Multiobjective Optimization Using Cross-Entropy Approach

Algorithm 1

A cross-entropy approach for the computation of the nondominated solutions.

Input: A number of performances functions 𝑀, a performances functions 𝑆₁,…,𝑆_𝑀: 𝒰→ℝ

where 𝒰 is the set of the feasible solutions, the starting CE's parameter 𝑣₀ and two parameters: 𝐶 and 𝜚.

Output: An element 𝑣_output∈ℝ⁺ and the Pareto front 𝑃_nbElite.

Algorithm:

      Step 1. Set 𝑡 equal to 1 and the components 𝑣_𝑡[𝑖] of the 𝑛-dimensional vector 𝑣_𝑡 equal to 𝑣₀.

   Set nbElite equal to the largest integer inferior or equal to 𝜚×𝐶×𝑛.

      Step 2. Set 𝑋_𝑡 equal to an empty set.

      Step 3. Draw independently 𝐶×𝑛 elements according to the Exponential pmf Exp_𝑛(⋅,𝑣_𝑡) and set them in 𝑋_𝑡.

      Step 4. Search all possible non-dominated solutions of 𝑋_𝑡 (Ω=Ω₁,Ω₂,…).

      Step 5. Set 𝑖 to 1 and the elite set of sample 𝑃_nbElite to the empty set.

      Step 6. Computation of 𝑃_nbElite Set based on non dominance and crowding operator.

      Step 6  a. Until 𝑃_nbElite+𝐹_𝑖≤nbElite.

      Step 6  b. Include 𝑖th non-dominated front in the elite set 𝑃_nbElite 𝑃_nbElite=𝑃_nbElite⋃𝐹_𝑖.

      Step 6  c. Check the next front for inclusion (𝑖←𝑖+1).

      Step 7. Calculate crowding distance in 𝐹_𝑖 and order the elements according to the ≺_𝑛 operator.

      Step 8. Choose the first nbElite−𝑃_nbElite elements of 𝐹_𝑖 and added its to the elite set

    𝑃_nbElite=𝑃_nbElite∪𝐹_𝑖1∶nbElite−𝑃_nbElite.

      Step 9. If stopping conditions are met, then return 𝑣_output=𝑣_𝑡 and stop. Otherwise go to Step 10.

      Step 10. Set 𝑣_𝑡∑[𝑖]=^{𝐶×𝑛𝑘=1}𝐼_{{(𝑆1(𝑋𝑘),…,𝑆𝑀(𝑋𝑘))∈𝑃nbElite}}𝑋_𝑘∑^𝐶×𝑛_𝑘=1𝐼_{{(𝑆1(𝑋𝑘),…,𝑆𝑀(𝑋𝑘))∈𝑃nbElite}}𝑊𝑋_𝑖;𝑢,𝑣_𝑡−1[𝑖] for 𝑖=1,2,…,𝑛 and 𝑡←𝑡+1. Go to Step 2.