the path from the source point to pin p;

[0078] Method A2, Improved Domination Criterion IDR:

[0079] The IDR (Independent Dependent Relationship) uses convergence indices as the primary criterion for determining the dominance relationship between solutions, aiming to enhance the convergence of the population. To improve population diversity, solutions are assigned to their respective niches based on cosine similarity. IDR utilizes a customized niche technique to balance the convergence and diversity of the non-dominated solution set. Specifically, in IDR, candidate solution x dominates candidate solution y if and only if:

[0080] Con(x)·λ<Con(y) Formula 4;

[0081] Where Con(x) is the convergence index, and λ is the adaptive parameter, calculated as follows:

[0082]

[0083]

[0084] in, The value is the size of the individual's microhabitat, and θ(x,y) is the angle between the solutions. The calculation formula is as follows:

[0085]

[0086] microhabitat size Different values ​​of will affect the convergence and diversity of the population; if If the value is too small, the convergence of the non-dominated solution set dominated by IDR will be poor; if If the value is too large, the diversity of non-dominated solution sets dominated by the IDR will be poor; therefore, this parameter... The following non-repeating angles are set to be the |P| / 2 smallest in the set P:

[0087]

[0088] Step two is performed using the following method;

[0089] In this example, the dominance region of IDR is as follows: Figure 2The shaded area is shown in the diagram. The dominance region of IDR is determined by Equations 4 and 6. The dark gray and light gray areas represent the dominance region of individual A, with the dark gray area located within individual A's territory and the light gray area located outside of individual A's territory. Since individual B is located within individual A's territory, and the convergence of individual A is better than that of individual B, B is a dominated solution. Although individual C is located outside of individual A's territory, its convergence is poor, and it is still dominated by individual A. Therefore, the non-dominated solution set based on IDR has both good convergence speed and distribution, thus demonstrating that IDR has a significant advantage in balancing the convergence speed and distribution characteristics of non-dominated solution sets.

[0090] Method A3, Fitness Function Calculation: A fitness function that simultaneously considers both convergence and diversity metrics is proposed, specifically as follows:

[0091] The dynamic fitness function Fitness(x) and adaptive parameters of individual x in the population are calculated using the following formulas IX and X:

[0092]

[0093]

[0094] Wherein, Con(x) is the convergence index, calculated as shown in Formula 5, R Con(x) R is the ascending ranking of all individuals Con(x) in the population; FE is the current evaluation count, and maxFE is the maximum evaluation count; Div(x) is the diversity index, calculated as shown in Formula 11 below. Div(x) It is a descending ranking of the Div(x) values ​​of all individuals in the population;

[0095]

[0096] Among them, ManhattanDist 1st and ManhattanDist 2nd These refer to the Manhattan distances to the nearest and second-nearest individuals in the evaluation of individual x, respectively.

[0097] Method A4, MaOEA-IDR environment selection, specifically:

[0098] First, merge the parent population with the offspring population, and use IDR to perform non-dominated sorting on the merged population to obtain several levels of non-dominated solution sets.

[0099] Secondly, according to the hierarchy of non-dominated solutions, non-dominated solutions are retained to the next generation from smallest to largest. If the number of retained individuals exceeds the population size n, then the last set of non-dominated solutions to be retained is taken as the critical layer F. l ;

[0100] Then, the convergence index Con(x) of the critical layer individuals is calculated according to Formulas 5 and 11. i ) and diversity index Div(x i ), calculate the dynamic fitness value of all critical layer individuals according to Formula 9; finally, retain the critical layer individuals to the next generation in order of increasing dynamic fitness value.

[0101] In step three, during the early search phase of the multi-objective evolutionary method, the population is far from the Pareto front, R Con(x) It has a larger weight in the dynamic fitness function Fitness(x), and preferentially selects individuals with better convergence to enter the next generation, so that the population can quickly approach the Pareto front;

[0102] In the later stages of the multi-objective evolutionary search, most individuals in the population are sufficiently close to the Pareto front. To ensure that the population can evenly and broadly cover the entire Pareto front, R at this point... Div(x) The method prioritizes individuals with higher diversity in the dynamic adaptive function Fitness(x), thus ensuring the diversity of the population.

[0103] In step three, the following method is used;

[0104] Method A5: The analog binary SBX operator is used as the crossover operator, and its definition is expressed by the formula:

[0105]

[0106] Where x1 and x2 represent two parent individuals, c1 and c2 represent offspring individuals, and β represents a random variable. The calculation formula is as follows:

[0107]

[0108] Where μ is a random number, uniformly distributed in the interval (0,1); η is a constant, representing the crossover parameter;

[0109] Method A6: To address the limited search range of β in the simulated binary crossover operator, which easily leads to unstable evolutionary processes and local optima, the t-distribution is introduced into the crossover operator, and its degrees of freedom are set as a dynamic parameter δ related to the number of evaluations, replacing the random variable β in the original formula 12, thus forming the adaptive t-distribution crossover operator ASDX; the formula is as follows:

[0110]

[0111] Among them, (x 1,i +x 2,i) / 2 represents the midpoint of the two parent individuals x1 and x2 in the i-th dimension, while (x 1,i -x 2,i ) / 2 represents the search step size in this dimension, and the formula for calculating the degrees of freedom δ is as follows:

[0112]

[0113] Among them, when the degrees of freedom are 50, the t-distribution tends to a Gaussian distribution.

[0114] In step three, the adaptive t-distribution crossover operator ASDX is combined with the advantages of Cauchy and Gaussian operators. In the early stage of evolution, the strong local exploitation ability of Gaussian distribution is used to ensure that the population can converge quickly. In the later stage of the method, the global search ability of Cauchy operator is used to maintain population diversity.

Claims

1. A high-dimensional multi-objective evolutionary method based on improved dominance criteria for optimizing wiring design in VLSI physical design, characterized by: Includes the following steps; Step 1: Based on the defined convergence index, the convergence of the non-dominated solution set is guaranteed. Adaptive parameters based on the niche of the genetic algorithm are combined to control the diversity of the solution set. The radius target and line length target of the wiring design are optimized by minimizing the MOP, and the dominance criterion is improved. Step 2: Design convergence and diversity indices so that they together constitute a dynamic fitness function for MaOEA-IDR environment selection; Step 3: Propose an adaptive t-distribution crossover operator to make the t-distribution crossover operator ASDX adaptive; Step one includes improving the Dominance Criterion (IDR): IDR utilizes a customized niche technique to balance the convergence and diversity of non-dominated solution sets. Specifically, in IDR, a candidate solution x dominates a candidate solution y if and only if: Formula 4; Where Con(x) is the convergence index, The adaptive parameter is calculated using the following formula: Formula 5; Formula Six; in, ; It refers to the size of the individual's microhabitat. The angle between the solutions is calculated using the following formula: Formula 7; parameter Set as the first in the following set P Small, non-repeating angles: Formula 8; Step two is performed using the following method; Method A3, fitness function calculation, specifically: The dynamic fitness function Fitness(x) and adaptive parameters of individual x in the population are calculated using the following formulas IX and X: Formula Nine; Formula 10; R Con(x) R is the ascending ranking of all individuals Con(x) in the population; FE is the current evaluation count, and maxFE is the maximum evaluation count; Div(x) is the diversity index, calculated as shown in Formula 11 below. Div(x) It is a descending ranking of the Div(x) values ​​of all individuals in the population; Formula 11; in, and These refer to the Manhattan distances to the nearest and second-nearest individuals in the evaluation of individual x, respectively. Method A4, MaOEA-IDR environment selection, specifically: First, merge the parent population with the offspring population, and use IDR to perform non-dominated sorting on the merged population to obtain several levels of non-dominated solution sets. Secondly, according to the hierarchy of non-dominated solutions, non-dominated solutions are retained to the next generation from smallest to largest. If the number of retained individuals exceeds the population size n, then the last set of non-dominated solutions to be retained is taken as the critical layer F. l ; Then, the convergence index Con(x) of the critical layer individuals is calculated according to Formulas 5 and 11. i ) and diversity index Div(x i ), calculate the dynamic fitness value of all critical layer individuals according to Formula 9; finally, retain the critical layer individuals to the next generation in order of increasing dynamic fitness value; In step three, the following method is used; Method A5: The analog binary SBX operator is used as the crossover operator, and its definition is expressed by the formula: Official Twelve; in, and This represents two parent individuals. and Represents offspring individuals. Represents a random variable; Method A6: Introduce the t-distribution into the crossover operator and set its degrees of freedom as dynamic parameters related to the number of evaluations. The random variable β in the original formula 12 is replaced to form the adaptive t-distribution crossover operator ASDX; Step one adopts the following method. Method A1, minimizing MOP, specifically: In the routing problem of very large-scale integrated circuits, considering both the radius and the line length, the minimization of the mean squared error (MOP) problem can be expressed as follows: Formula 1; in, This indicates that taking n possible wiring solutions is feasible. It is a solution space; Compose two objective functions, It is the target space; For two candidate solutions x1 and x2 in two decision spaces, if and only if , and , When x1 dominates x2, the candidate solution x1 is said to be Pareto optimal. If the candidate solution x1 is not dominated by any other candidate solution, then the candidate solution x1 is called a Pareto optimal solution. The set of all Pareto optimal solutions is called the Pareto optimal solution set, and its mapping on the target space is called the Pareto front. function Here is the formula for calculating the wire length of the wiring, and the calculation process is as follows: Formula 2; in, Indicates the edge between two pins. express The length of the line, This indicates the length of the repeating edge.

2. The high-dimensional multi-objective evolutionary method based on improved dominance criteria according to claim 1, characterized in that: In step three, during the early search phase of the multi-objective evolutionary method, the population is far from the Pareto front, R Con(x) It has a greater weight in the dynamic fitness function Fitness(x), and prioritizes individuals with good convergence to enter the next generation, so that the population can quickly approach the Pareto front. In the later stages of the multi-objective evolutionary search, most individuals in the population are sufficiently close to the Pareto front. To ensure that the population can evenly and broadly cover the entire Pareto front, R at this point... Div(x) The method prioritizes individuals with higher diversity in the dynamic adaptive function Fitness(x), thus ensuring the diversity of the population.

3. The high-dimensional multi-objective evolutionary method based on improved dominance criteria according to claim 1, characterized in that: In step three, the adaptive t-distribution crossover operator ASDX is combined with the advantages of Cauchy and Gaussian operators. In the early stage of evolution, the strong local exploitation ability of Gaussian distribution is used to ensure that the population can converge quickly. In the later stage of the method, the global search ability of Cauchy operator is used to maintain population diversity.

4. The high-dimensional multi-objective evolutionary method based on improved dominance criteria according to claim 1, characterized in that: The calculation formula is as follows: Formula Thirteen; Where μ is a random number, uniformly distributed in the interval (0,1); η is a constant, representing the crossover parameter; Step two also employs the following method; Method A6 uses the following formula: Formula Fourteen; in, Let x1 and x2 represent the midpoints of the two parent individuals x1 and x2 along the i-th dimension, and This represents the search step size and degrees of freedom in this dimension. The calculation formula is as follows: Formula 15; Among them, when the degrees of freedom are 50, the t-distribution tends to a Gaussian distribution.

Images