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] 2 2.. F5: Six-Hump Camel Back (2D) 2 x [.9,.9]; x 2 [,,.].363 F6: Shubert (2D) 8 [, ] 2 86.73 F7: Vincent (2D) 36 [5, ] 2. F8: Shubert (3D) 8 [, ] 3 279.935 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
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 F2........ 3........ 5........ F3 F4........ 3........ 5....... 9 F5 F6...... 98 63 3...... 92 58 5........ F7 F8... 46. 87 54 4 3... 8. 8 22 4 5.... 8 7 39 F9 F.. 78 22.... 3 56. 78 84.... 5 54 8 78 79... 8 F F2. 97 9 8. 85 95 53 3. 3 9 67. 55 95 43 5. 73 9 6. 88 9 33 F3 F4. 57 9 77. 3 7 3 3. 77 83 6 23 67 23 5. 67 83 47 23 67 2 F5 F6 9 78 5 43. 43 6 73 3 8 38 42 2 67 43 6 73 5 8 2 32 3 67 43 6 73 F7 F8 95 5 8 95 7 87 5 67 3.338 5 7 95.33 87 5 67 5.338 85 6 95.33 87 5 63 F9 F2 37.37 6 67.33 5.363 3.37 57 67 5 5 2.36 5.37 37 67 5 5 2.35
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 F2........ 3........ 5........ F3 F4........ 3........ 5........ F5 F6..... 8 6 8 3..... 6 8 4 5........ F7 F8.... 6.6. 3....6. 4.2. 5....4. 4.. F9 F. 8..... 3. 8..... 5..4.... 6 F F2. 8 4 8. 8 6 3. 2 4. 6 6 5.. 4 6..8 2 F3 F4. 6 4 6...2 3.. 6.6..2.8 5...6...6 F5 F6 2...2.... 3........ 5........ F7 F8 4...... 3........ 5........ F9 F2.4....... 3........ 5........
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..5.5.5.5 (a) F (b) F2 (c) F3 (d) F4.5 5 5 5 5 (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 =.
IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX 5.5.5.5.5 (a) F (b) F2 (c) F3 (d) F4.5 5 5 5 5 (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) F6.3.3. (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.
IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX 6.5.5.5 5 5 (a) F (b) F2 (c) F3 5 (d) F4.5 5 5 5 5 (e) F5 (f) F6 (g) F7 (h) F8 8 6 4 2 (i) F9 (j) F (k) F (l) F2 5 5 (m) F3 (n) F4 (o) F5 (p) F6.3.3. (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.
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 F4........ 3........ 5........ F5 F6 F7 F8..... 39. 5 3..... 39. 5 5..... 33. 5 F9 F F F2. 79..... 88 3 56 98..... 88 5 54 97..... 88 F3 F4 F5 F6. 33. 5 9 3. 67 3. 7 23 67 8 38 67 67 5. 7 23 67 8 38 67 67 F7 F8 F9 F2 95.325 7 7 37 7.33 3.338.33.33 5 7 5 5 5.338.33.33 5 7 5 5 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 F4........ 3........ 5........ F5 F6 F7 F8..... 87. 9 3..... 87. 9 5..... 86. 9 F9 F F F2. 67...... 3 56 3...... 5 54 8...... F3 F4 F5 F6. 43.. 9 95.. 3. 63 23 7 8 9 67 67 5. 63 23 7 8 9 67 67 F7 F8 F9 F2 95 65 7 53 37 7.33 3.338 23.33 73 2 5 3 5.338 23.33 73 7 5 3
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) F4.5 95 8 - - 9 85 8 75 6 4 2 (e) F5 (f) F6 (g) F7 (h) F8 5.2 5 8 5 6-4 - (i) F9 (j) F (k) F (l) F2. (m) F3 (n) F4 (o) F5 (p) F6 - - - - -2-2 -2-2 (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.
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