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Self-Adaptive Root Growth Model for Constrained Multi-Objective Optimization
Author: Update times: 2013-08-21                          | Print | Close | Text Size: A A A

Multi-objective optimization is a research field that has raised great interest over the past two decades. Many real world optimization problems in engineering, finance, and science involve simultaneous optimization of several objective functions to be simultaneously optimized. Generally, these objective functions are non-commensurable and often competing and conflicting.

To investigate a novel biologically inspired methodology for complex system modeling and computation, particularly for constrained multi-objective optimization, researchers from Shenyang Institute of Automation (SIA), the Chinese Academy of Sciences designed a novel method called “multi-objective root growth algorithm” (MORGA) for constrained multi-objective optimization. It was inspired from the growth behaviors of plant root system. A self-adaptive strategy is adopted to tie this model closer to plant root growth behaviors in nature, as well as improve the robustness of MORGA.

The root growth model proposed is instantiated as multi-objective root growth algorithm (MORGA) for multi objective function optimization. The threshold of the distance between root tips and the growth length of each root hair are important parameters for MORGA. The flowchart of the MORGA is presented in Fig. 1. The pseudo code for the MORGA is listed in Table I.

                                                                                                 

            Fig. 1. The flowchart of MORGA (Image provided by ZHANG Hao, et.al.)

 

Step 1: SET PARAMETER

Set maximum number of cycles (MNC);

Set the number of the loop T = 1;

Set values of the other parameters.

Step 2: INITIALIZE

1) Randomize seed positions;

2) Sort all individuals based on the nondomination. Each solution is assigned a fitness (or rank) equal to its nondomination level.

3) Modify the fitness values using constraint handling approach.

Step 3: LOOP

WHILE (T <= MNC)

FOR (each root tip)

Select root tips based on nonominated sorting procedure and crowding distance.

END FOR

FOR (each root tip selected)

     Produce new growing points (new solutions)

END FOR

/*Growth Phase*/

FOR (each root tip)

IF (the fitness of the new solution > the fitness of the old one)

1)   Produce next new solution from the new solution.

2)   Calculate modified objective function values using the selected constraint handling method for all individuals. Then all the individuals are sorted according to nondomination.

3)   Update the growth length of each root hair μi.

ELSE

     Break.

END IF

Calculate modified objective function values using the selected constraint handling method for all individuals. Then all the individuals are sorted according to nondomination.

IF (the number of individuals > the number of population)

Select exactly m best individuals from all individuals using the crowded-comparison operatorn.

END IF

END FOR

T = T + 1.

END WHILE

 
 TABLE I. PSEUDOCODE OF THE MORGA
 
The researchers conducted simulation on the five benchmark functions to compare the performances of MORGA with non-dominated sorting genetic algorithm (NSGA) and multi-objective particle swarm optimization (MOPSO). The results were obtained from thirty independent runs of MORGA, NSGA and MOPSO. Tableand shows maximum, minimum, average value and standard deviation of the GD metric using the three algorithms MORGA, NSGA and MOPSO for five benchmark functions. The best values have been marked in bold. The numerical results demonstrate MORGA approach is a powerful search and optimization technique for constrained multi-objective optimization.  
 

Function

Item

MORGA

NSGAII

MOPSO

SRN

Max.

4.1709

4.3876

4.5091

Min.

3.5130

3.9003

3.7424

Avg.

4.0100

4.0519

4.1239

Std.

0.1306

0.0989

0.1338

BNH2

Max.

0.3088

0.5193

0.4992

Min.

0.2570

0.4319

0.4498

Avg.

0.2901

0.4718

0.4819

Std.

0.0288

0.0208

0.0192

KITA

Max.

0.0141

0.1623

0.0538

Min.

0.0076

0.0082

0.0101

Avg.

0.0113

0.0322

0.0152

Std.

0.0034

0.0431

0.0194

OSY2

Max.

2.5797

3.2348

2.3814

Min.

1.5708

1.4331

1.5036

Avg.

2.0416

2.5432

1.9935

Std.

0.5078

1.5275

0.9641

TNK

Max.

0.0044

0.0382

0.0201

Min.

0.0014

0.0019

0.0053

Avg.

0.0029

0.0091

0.0089

Std.

0.0034

0.0127

0.0101

 
 TABLE II. THE COMPARISON RESULT OF THE GD METRIC
 

Function

Item

MORGA

NSGAII

MOPSO

SRN

Max.

0.7093

0.7373

0.7233

Min.

0.5817

0.5734

0.5648

Avg.

0.6363

0.6305

0.6283

Std.

0.0386

0.0417

0.0671

BNH2

Max.

0.6780

0.7112

0.7003

Min.

0.5694

0.6029

0.5817

Avg.

0.6184

0.6714

0.6326

Std.

0.0550

0.0391

0.0326

KITA

Max.

0.7650

0.7325

0.7217

Min.

0.5682

0.5734

0.5835

Avg.

0.6132

0.6623

0.6205

Std.

0.1210

0.0543

0.0329

OSY2

Max.

1.0819

0.8390

0.8924

Min.

0.9181

0.6318

0.7002

Avg.

0.9916

0.7462

0.8193

Std.

0.0832

0.0904

0.1153

TNK

Max.

0.7583

0.7413

0.7138

Min.

0.5219

0.5438

0.5321

Avg.

0.6284

0.6410

0.6374

Std.

0.0493

0.0801

0.0597

 
 TABLE III. THE COMPARISON RESULT OF THE Δ METRIC 
 

This work was presented on IEEE Congress on Evolutionary Computation in Cancun, Mexico in Jun 2013. The annual IEEE Congress on Evolutionary Computation is one of the leading events in the area of evolutionary computation. It covers all topics in evolutionary computation. CEC was held annually since 1999 and was the fifteenth occurrence of this major event in the evolutionary calendar this year. CEC2013 was sponsored by the IEEE Computational Intelligence Society and cosponsored by the Evolutionary Programming Society, and the Institution of Engineering and Technology.

 

 

CONTACT:

Professor: Yunlong Zhu

Doctoral candidate: Hao Zhang

Shenyang Institute of Automation, Chinese Academy of Sciences

Email: Zhanghao@sia.cn

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