A PS based DE method for constrained global opti- opti-mization (PSDEC)opti-mization (PSDEC)

The aim of the PS based DEC method is to provide a local search mechanism within the scope of the DEC algorithm. This hybrid will have the global properties of the DEC algo-rithm while the PS method will allow for quicker convergence to local minima.

6.3.1 The PS based local exploration algorithm

We first look at the PS method that will be used for the PSDEC algorithm. Our aim is not to carry out a full local search but just a simple limited local exploration. Hence we will not implement the full PS method but a limited version of it. We denote this method by PPS or partial pattern search. A detailed description of the general PS algorithm is presented in Chapter 5 and by algorithm 3. Here we briefly describe the method: A single point xk

referred to as the current iterate is evaluated, k being the iteration number. Then a set of points that lie on a mesh neighboring the current iterate are considered. These points are referred to as the poll set. The mesh size is dependent on a parameter ∆_k. If an improving point is found in the poll set then the mesh is either coarsened or kept the same. If no improving point is found the current iterate is considered to be locally optimal for that mesh and hence the mesh is refined. The parameter ∆_kis used to either coarsen or refine the mesh.

Equations (5.2) to (5.5) give the mathematically formulation for the mesh, poll set and mesh update.

What we propose for the PPS method is a non-iterative process using a single iterate.

Hence the PPS method uses a single value for ∆ to produce one poll set around the current iterate. If an improving point is found the algorithm is stopped and considered successful. A point will be considered to be improved if a lower fitness function value is found. If none of the points in the poll set improve the current poll center is considered to be a minimum for the current poll set and the PPS algorithm is ended. This will allow us to have a partial local

search within our DEC algorithm. We will fully describe the PPS algorithm below.

Firstly five points are selected randomly from the best α% of the population¹. One of the five points is randomly selected to be the current iterate xkand the poll centre. The most important feature of PPS is the calculation of ∆. The average distance (AvgDis) between the five points that have been selected is calculated. ∆ will be set as a fraction of the AvgDis i.e. ∆ = β × AvgDis, where β < 1². Since atleast one of the directions in the poll set is a descent direction, we set ∆ as a small value in order to find a better point than x_k. The value for ∆ will vary depending on the distance between the points. At early stages the population will be diverse and this will result in larger values for ∆. As the population converges the distance between points will decrease and so will the value for ∆. Clearly, ∆ will no longer be a user defined parameter and will adapt with the population. Since the algorithm is to be tested on a wide range of problems whose search areas vary dramatically in size, using a fixed value such as ∆ = 1 will results in poor performance. The PPS algorithm is presented below by algorithm 5.

Algorithm 7 The PPS algorithm

1. Randomly select five point within best 10 % of the population.

2. Calculate the average distance, AvgDis, between the five points and set ∆ = 0.1 × AvgDis

3. Uniformly select one point say x_r,gamongst the five points be the current iterate x_k = xr,g

4. POLL STEP: Evaluate objective function value at trial points in poll set, xⁱ_k = (x_k+

∆_kd_i).

1For our problem set this percentage was determined empirically. We tested a range of values between 5%

and 15% and found 10% to be most optimal.

2Empirical testing showed that β = 0.1 is a good choice for the problem sets.

6.3.2 The PSDEC algorithm

To implement the PPS method in DEC, the underlying DEC algorithm need not change. The reader is referred to section 6.1 for a full description of the DEC algorithm. The only change would be to call PPS after the population is updated at the end of an iteration. One of the drawbacks of the PS method is that its efficiency is extremely dependent on the dimension of the problem. Higher dimensions imply larger poll sets and thus a larger number of function evaluations. In order to limit the number of function evaluations for the PPS method we only invoke the method after every 10 iterations. The PPS method will be carried out using a random point selected as the poll center. The poll center is therefore a point within the best 10% of the current population. If the PPS method finds an improving point it will replace the original point with the improved point in the population, otherwise the population remains unchanged.

The most noteworthy feature of this algorithm is that it does not require any additional parameters. The only parameters are those of the DEC algorithm, while the PPS algorithm is self contained and requires no user defined input. The PSDEC algorithm is presented below:

Algorithm 8 The PSDEC algorithm

1. Set control parameters N , cr and g = 0 as well as penalty parameter R for the SFP method.

2. Initialize population S₀ = {x_1,0, x_2,0, . . . , x_N,0} uniformly using (2.1)

3. Evaluate objective function value and constraint violation of each member in the pop-ulation

4. Determine penalty function Θ0 and evaluate fitness of initial population using (3.7)-(3.9) for SFP method or (3.11) and (3.12) for PFP method

5. IF Stopping Criteria not met

(a) Generate Fg ∈ [−1, −0.4] ∪ [0.4, 1] uniformly (b) FOR i = 1 TO N ,

i. generate trial point y_i,g via:

• Mutation using (2.2),(2.3) or (2.4)

• Crossover using (2.5) ii. Evaluate f (y_i,g) andPm

j=1hg_j(y_i,g)i

6. Determine penalty function Θgand evaluate fitness of trial population using (3.7)-(3.9) for SFP method or (3.11) and (3.12) for PFP method

7. Update population using (6.1)

8. If (g mod 10 = 0), then call PPS and update population if required 9. Set g = g + 1, calculate stopping criteria and go to 5.

6.3.3 Implementational issues

The only implementational issue to consider for this method is that the PPS method requires additional fitness function evaluations. These must be considered and they should contribute to the total number of fitness function evaluations for the algorithm. The implementational issues of the DEC algorithm are also applicable for the PSDEC algorithm.

6.4 A PS filter-based DE method for constrained global