4.2.1 Single-loop formulation using shifting vector approach and DE vectors The formulation for single-loop method was given in equation (3.6) in which KKT conditions were used to approximate reliability analysis. The modified single-loop formulation using shift- ing vector approach is given as
Min. f(d,µX),
s.t.: Gi(d,Ψ(k))≤0, i= 1, . . . , I, whereΨ(k) =
X(k)i,M P T P, ∀ target vectors, µ(k+1)U −S(k+1)i , ∀ trial vectors, S(k+1)i =µ(k)X −X(k)i,M P T P,
X(k)i,M P T P =µ(k)X +βitσXα(k)i,X, µ(L)X ≤µX ≤µ(U)X ,d(L)≤d≤d(U)
(4.4)
Here,S(k+1)i is the shifting vector ofi−th performance function at (k+ 1)−th iteration. It can be seen thatGi(d,Ψ(k)) includes bothX(k)i,M P T P and (µ(k+1)U −S(k+1)i ). X(k)i,M P T P is used when evaluating the population of target vectors, whereas (µ(k+1)U −S(k+1)i ) is used for shifting the violated constraint/performance function towards the feasible direction for the population of trial vectors. This shift can help DE in guiding the vectors for better convergence.
4.2.2 Adaptive mutation scheme
Different variants of DE are available in the literature. For example, Section 4.1 presented
“DE/rand/1/bin” in which ‘rand’ represents r1 in equation (4.1), ‘1’ represents a single differ- ence of vectors (µ(k)r2 (t)−µ(k)r3 (t)) in the equation, and ‘bin’ represents crossover according to binomial experiments in equation (4.2). This variant of DE has a good capability of exploring the search space for locating the optima. However, the convergence of DE can be slower due to r1 because the mutation is done with respect to a random vector. The other variant such as “DE/best/1/bin” can be faster since it uses the ‘best’ vector (µ(k)best(t)) in terms of fitness in place of a random vector (µ(k)r1 (t)) in equation (4.1). However, this variant can converge to a local optima since all target vectors are mutated with respect to only the ‘best’ vector in the current generation (k). Since both the variants have their own merits and demerits, a
heuristic convergence parameter (ζ) is proposed that helps DE to use either of these variants.
The parameterζ is defined as the difference between the ratio of the best fitness to the worst fitness of the target vectors in the current generation (k) to its previous generation (k−1), that is,
ζ = fbest(k) fworst(k)
−fbest(k−1) fworst(k−1)
. (4.5)
The adaptive mutation thus can be written as
µ(k+1)V (t) =
µ(k)r1(t) +F×(µ(k)r2(t)−µ(k)r3(t)), ζ > ϵ, µ(k)best(t) +F×(µ(k)r2(t)−µ(k)r3(t)), otherwise,
(4.6)
whereϵis a user defined parameter. The condition (ζ > ϵ) signifies that mutation of “DE/rand/1/bin”
variant can help DE in exploring the search space in the beginning. After some generations, mutation of ‘DE/best/1/bin” variant can exploit the promising search space for faster locating the optima.
4.2.3 Constraint handling with DE
The penalty-parameter less approach (Deb, 2000) is used for handling the constraints in which the fitness of each vector in a population of DE is calculated as
F(µ(k)X (t)) =
f(µ(k)X (t)), ifµ(k)X (t) is feasible;
fmax+PI
i=1CVi, Otherwise.
(4.7)
Here, f(µ(k)X (t)) is the objective function value, fmax is the objective function value of the worst feasible vector in the current population, and CVi represents constraint violation of i−the performance function.
4.2.4 Selection scheme for next generation target vectors
The (µ+ λ)−strategy is used for selecting the next generation target vectors, instead of using equation (4.3). In this strategy, the target vectors and trail vectors are combined (µ(k)X S
µ(k+1)U ). The bestN vectors are selected based on the fitness values, and these vectors are copied toµ(k+1)X .
Input
k= 1, Initialize population
For each target vector Calculate f(µ(1)X ),α(1), MPTP,
shifting vector, constraint violation and fitness using
equation (4.7)
Checkk > K?
Generate mutant vectors using equation (4.6)
Generate trial vectors using equation (4.2)
For each trial vector Calculatef(µ(k+1)U ),α(k), MPTP,
shifting vector, constraint violation and fitness using
equation (4.7)
Select next generation target vectors and update corresponding
MPTPs and shifting vector
Stop
k=k+ 1
N, n, K, F, pc,σ, βt
No
Yes
Figure 4.1: Flowchart of SLADE
4.2.5 Steps for implementation
In this section, the procedure for implementing the proposed method is presented. As the proposed method uses a single-loop structure using adaptive differential evolution, it is abbre- viated as SLADE. In SLADE, the reliability analysis loop is replaced by approximate MPTP.
Figure 4.1 shows the flowchart of SLADE. It can be seen that after initialization, for each target vector, equation (4.7) is used for calculating objective function using the random variables and fitness is assigned using MPTP. The standard loop over number of generations (k) then starts in which mutant and trail vectors are generated. Again, for each trial vector, objective function is calculated and fitness is assigned using equation (4.7). Thereafter, the next generation target vector is constructed, and MPTPs and shifting vectors are updated. The loop continues until the termination criterion is met. The complete details of every step for implementing SLADE are as follows.
Step 1 Input: population size (N), number of variables (n), total number of generations (K), scaling factor (F), probability of crossover (pc), standard deviation (σ) for random vari- ables, and target reliability index for constraints (βt), generation counter (k= 1).
Step 2 Initialize random population (P(k)).
Step 3 For each target vector (µ(k)X (t)) of (P(k)) Step 3.1 Calculate objective function, f(µ(k)X (t)).
Step 3.2 Calculate the steepest descent direction, α(k)= ∥σ∇G∥σ∇G µ(k)X (t).
Step 3.3 Calculate MPTP for each performance function (i) using equation (4.4) and estimate shifting vector S(k+1)i,µ
X =µ(k)X −X(k)i,M P T P.
Step 3.4 Calculate the fitness using equation (4.7). It is noted that constraint violation is calculated for each performance function,Gi(d,X(k)i,M P T P).
Step 4 If (k > K), terminate. Otherwise, continue to Step 5.
Step 5 Generate mutant vectors (µ(k+1)V ) using the proposed modified scheme as given in equa- tion (4.6).
Step 6 Generate trial vectors (µ(k+1)U ) as given in equation (4.2).
Step 7 For each trail vector
Step 7.1 Calculate objective function,f(µ(k+1)U (t)).
Step 7.2 Calculate the steepest descent direction,α(k) = ∥σ∇G∥σ∇G
µ(k+1)U (t).
Step 7.3 Calculate MPTP ( ˆX(k+1)i,M P T P) and shifting vector for each performance function (i) using equation (4.4) andS(k+2)i,µ
U =µ(k+1)U −Xˆ(k+1)i,M P T P and estimate constraint viola- tion ofGi(d,µ(k+1)U −S(k+1)i,µ
X ).
Step 7.4 Calculate the fitness using equation (4.7).
Step 8 Select target vectors (µ(k+1)X ) for the next generation using (µ+λ)−strategy. It is noted that the corresponding MPTPs and shifting vector are stored for utilizing them in Step 7.3. Set k=k+ 1, and go to Step 4.