Supplementary Materials for Evolutionary Multiobjective Optimization Based Multimodal Optimization: Fitness Landscape Approximation and Peak Detection

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1 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX Supplementary Materials for Evolutionary Multiobjective Optimization Based Multimodal Optimization: Fitness Landscape Approximation and Peak Detection Ran Cheng, Miqing Li, Ke Li, Xin Yao, Fellow, IEEE I. EXPERIMENTAL SETTINGS TABLE I MAIN OPERTIES OF TEST FUNCTIONS IN THE CEC 23 TEST SUITE Function No. of global optima Variable range Peak height r F: Five-Uneven-Peak Trap (D) 2 [, 3] 2.. F2: Equal Maxima (D) 5 [, ].. F3: Uneven Decreasing Maxima (D) [, ].. F4: Himmelblau (2D) 4 [ 6, 6] F5: Six-Hump Camel Back (2D) 2 x [.9,.9]; x 2 [,,.].363 F6: Shubert (2D) 8 [, ] F7: Vincent (2D) 36 [5, ] 2. F8: Shubert (3D) 8 [, ] F9: Vincent (3D) 26 [ 5, 5] 3. F: Modified Rastrigin 2 [ 5, 5] 2-2. F: Composition Function (2D) 6 [ 5, 5] 2. F2: Composition Function 2 (2D) 8 [ 5, 5] 2. F3: Composition Function 3 (2D) 6 [ 5, 5] 2. F4: Composition Function 3 (3D) 6 [ 5, 5] 3. F5: Composition Function 4 (3D) 8 [ 5, 5] 3. F6: Composition Function 3 (5D) 6 [ 5, 5] 5. F7: Composition Function 4 (5D) 8 [ 5, 5] 5. F8: Composition Function 3 (D) 6 [ 5, 5]. F9: Composition Function 4 (D) 8 [ 5, 5]. F2: Composition Function 4 (2D) 8 [ 5, 5] 2. Peak height: function (fitness) value of global optimal solution(s) r: niche radius to distinguish two neighboring global optimal solutions TABLE II SECTION IV-A: MAXIMUM NUMBER OF FES FOR EACH TEST FUNCTION Test Function Maximum Number of FEs F F5 5 4 F6, F7 2 5 F8, F9 4 5 F F3 2 5 F4 F2 4 5 TABLE III SECTION IV-A: POPULATION SIZE SETTINGS OF MOMMOP Test Function Population Size F-F5 8 F6 F7 3 F8-F9 3 F F-F3 2 F4-F2 2

2 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX 2 II. STATISTICAL RESULTS OF THE COMPARISONS BETWEEN EMO-MMO, MOMMOP, NMMSO AND NEA2 ON THE IEEE CEC 23 MULTIMODAL OPTIMIZATION TEST SUITE. TABLE IV SECTION IV-A: THE MEAN PEAK RATIOS AVERAGED OVER 5 RUNS OBTAINED BY EMO-MMO, MOMMOP, NMMSO AND NEA2, AT THE ACCURACY LEVEL OF =, = 3, = 5, RESPECTIVELY. BEST RESULTS ARE LIGHTED. F F F3 F F5 F F7 F F9 F F F F3 F F5 F F7 F F9 F

3 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX 3 TABLE V SECTION IV-A: THE MEAN SUCCESS RATES AVERAGED OVER 5 RUNS OBTAINED BY EMO-MMO, MOMMOP, NMMSO AND NEA2, AT THE ACCURACY LEVEL OF =, = 3 AND = 5, RESPECTIVELY. BEST RESULTS ARE LIGHTED. F F F3 F F5 F F7 F F9 F F F F3 F F5 F F7 F F9 F

4 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX 4 III. BOXPLOTS OF THE RESULTS OBTAINED BY EMO-MMO, MOMMOP, NMMSO AND NEA2 ON EACH TEST FUNCTION IN THE IEEE CEC 23 MULTIMODAL OPTIMIZATION TEST SUITE IN 5 RUNS (a) F (b) F2 (c) F3 (d) F (e) F5 (f) F6 (g) F7 (h) F8.5 (i) F9 (j) F (k) F (l) F2 (m) F3 (n) F4 (o) F5 (p) F6 (q) F7 (r) F8 (s) F9 (t) F2 Fig.. Section IV-A: Boxplots of the results obtained by each algorithm in 5 runs at accuracy level =.

5 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX (a) F (b) F2 (c) F3 (d) F (e) F5 (f) F6 (g) F7 (h) F8.5 (i) F9 (j) F (k) F (l) F2 5 5 (m) F3 (n) F4 (o) F5 (p) F (q) F7 (r) F8 (s) F9 (t) F2 Fig. 2. Section IV-A: Boxplots of the results obtained by each algorithm in 5 runs at accuracy level = 3.

6 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX (a) F (b) F2 (c) F3 5 (d) F (e) F5 (f) F6 (g) F7 (h) F (i) F9 (j) F (k) F (l) F2 5 5 (m) F3 (n) F4 (o) F5 (p) F (q) F7 (r) F8 (s) F9 (t) F2 Fig. 3. Section IV-A: Boxplots of the results obtained by each algorithm in 5 runs at accuracy level = 5.

7 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX 7 IV. STATISTICAL RESULTS OF THE COMPARISONS BETWEEN ORIGINAL EMO-MMO AND ITS MODIFIED VERSIONS ON THE IEEE CEC 23 MULTIMODAL OPTIMIZATION TEST SUITE. TABLE VI SECTION V-A: MEAN PEAK RATIOS AVERAGED OVER 5 RUNS OBTAINED BY ORIGINAL EMO-MMO AND THE MODIFIED EMO-MMO USING THE REAL COORDINATE SYSTEM FOR DIVERSITY MEASUREMENT (EMO-MMO-R), AT THE ACCURACY LEVEL OF =, = 3, = 5, RESPECTIVELY. BEST RESULTS ARE LIGHTED. F F2 F3 F F5 F6 F7 F F9 F F F F3 F4 F5 F F7 F8 F9 F TABLE VII SECTION V-B: THE MEAN PEAK RATIOS AVERAGED OVER 5 RUNS OBTAINED BY ORIGINAL EMO-MMO AND THE MODIFIED EMO-MMO WITH LOCALIZED DE OPERATOR (EMO-MMO-DE), AT THE ACCURACY LEVEL OF =, = 3, = 5, RESPECTIVELY. BEST RESULTS ARE LIGHTED. F F2 F3 F F5 F6 F7 F F9 F F F F3 F4 F5 F F7 F8 F9 F

8 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX 8 V. SENSITIVITY ANALYSES OF CHANGING RATE (α) OF THE GRID-BASED DIVERSITY INDICATOR d grid (x) (a) F (b) F2 (c) F3 (d) F (e) F5 (f) F6 (g) F7 (h) F (i) F9 (j) F (k) F (l) F2. (m) F3 (n) F4 (o) F5 (p) F (q) F7 (r) F8 (s) F9 (t) F2 Fig. 4. Section III-B: Error bars of the results obtained using different changing rate (i.e., settings of α) of the grid-based diversity indicator d grid (x). Results are obtained via 5 runs on the IEEE CEC 23 multimodal optimization test suite.

9 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX 9 VI. LOCALIZED DE OPERATOR FOR MOFLA Algorithm Section IV-B: Localized DE Operator : Input: parent population X = (x, x 2,..., x N ); 2: Output: offspring population X = ( x, x 2,..., x N ); 3: /*Mating Selection*/ 4: σ max{min x i x j }; // adaptive mating neighborhood size i j i 5: Γ = (γ, γ 2,..., γ N ); // mating neighborhoods 6: for i = to X do 7: γ i = {j : x i x j σ}; 8: end for 9: /*DE Recombination*/ : CR =, F = ; // parameters in DE : for i = to N do 2: /*randi3(a): randomly return 3 elements in A*/ 3: if γ i < 3 then 4: (r, r 2, r 3 ) = randi3({, 2,..., N}); 5: else 6: (r, r 2, r 3 ) = randi3(γ i ); 7: end if 8: x i = ( x r,, x r,2,..., x r,d); // i-th offspring solution 9: for j = to D do 2: /*randr(a, b): randomize a real number in [, ]*/ 2: if rand(, ) < CR then 22: x i,j = x i,j + F ( x r2,j x r3,j); 23: end if 24: end for 25: end for

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