Self-Adaptive Root Growth Model for Constrained Multi-Objective Optimization----Shenyang Institute Of Automation ,Chinese Academy Of Sciences

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:SETPARAMETER

Set maximum number of cycles (MNC);

Set the number of the loopT= 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 andcrowding 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 hairμ_i.

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 exactlymbest individuals from all individualsusing the crowded-comparison operator_n.

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 algorithmⅡ (NSGAⅡ) and multi-objective particle swarm optimization (MOPSO). The results were obtained from thirty independent runs of MORGA, NSGAⅡ and MOPSO. TableⅡand Ⅲ shows maximum, minimum, average value and standard deviation of the GD metric using the three algorithms MORGA, NSGAⅡ 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