4.3 Minimization using River Formation Dynamics Scheme
4.3.10 Performance Assessment
4.3 Minimization using River Formation Dynamics Scheme
as [126],
SP =fmean× Runtotal
Runsuccess, (4.25)
wherefmeandenotes mean of successful runs achieved by the algorithm;RuntotalandRunsuccess denote number of total runs and number of successful runs achieved by the algorithm, respec- tively.
effort (i.e., number of FEs). For algorithms, which fail to converge to a global optimum so- lution (number of FEs is not available), rank of median final function value is obtained and listed within angle brackets in Table 4.2. It can be observed that MRFD is able to locate near-optimal solution for all35test functions with relatively better success rate and minimum FEs, indicating effectiveness and statistical stability. During numerical analysis, mean of best fitness is considered out of 25runs to observe the evolution in the movement of sediments and the standard deviation of best fitness is evaluated to observe the consistency in achieving global optimum solution.
It can be observed from Table 4.2 that proposed MRFD method demonstrates sufficient improvement in solving these test functions with minimum number of FEs over traditional RFD method in solving unimodal functions (i.e., F1-F5, Sphere and Rosenbrock). The success rate for completing 25independent runs is100% in all the cases. The MRFD takes 1e+ 05,1.5e+ 05,1e+ 05,1.1e+ 05and1e+ 05function evaluations forF1,F2, F3, F4and F5test functions, respectively, to achieve convergence using ten decision variables. However, except forF2andF4test functions, MRFD demonstrates superior performance over SPSO and G-CMA-ES in solving unimodal test functions in terms of number of FEs. UMOEAII shows better performance for F2 test function and L-SHADE demonstrates better perfor- mance while solving F4 test function. On the other hand, for functions, Rosenbrock and Sphere, the results listed in Table 4.2 show a competitive performance of MRFD as compared to all peer algorithms. BothSphere andRosenbrockfunctions are implemented for five de- cision variables (dimension=5). It takes10400and100function evaluations forRosenbrock and Sphere functions, respectively, to achieve satisfactory accuracy. To demonstrate the effectiveness of MRFD over unimodal functions, plots of mean and standard deviations of fitness values and decision variables over25runs are shown in Figure 4.4 forF3function. It can be observed from Figure 4.4(a) that MRFD quickly converges to global optimum solution within a relatively small number of FEs (<1e+ 05) forF3function. However, the decision variables converge at1e+05FEs (refer Figure 4.4(b) and Figure 4.4(c)). Therefore, minimum
4.3 Minimization using River Formation Dynamics Scheme
FEs are evaluated after verification of convergence of all decision variables in25independent runs.
Table 4.2: Performance analysis of MRFD on different single objective test functions.
Test Functions SP SO U M OEAII LSHADE G-CM A-ES RF D M RF D min. FEs Cost Function Property/Shape/Number of constraints Unconstrained Functions
3-hump camel 1(100%) 1(100%) 1.2(100%) 8(100%) 4(100%) 1(100%) 100 Multimodal, three local minima, valley-shaped, distributed
Ackley 1(100%) 1(100%) 1(100%) 8.5(100%) 3(80%) 1(100%) 210 Multimodal
Beale 1(100%) 1(100%) 1(100%) ⟨6⟩ 4(100%) 1(100%) 100 Multimodal, sharp peaks at corner
Booth 1(100%) 1(100%) 1(100%) 8(100%) 2(100%) 1(100%) 100 Plate-shaped
Cross-in-tray 1(100%) 1(100%) 1(100%) 7(100%) 6(100%) 1(100%) 100 Multimodal, multiple global minima Easom 1(100%) 1(100%) 1(100%) ⟨6⟩ 3(100%) 1(100%) 100 Multimodal, multiple local minima, steep ridges/drops Eggholder 2(100%) 1.3(92%) 1(100%) ⟨6⟩ 1(100%) 1(100%) 100 Multimodal, multiple local minima Goldstein-Price 1(100%) 1(100%) 2.4(100%) 13(100%) 1(100%) 1(100%) 100 Multimodal, multiple local minima
Holder Table 1(100%) 1(100%) 1(100%) ⟨6⟩ 1(100%) 1(100%) 100 Multimodal
Levy-13 1(100%) 1(100%) 1(100%) 14(100%) 1(100%) 1(100%) 100 Multimodal
Matyas 1(100%) 1(100%) 1(100%) 7(100%) 1(100%) 1(100%) 100 Multimodal, one global minima, plate-shaped
McCormick 1(100%) 1(100%) 1(100%) 10(100%) 3(100%) 1(100%) 100 Multimodal, plate-shaped
Rastrigin ⟨5⟩ 3.1(92%) 2.6(88%) ⟨4⟩ ⟨6⟩ 1(100%) 250 Multimodal
Rosenbrock 2.4(80%) 1.3(92%) 1.56(88%) 1.83(100%) 2.67(72%) 1(96%) 10400 Unimodal, valley-shaped
Schaffer-2 1(100%) 1.1(100%) 1.1(100%) ⟨6⟩ 1.1(92%) 1(100%) 100 Multimodal
Schaffer-4 2(100%) 1.8(100%) 1(100%) ⟨6⟩ 2.3(96%) 1(100%) 100 Multimodal
Sphere 3(100%) 1.2(100%) 1(100%) 58(100%) 1.4(100%) 1(100%) 100 Unimodal, bowl-shaped
Styblinski-Tang 1(100%) 1(100%) 1(100%) ⟨6⟩ 1(100%) 1(100%) 300 Multimodal
CEC-2014 Unconstrained Benchmark Functions (Dimension=10)
F1 60(100%) 1(100%) 1(100%) 8(100%) ⟨6⟩ 1(100%) 1e+05 Unimodal, shifted, separable, scalable F2 5(100%) 1(100%) 1.3(100%) 6.1(100%) ⟨6⟩ 1.2(100%) 1.2e+05 Unimodal, shifted, non-separable, scalable F3 117(100%) 1(100%) 1.4(96%) 65(100%) ⟨6⟩ 1(100%) 1e+05 Unimodal, shifted, rotated, non-separable, scalable F4 ⟨5⟩ 1.5(100%) 1(100%) 12.3(100%) ⟨6⟩ 1.3(100%) 1e+05 Unimodal, shifted, non-separable, scalable, noise in fitness F5 548.70(80%) 1(100%) 1(100%) 29.50(100%) ⟨6⟩ 1(100%) 1e+05 Unimodal, non-separable, scalable
Avg. rank 2.65 1.6 1.7 5.26 4.13 1.1 - Rank on all unconstrained functions
Constrained Functions
Rosenbrock-1 ⟨6⟩ 1.3(88%) 1(100%) ⟨5⟩ 1.36(80%) 1(100%) 2200 Two constraints, constrained with a cubic and a line Rosenbrock-2 1(100%) 3.3(100%) 2(100%) 20(100%) 4(100%) 1(100%) 150 Single constraint, constrained to a disk
Simionescu ⟨6⟩ 2.6(92%) 1.1(100%) ⟨5⟩ 5(100%) 1(100%) 300 One constraint
Mishra 1(100%) 1.1(100%) 1(100%) 17(100%) 1.8(100%) 1(100%) 220 Single constraint
g11 2(100%) 1.75(100%) 1.2(100%) 3.1(100%) 4.6(90%) 1(100%) 200 Single equality constraint
g12 2.1(100%) 2(88%) 1(100%) 4(100%) 7.1(80%) 1(100%) 300 Eight inequality constraints
g13 10(100%) 1.66(100%) 1.1(100%) 7.2(100%) 12.3(70%) 1(100%) 600 Exponential function, three equality constraints CEC-2017 Constrained Benchmark Functions (Dimension=10)
C01 15.6(92%) 2.2(100%) 2(100%) 12.8(100%) ⟨6⟩ 1(100%) 1e+05 Non-separable, single inequality constraint C02 12.2(92%) 2.5(96%) 1(100%) 5.1(100%) ⟨6⟩ 2.25(100%) 2e+05 Non-separable, rotated, single inequality constraint C03 ⟨6⟩ 1.8(72%) 1.5(92%) 45(40%) ⟨5⟩ 1(100%) 1.3e+05 Non-separable, one inequality and one equality constraints C04 22.4(92%) 2(96%) 1(100%) 14.3(100%) ⟨6⟩ 1.90(100%) 2e+05 Separable, two inequality constraints C05 15.5(88%) 1.5(72%) 1.1(100%) 8.2(92%) ⟨6⟩ 1(100%) 1.8e+05 Non-separable, two inequality constraints
Avg. rank 4.3 3.16 1.58 4.67 5.42 1.16 - Rank on all constrained functions
For multimodal functions listed in Table 4.2, the results clearly indicate that MRFD can achieve convergence while optimizing these test functions with satisfactory accuracy. It can be observed that MRFD shows1.1(minimum) to 58(maximum) folds improvement in solv- ing multimodal functions with minimum number of FEs over traditional RFD and other peer algorithms. Figure 4.5 shows typical evolution curves ofRastriginfunction having four de- cision variables (dimension=4), which demonstrates a stable convergence behavior of MRFD over25runs, despite the occurrence of numerous local optima in this function.
0 5E+04 1E+05 Function Evaluation (FE)
0 2 4 6 8
Fitness value
×10-3
Mean Standard deviation
(a)
0 5E+04 1E+05
Function Evaluation (FE) -100
-50 0 50 100
Decision variables (Mean)
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
(b)
0 5E+04 1E+05
Function Evaluation (FE) 0
5 10 15 20 25 30 35
Decision variables (SD)
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
(c)
Figure 4.4: (a) Evolution of global fitness values, (b) mean and (c) standard deviation (SD) of ten variables ofF3test function at different function evaluations.
0 50 100 150 200 250
Function Evalution (FE) 0
5 10 15 20
Fitness value
Mean Standard deviation
(a)
0 50 100 150 200 250
Function Evaluation (FE) -0.3
-0.2 -0.1 0 0.1 0.2
Decision variables (Mean)
x1 x
2 x
3 x
4
(b)
0 50 100 150 200 250
Function Evaluation (FE) 0
0.5 1 1.5 2
Decision variables (SD)
x1 x2 x3 x4
(c)
Figure 4.5: (a) Evolution of global fitness values, (b) mean and (c) standard deviation (SD) of four variables ofRastrigintest function at different function evaluations.
Further, to demonstrate the applicability of proposed MRFD method on single objec- tive constrained optimization problems, we consider 12 different constrained test functions (Rosenbrock-1,Rosenbrock-2,Simionescu,M ishra,g11,g12,g13andC01-C05) as listed in Table 4.2. To handle any violations in constraints, Deb’s rules described in [127] are fol- lowed to separate the occurrence of infeasible solutions from feasible ones. It can be observed that MRFD shows superior performance over traditional RFD and peer algorithms in solving all seven constrained test functions in terms of number of FEs. The proposed MRFD method solves M ishra, Rosenbrock-1, Rosenbrock-2, Simionescu, g11, g12 and g13 functions within 220, 2500, 150, 300, 200, 300 and 600 number of FEs, respectively. Moreover, it can be observed from Figure 4.6 that the evolution curves (mean and standard deviation) of fitness value and decision variables for Rosenbrock-1function demonstrate a stable behav-
4.3 Minimization using River Formation Dynamics Scheme
0 500 1000 1500 2000 2500
Function evaluation (FE) 0
0.2 0.4 0.6 0.8 1
Fitness value
Mean Standard deviation
(a)
0 500 1000 1500 2000 2500
Function Evaluation (FE) 0
0.2 0.4 0.6 0.8 1 1.2
Decision variables (Mean)
x1 x 2
(b)
0 500 1000 1500 2000 2500
Function evaluation (FE) 0
0.1 0.2 0.3 0.4 0.5 0.6
Decision variables (SD)
x1 x2
(c)
Figure 4.6: (a) Evolution of global fitness values, (b) mean and (c) standard deviation (SD) of four variables ofRosenbrock-1test function at different function evaluations.
1 1E+01 1E+02 1E+03 1E+04 1E+05
Function Evaluation (FE) 0
2000 4000 6000 8000 10000 12000
Fitness value
Mean Standard deviation
(a)
1 1E+01 1E+02 1E+03 1E+04 1E+05 Function Evaluation (FE) -50
-40 -30 -20 -10 0 10 20 30 40
Decision variables (Mean)
x 1 x2 x3 x
4 x5 x
6 x7 x8 x
9 x10
(b)
1 1E+01 1E+02 1E+03 1E+04 1E+05 Function Evaluation (FE)
0 10 20 30 40 50 60 70
Decision variables (SD)
x 1 x2 x
3 x4 x
5 x6 x
7 x8 x
9 x10
(c)
Figure 4.7: (a) Evolution of global fitness values, (b) mean and (c) standard deviation (SD) of ten variables ofC01test function at different function evaluations (FEs). FE is considered in log scale.
ior over several FEs, which showcase five fold improvement over traditional RFD method in terms of number of FEs. Moreover, for functionsC01-C05, the results shown in Table 4.2 indicate that MRFD can solve these functions with higher success rates as compared to other peer algorithms. In addition, MRFD exhibits minimum FEs while solving these CEC-2017 constrained test functions as compared to other peer algorithms except forC02andC04test functions. Figure 4.7 shows typical evolution curves of C01 test function having ten deci- sion variables (dimension=10), which demonstrates a convergence behavior of MRFD over 25runs. It can be observed from Figure 4.7 that evolution curves ofC01test function achieve optimum values with minimum FEs (i.e.,≤1E+ 05).
As the evolution curves (mean and standard deviation) of fitness approach towards opti- mum value (i.e., zero), there appears to be a stable decrease in the fitness value with increase
in FEs. Such stable behavior of MRFD enables to achieve consistency in achieving global optimum solution. A composite ranking8is also evaluated to test the performance of all algo- rithms on test functions listed in Table 4.2 and it can be observed that MRFD has an average rank of1.1and1.16for unconstrained and constrained test functions, respectively, which are found to be the smallest among all. This demonstrates that MRFD maintains a consistent and satisfactory performance while solving the test functions.
The MRFD method can be extended in future to solve other complicated unimodal and multimodal functions [128–130] to demonstrate its applicability and effectiveness. However, as one of the motivations of this proposed work is to demonstrate the applicability of MRFD method in optimizing real-life single objective problems, we focus our discussion to minimize IR drops in VLSI power distribution networks.