The W, C, & H Methods 4 Generating Pareto Solutions

1. The W, C, & H Methods 4 Generating Pareto Solutions Prepared by: MA A-Sultan March 20, 2006 Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

2. MO Optimization Methods The Weighting Method A 1st standard technique for MOO is to optimize a positively weighted convex sum of the objectives, that is, For bi-objective example, will create a table like this: 0.8, 0.2 1.0, 0.0 0.0, 1.0 0.2, 0.8 0.35, 0.65 0.225, 0.775 Etc. Optimize Cause Headache? What if you have 3, 5, … 25 objs.? Not adequate alone! Non-Convex Pareto solutions can’t be generated; WHY?

3. F2(X*) F1(X*) Convexity & Concavity

4. Convexity & Concavity

5. -w1/w2 -w1/w2 -w1/w2 The Weighting Method Weights don’t reflect objective importance at all!

6. MO Optimization Methods The Constraint Method A 2nd standard technique for MOO is to optimize one objective while setting remaining ones to variable feasible bounds, that is: Virtually, there are weights here too! Cause Headache? What if you have 3, 5, … 25 objs.? Optimize S.t: Values must be chosen so that feasible solutions to the resulting single-objective problem exist.

7. Cost Effec. The Constraint Method Specifying a bound on Effec., defines a new feasible region which does allow Cost to be minimized to find a Parerto point.

8. Generation of Pareto Solutions by Entropy-based Methods Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(115 – 118), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

9. The Weighting Method The Entropy-based Weighting Method The Entropy-based Weighting Method V = U = Weights follow Boltzmann’ Distribution! We have proved that: where is an optimal solution Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(115 – 118), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

10. The Constraint Method The Entropy-based Constraint Method The Entropy-based Constraint Method Weights follow Boltzmann’ Distribution! Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(115 – 118), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

11. Bi-Obj Optimization Example 1 No more than 2 solutions can be generated since those in-between are CONCAVE! Better Representative Pareto Set Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(115 – 118), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

12. Bi-Obj Optimization Example 2 Grouped Pareto set with a large GAP between the two! Better Representative Pareto Set Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(119 – 122), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

13. Bi-Obj Optimization Example 3 Large portion of Pareto set is MISSING! The shape of the whole Pareto set is generated Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(123 – 128), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

14. A A B B C C Bi-Obj Optimization Example 4 1) A to B Pareto set has no GAP 2) B to C Pareto set has large GAP! 1)B to C Pareto set is better representative! Reference: Entropic Vector Optimization and Simulated Entropy: Theory & Applications; pp.(130 – 134), Ph.D. Thesis, Liverpool University, 1991. http://library.liv.ac.uk/search/a?searchtype=t&searcharg=Entropic+Vector+Optimization&searchscope=0&SORT=&SUBMIT=Search

15. Generation of Pareto Solutions by Entropy-based Methods (MOPGP94 Report)Portsmouth, UK, 1994 The authors present in this paper a method to compute the set of Pareto points for some multi-valued optimization problems. The paper is interesting and it must be published* since: 1) The Authors apply the entropy method to the study of Pareto points which is a new idea used also by other authors to the study of scalar optimization problems; 2) The method presented by the authors can be used to compute not only a Pareto point but the set of Pareto points, which is an important aspect; 3) The applications to engineering problems are interesting. * http://www.amazon.com/gp/product/3540606629/qid=1143034880/sr=1-5/ref=sr_1_5/002-0382939-5305662?s=books&v=glance&n=283155 pp. 164 - 194

16. Generation of Pareto Solutions by the Hybrid Method

17. The Hybrid Method Optimize Subject to:

18. The W-Method finds NO Concave solutions The C-Methods finds Convex & Concave solutions Both Need to be used! The H-Method is far more powerful than the W-Method and C-Method; yet rarely addressed! All 3 Methods have NO Problem with # of objectives The Entropy-based W, C, & H Methods

19. How the W, C, & H Methods were Incorporated into GAME ACSOM Prepared by: Michael M Sultan March 28, 2006

20. The Weighting Method A 1st standard technique for MOO is to optimize a positively weighted convex sum of the objectives, that is: The Constraint Method A 2nd standard technique for MOO is to optimize one objective while setting remaining ones to variable feasible bounds, that is: MO Optimization Methods Optimize Optimize

21. The Weigh & Constrain Method

22. How The Weigh & Constrain Method Work?