Biobjective Optimization Algorithms Using Neumann Series Expansion for Engineering Design

Algorithm 2

  (Initialization)

Step 1 Choose and a set of weighting factors ,give stopping criteria and , coefficients 、 、 .

Step 2 Establish each logarithmic penalty function of objectives, i.e., .

  (Main loop)

Step 3 Calculate , , if , stop and output ,then go to Step 8. Else, go to Step 4.

Step 4 Calculate the iterative direction ,

  If and , then

     ;

  If and , adjust weighting factors and to satisfy with .

  Else, .

Step 5 Calculate iteration step size .

  If , for all , take , and go to Step 7. Else, go to Step 6.

Step 6 Take ,go to Step 5.

  (Update)

Step 7 Calculate , define , and go to Step 3.

Step 8 Calculate , and if , stop and the is the Pareto optimal solution, output . Else, , go to Step 1.