4.3 Minimization using River Formation Dynamics Scheme
4.3.9 Performance on Single Objective Test Functions
4.3.8.2 Case II
ForD = 0 (bothR = 0and V = 0), all eigenvalues are real, where at least two eigen- values are found to be equal. This condition generates negative values of w for R = 0 (i.e.,−1/27(13w2+ 4w+ 1)(1 + 2w) = 0) andV = 0(i.e.,−1/9(w2 + 4w+ 1) = 0).
4.3.8.3 Case III
ForD < 0, all eigenvalues are found to be real and unequal. Further, forD < 0, V < 0 and |V| > R, considering V = −1andR = 0, −19(w2 + 4w+ 1) = −1condition can be employed to evaluate a stable bound forw. Solving this, a fixed lower bound of w = 1.464 can be obtained while evaluating a scheduled bound.
We conclude this analysis by summarizing that an estimate of constriction coefficient, i.e., 1< w < 2can be used as an estimate of evaluating the points with the best estimated objec- tive function values3. Although the proposed developments in the model described in section 4.3.6 follow a random probabilistic search, a scheduled bound for constriction coefficients (w1, w2, w3) in (4.21) supports the direction of return of estimated objective function value at the estimated optimal solution in each iteration. This favors the iteration to converge faster for such a random procedure.
4.3 Minimization using River Formation Dynamics Scheme
4.3.9.1 Test Problems
In this section, we provide insight into different sets of both constrained and unconstrained problems solved during analysis. These problems have been used in the literature as bench- marks. We have selected35different single objective problems for evaluation, in which23 are unconstrained and12are constrained test problems. The first set of18unconstrained test functions4 are based on different genre and five set of unconstrained scalable test problems (F1toF5) are selected from CEC-2014single objective benchmark suite (having dimension5 10) [121]. The second set of12constrained problems consist of seven constrained test func- tions of different genre4, i.e.,Rosenbrock1, Rosenbrock2, M ishra,Simionescu, g11,g12 andg13, and five constrained test problems (C01toC05) of dimension10. These five con- strained test problems (C01toC05) are selected from CEC-2017single objective benchmark suite [122]. The basic properties of all test functions are listed in Table 4.2. All benchmark functions can also be categorized into unimodal and multimodal with increase in test function dimension. For unimodal functions, the convergence rates are crucial as optimum solutions of these functions can be obtained without much difficulty. On the other hand, solving mul- timodal functions reflect the ability of MRFD in avoiding local optima and converging to desired near-global solution.
4.3.9.2 Peer algorithms and parameter settings
With aim of giving a complete overview on the performance of proposed MRFD, four different state-of-art optimization techniques, i.e., SPSO [67], G-CMA-ES [123], UMOEAII [124] and L-SHADE [65, 66] are considered for evaluation on single objective test problems.
For the purpose of comparison, we have used the authors implementation of all these algo- rithms available in public domain.
Standard particle swarm optimization (SPSO) algorithm presented in [67] is used as a
4These are standard single objective optimization test functions used in literature. They are also avail- able at https://en.wikipedia.org/wiki/Test functions for optimization and at http://www-optima.amp.i.kyoto- u.ac.jp/member/student/hedar/Hedar files/TestGO files/Page422.htm
5‘Dimension’ means number of decision variables.
baseline for performance testing and to represent PSO in literature. During analysis using SPSO, constriction factor (χ) is set to0.72984, and constriction coefficients (c1 andc1) are kept at2.05each to ensure convergence [125]. G-CMA-ES6 is a variation of Covariant Ma- trix Evolutionary Strategy (CMA-ES) that avoids premature convergence by restart strategy, which doubles the population size on each restart and search for global solution in the in- creased search space. We have considered the values for different parameters of G-CMA-ES as suggested by Auger and Hansen [123]. The initial solution in G-CMA-ES is chosen uni- formly from the domain and the initial distribution size is set to one third of domain size [123].
UMOEAII7is a united multi-operator evolutionary algorithm framework, which incorporates multi-operator differential evolution (DE) algorithm, multi-operator genetic algorithm (GA) and CMA-ES algorithm. The different parameters used during implementation including the respective parameter values are considered according to the authors implementation as de- scribed in [124]. Further, L-SHADE algorithm is a variant of SHADE algorithm, which uses a strategy to linearly reduce the size of population in each generation. L-SHADE algorithm is used to solve unconstrained test functions considered in this thesis and an enhanced version of L-SHADE (i.e., L-SHADE44) algorithm is considered during evaluation of constrained test functions only. It should be noted that L-SHADE44 is the winner of CEC-2017 competition, which is performed on single objective constrained real parameter optimization problems.
The different parameters used during implementation of both versions of L-SHADE algo- rithm are considered according to the authors implementation described in [65, 66].
As the choice of population size plays a prime factor in performance of algorithms, a minimum population size of100is considered during evaluation of all these algorithms. Fur- ther, to assess the performance of algorithms on single objective test problems quantitatively, success performance (SP or FEs) is used as a measure for the expected number of function evaluations (FEs) to reach a target function value presented in [126]. SP is an empirical mea- sure on the ability to generalize performance results of algorithms and it can be evaluated
6G-CMA-ES is the winner of CEC-2005 competition on single objective real parameter optimization.
7UMOEAII is the winner of CEC-2016 competition, which is performed on CEC-2014 single objective
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.