Tomtat luanan tienganh nghiên cứu, phát triển một số kỹ thuật Điều khiển thích Ứng nhằm cải thiện chất lượng giải thuật tiến hóa tối Ưu Đa mục tiêu

Objectives of the thesis Developing some adaptive control techniques to maintain a balance between exploration and exploitation capabilities in the evolution process of multi-objective

MINISTRY OF EDUCATION AND TRAINING MINISTRY OF NATIONAL DEFENCE

ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY

TRAN BINH MINH

RESEARCHING AND DEVELOPING SOME ADAPTIVE CONTROL TECHNIQUES TO IMPROVE THE QUALITY OF MULTI-OBJECTIVE EVOLUTIONARY ALGORITHMS

Specialization: Mathematical foundation for informatics

SUMMARY OF PhD THESIS IN MATHEMATICS

Hanoi, 2024

ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY

Scientific Supervisors:

1 Assoc Prof Dr Nguyen Long

2 Dr Thai Trung Kien

Reviewer 1: Prof Dr Nguyen Hieu Minh

Academy of Cryptography Techniques

Reviewer 2: Assoc Prof Dr Nguyen Quang Uy

Military Technical Academy

Reviewer 3: Dr Do Viet Binh

Academy of Military Science and Technology

The thesis was defended by the Dissertation Evaluation Committee

at the Academy level, held at the Academy of Military Science and Technology at ………, 2024

The thesis can be assessed at:

- The Library of the Academy of Military Science and Technology

- The National Library of Vietnam

INTRODUCTION

1 The necessity of the thesis

The quality and effectiveness of the multi-objective evolutionary algorithm (MOEA) are evaluated on two aspects: The solution set quality and the algorithm search efficiency Maintaining the balance between exploration and exploitation capabilities has a significant impact on the algorithm’s search efficiency and the solution set quality The reference information that has often been used to keep the equilibrium is implemented from the beginning and independent of the evolutionary process Some important information can

be extracted from the evolution process, such as the variation trend of population’s convergence and diversity indicators over periods or empty regions appearing in the population’s distribution have not been used or not fully used Therefore, researching and appropriately using the information mentioned in the adaptive control mechanism to balance the exploration and exploitation capabilities of the algorithm is necessary and has high scientific and practical significance

2 Objectives of the thesis

Developing some adaptive control techniques to maintain a balance between exploration and exploitation capabilities in the evolution process of multi-objective evolutionary algorithms Applying them to improve some typical multi-objective evolutionary algorithms to demonstrate the effectiveness and universality of the developed techniques

3 Object and scope of the research

Objects of the research: The adaptive control technique to balance

exploration and exploitation abilities in the evolutionary process of multi-objective evolutionary algorithms

Scope of the research: Typical multi-objective evolutionary

algorithms in the case of the Pareto optimal front of the multi-objective optimization problem are continuous

4 Content of the research

- Overview of multi-objective evolutionary algorithms, evaluating the algorithm quality balancing exploration and exploitation capabilities

- Researching and developing the adaptive control technique based on the variation trend of solution set’s convergence and diversity indicators and applying

it to improve DMEA-II, MOEA/D, MOEA/D-DE, and NSGAII-DE algorithms

- Researching and developing the adaptive control technique based on the population’s distribution and applying it to improve DMEA-II and MOEA/D algorithms

5 Research method

The analysis-synthesis method is used to synthesize and classify adaptive control techniques The inductive-deductive method is used to analyze existing problems The experimental method is used to test and evaluate the quality and effectiveness of improved algorithms The analysis - summary method is used to make assessments on advantages and disadvantages, draw conclusions, and future works

6 The scientific and practical significance of the thesis

Scientific significance: The thesis contributes two new adaptive control

techniques to balance the exploration and exploitation capabilities and some improved algorithms based on these techniques to the research field

Practical significance: Proposed techniques and improved algorithms can

be well applied to solve multi-objective optimization problems in practice

7 Structure of the thesis

The main content of the thesis is presented in 3 chapters, with 43 illustrative drawings and 17 tables, using 104 reference documents in two languages (Vietnamese and English) The thesis structure is as follows: Introduction, three chapters, conclusion, references, and appendices

Chapter 1 OVERVIEW OF MULTI-OBJECTIVE EVOLUTIONARY

ALGORITHMS 1.1 Multi-objective optimization problem

Is the person who makes the final decision on the MOP’s solutions

1.1.4 Application of MOP in practice

MOP is widely applied in many different fields from economics - society, science and technology, security - defense

1.1.5 MOP solving method

Including traditional methods and evolutionary principles-based methods,

in which MOEA is an effective method to solve MOP

1.2 Multi-objective evolutionary algorithm

1.2.1 Overview of the algorithm

MOEA uses evolutionary principles to find globally optimal solutions for MOP MOEA has gone through four stages of development, divided into

dominance-based, decomposition-based, direction-based, indicator-based and hybrid groups The basic operation scheme of MOEA includes three main phases: Initialization, evolution loop, and termination.

1.2.2 Some typical MOEA

Some typical algorithms to generalize and aim to apply proposed adaptive control techniques are direction-based DMEA-II, decomposition-based MOEA/D, partitioning and differentiation-based MOEA/D-DE, dominance and differentiation-based NSGAII-DE

1.3 Evaluating the quality and effectiveness of MOEA

1.3.1 Quality assessment of the solution set

Convergence and diversity are two core elements in MOEA's quality assessment Convergence is solution approaching the Pareto optimal front (PF), while diversity is solution widely and evenly distributed according to the

PF The solution set quality is evaluated quantitatively by the indicators with some standard metrics such as GD, IGD, and HV The experimental data sets are benchmarks, and some typical classes are ZDT, UF, DTLZ, and WFG

1.3.2 Search effectiveness assessment of MOEA

Aiming to achieve a good quality solution set in terms of convergence and diversity, it is necessary to both locally search to obtain solutions set that are asymptotic to the PF and widely search to ensure global optimality Therefore, MOEA needs the ability to exploit the vicinity of the obtained solutions and the ability to explore new regions in the objective space

1.3.3 Other criteria assessment

Including robustness and computational complexity of MOEA

1.4 Some issues in evaluating the quality and effectiveness of MOEA

1.4.1 Balancing between the convergence and diversity of the solution set

If convergence quality is low, the objective function value is relatively poor in at least one objective On the contrary, when diversity quality is low, the solution set has a high similarity in value across objectives but may ignore globally optimal choices According to the principle of evolution, the quality

of the solution set in the previous generation will directly affect the next generation, so it is necessary to simultaneously achieve the quality of convergence and diversity in previous generations

1.4.2 Balancing between the exploration and exploitation capabilities

If MOEA prioritizes exploration ability, the search will be carried out widely in the search space, leading to ensuring globality and improving diversity quality, but the speed of obtaining solutions approaches to the PF will be slow and will not improve convergence quality On the contrary, when MOEA is biased towards exploitation ability, the search will be performed narrowly around the explored regions, leading to improved convergence

quality, but the ability to expand the search space is limited and has not improved the diversity quality The imbalance between the ability to explore and exploit in the previous stage greatly affects the search process in the later stage, so it is necessary to maintain balance in the early stage to ensure the search efficiency of the algorithm throughout the entire evolutionary process

1.4.3 Adaptive control techniques to keep balance between exploration and exploitation capabilities

Some typical groups of the adaptive control techniques to keep a balance between exploration and exploitation capabilities are spatial partitioning-based, indicators-based, and algorithm parameters-based techniques

1.5 Proposing research content of the thesis

1.5.1 Existing problems in the research field

- The utilization of the variation trend of solution set’s convergence and diversity indicators in time segments as a quantitative basis to evaluate the search trend of the algorithm and use correlation information between indicators as reference information to control the search process to maintain a balance between exploration and exploitation capabilities in the next generations has not yet been focused

- If empty regions are defined as areas in the objective space between selected solutions, then information about empty regions in the population distribution has not been focused on and used as reference information in adaptive control techniques to keep a balance between the exploration and exploitation capabilities of the algorithm during the evolution process

1.5.2 Research hypothesis

- If using information about the algorithm’s search trend determined by the variation trend of solution set’s convergence and diversity indicators according to time segments and correlation information between the algorithms indicator as reference information for adaptive control in the direction of enhancing under-priority capability will help better maintain the balance between exploration and exploitation capabilities in the evolutionary process, contributing to improving the quality and efficiency of MOEA

- If empty region information in the population distribution is used as reference information to prioritize searching in those areas, it will help expand the search area and avoid ignoring reasonable solutions, thereby better maintaining the balance between exploration and exploitation capabilities, contributing to improving the quality and efficiency of MOEA

1.5.3 Research content of the thesis

- Researching and developing the adaptive control technique based on the variation trend of solution set’s convergence and diversity indicators and applying

it to improve DMEA-II, MOEA/D, MOEA/D-DE, and NSGAII-DE algorithms

- Researching and developing the adaptive control technique based on the population’s distribution and applying it to improve DMEA-II and MOEA/D algorithms

1.6 Conclusion of Chapter 1

Chapter 1 has presented background knowledge, focusing on the role of maintaining a balance between MOEA’s exploration and exploitation capabilities Some shortcomings were identified as a basis for hypothesizing and determining the contents that need to be researched in the thesis

Chapter 2 RESEARCHING AND DEVELOPING THE ADAPTIVE CONTROL TECHNIQUE BASED ON THE VARIATION TREND OF SOLUTION SET’S CONVERGENCE AND DIVERSITY INDICATORS 2.1 The adaptive control technique based on the variation trend of solution set’s convergence and diversity indicators

2.1.1 The relationship between solution indicator quality and the search trend of MOEA

The convergence metric reflects the algorithm’s exploitation ability, and the diversity measure reflects the algorithm’s exploration ability Therefore, the variation trend of solution set’s convergence and diversity indicators towards improvement in a time segment represents the enhancement of the algorithm's exploitation/exploration ability and search trend, the correlation variation of indicators can be used as reference information for adaptive control to maintain

a balance between the exploration and exploitation capabilities of MOEA

2.1.2 Proposing the adaptive control technique based on the variation trend

of solution set’s convergence and diversity indicators

Analyzing the variation trend of solution set’s convergence and diversity indicators over time segments to determine the search trend of the algorithm, using the correlation variation of measures as reference information to adaptive control the evolutionary process The chosen metrics are GD and IGD When embedded in the basic MOEA scheme, The basic steps of the technique are presented in Figure 2.1

(1) Initialization: Set up the parameters of the algorithm and technique,

initialize the population P 0; calculate the GD and IGD values of the

population P 0 ; initialize the value of maxGD and maxGD as a reference when

comparing correlation variation:

of GD and IGD up to the cGen generation

- If cGen is the last generation of adjustment cycle:

+ Calculate changes in GD and IGD during the cycle:

+ Adjust the algorithm control parameter value by Δ value

- Following the original algorithm’s scheme, the control parameter with varying values will adaptively control the evolution process in the next time segment

(3) Termination: The solution set provided to the decision maker

2.1 Calculate GD, IGD values

2.2 Calculate maximum values of D and IGD to the current generation

3 Termination: Good solutions to provide to DM

Yes No

Steps in the adaptive control technique based on the variation trend of solution set s convergence indicators

Algorithm 2.1 CalculateCorrelation

Input: cGen: current generation; P: population of the cGen

generation; Ge: number of generations in the adjustment process; G c : number of generations in an adjustment cycle; GD e , IGD e : sets store GD, IGD value

in the evolution process; maxGD, maxIGD: maximum GD, IGD value up to cGen; dt old : delta value of the previous cycle

Output: The correlation variation value dt

1: if (cGen ≤ G e) then

2: gd  GD(P)

3: igd  IGD(P)

4: GD e (cGen)  gd; IGD e (cGen)  igd

5: if (maxGD < gd) then maxGD  gd end if

6: if (maxIGD < igd) then maxIGD  igd end if

2.1.3 Applying to enhance MOEA

The application process must be consistent with the characteristics and based on a survey of control parameters affecting MOEA’s exploration/exploitation capabilities

2.2 Proposing to improve some typical MOEA

2.2.1 The DMEA-II++ algorithm

The original DMEA-II algorithm selects child solutions in the direction of convergence and dispersion based on the ratio of non-dominant solutions in the solution set However, the large number of non-dominant solutions is not synonymous with poor diversity and vice versa, so it cannot show the algorithm’s search trend The thesis uses a flexible ratio based on the correlation between indicators and the number of solutions generated in selected directions as follows:

(0, 5 ) x ;

In which: n CD and n SD are number of solutions selected by convergence and

dispersion directions; Normalize() to ensure n is between 10% and 90% of

the individuals in the population to avoid over-prioritisation in one direction When Δ > 0 (the algorithm is prioritizing exploitability), the number of

child solutions generated in the convergent direction n CD is smaller than the

number of child solutions generated in the dissipative direction n SD, demonstrating that adjustment is in the direction of increase exploration In contrast to the case of Δ < 0 In the case Δ = 0, the number of child solutions

in both directions is balanced

Algorithm 2.2 DMEA-II++ [CT2]

Input: MOP with M is objective number; stop criteria; N:

population size; Ge: number of generations in the adjustment process; G c : number of generations in an adjustment cycle

Output: The population P contains the solutions of MOP

1: Initialize P; C  ∅; CD  ∅; SD  ∅;

4: while (stopping criteria is not satisfied) do

2.2.2 The MOEA/D+ algorithm

Algorithm 2.3 MOEA/D+ [CT3]

Input: MOP with M is objective number; stop criteria; N:

sub-problem number; weight vector set λ = (λ 1 ,…,λ N);

T: neighbor set size; Ge: number of generations in the adjustment process; G c : number of generations in an

adjustment cycle; ε: changing threshold

Output: The population EP contains the solutions of MOP

8: while (stopping criteria is not satisfied) do

9: for i = 1 to N

10: y  Reproduction(B(i)); y’  Enhance(y)

11: for j = 1 to M /* Update reference point */

12: if z j > f j (y’) then z j  f j (y’) end if

In the original MOEA/D algorithm, the large number of neighboring

vectors of a weight vector (T) leads to parent solutions selected to be put into

the matting pool being far apart, from which child solutions can be very far from the parent solutions, leading to increasing the diversity In contrast to the

case T has a small value Thus, if the T parameter value is adjusted in an

appropriate range, it can direct enhance the ability to explore or exploit

The parameter T in the MOEA/D algorithm is fixed at the initialization

step, independent of the evolutionary process, there is no mechanism to use flexible parameters based on the algorithm’s search trend The thesis uses a dynamic number of neighbor vectors based on the correlation variation between measures during the evolutionary process as follows:

11

niche niche

algorithm is prioritizing exploitation ability), the niche value will increase,

leading to an increase in the number of neighboring vectors to show that the adjustment direction is to prioritize exploration In contrast to the case where Δ exceeds the lower threshold In case Δ is under the threshold, the number of neighboring vectors does not change

2.2.3 The MOEA/D-DE+ algorithm

In the original algorithm MOEA/D-DE, the step length parameter of differential variation in the mutation strategy “DE/rand/1” affects the

convergence and diversity quality of the solution set When ꞵ is small, the

amplification level is low, leading to reduced diversity and vice versa

The original algorithm MOEA/D-DE sets a fixed value of ꞵ (equal to 0.5)

from the initialization stage, independent of the evolution process The thesis uses a flexible step length parameter of the differential variation calculated based on the correlation variation between indicators

The calculating formula of the flexible step length parameter:

0, 5 x )(

Normalize

In which: ω is the step length to adjust ꞵ value, the experimental value is 20; Normalize() to ensure ꞵ belongs to the [0.1; 0.9] range to avoid over-

prioritisation in one direction

When Δ > 0 (the algorithm is prioritizing exploitability), the amplification level is high to prioritize exploration ability In contrast to the case where Δ < 0

In the case Δ = 0, ꞵ = 0.5, like the original algorithm

Algorithm 2.4 MOEA/D-DE+ [CT5]

Input: MOP with M is objective number; stop criteria; N:

sub-problem number; weight vector set λ = (λ 1 ,…,λ N);

T: neighbor set size; n r : maximum number of parent replaced by child; Ge: number of generations in the adjustment process; G c : number of generations in an adjustment cycle; ω: step length

7: while (stopping criteria is not satisfied) do

a d

B i N

13: y  Reproduction(B(i)); y’  Enhance(y)

14: for j = 1 to M /* Update reference point */

15: if z j > f j (y’) then z j  f j (y’) end if