1. Department of Computer Science
and Information Technology
بهینه‌یابی مسائل چندهدفه با استفاده از الگوریتم‌های تقریبی Multi-Objective Optimization Using Approximation Algorithms
منصور داودي منفرد
Mansoor Davoodi
آذر ماه 1395

2. Multi-Objective Optimization (MOO)
Cost
Comfort

3. Multi-Objective Optimization (MOO)
Solution space c f2
Objective space
Pareto-optimal f1

4. Multi-Objective Optimization (MOO)f1 f2

5. General Formulation of MOO

6. Goals in MOO f2 f2 f1 f1
Pareto-Optimality
Diversity

7. Weighted Sum method

8. Weighted Sum method f2 f1

9. Weighted Sum method f2 f1

10. Multi-Objective Evolutionary Algorithms
Non-dominated Sorting Genetic Algorithm-II
Strength Pareto EA
Rudolph’s Elitist MOEA
Distance-Based Pareto GA
Thermodynamical GA
Pareto-Archived Evolution Strategy ….
Benchmark Problems
and
Comparison Metrics

11. Advantages and disadvantages of EAs
General framework
Reasonable resource consumption
Parallel evolution
Disadvantages
No guarantee in finding optimal solutions
Need to user parameter setting

12. Approximation Algorithms
Guaranteed to run in polynomial time for all instances.
Guaranteed to find solution within ratio 𝛼 of optimum.
Challenge. Need to prove a solution's value is close to optimum, without even knowing what optimum value is!

13. Approximation Algorithms-MCC

14. Approximation Algorithms-TSP
Input: Graph G, a complete weighted graph.
Output: Minimum weight tour that visits all the nodes C B A D F E G

15. Approximation Algorithms-TSPC B A D F E G
Double MST C B A D F E G
MST C B A D F E G
TSP Tour
MST<OPTTSP
TSP tour ≤2MST
𝛼=2

16. Schematic view of approximation factor
Minimization objective f
f(Opt)
𝑓 𝑥 ≤𝛼. 𝑓(𝑂𝑝𝑡)
Maximization objective f
𝑓 𝑥 ≥𝛼. 𝑓(𝑂𝑝𝑡)
f(Opt)

17. Approximation of MOO What is the meaning of “MOO approximation” ?
What is the difficulty of “MOO approximation” ? f2 f2 f1 f1

18. Approximation of MOO 𝑓 1 𝑥′ = 𝑓 1 𝑥 𝛼 𝑓 2 𝑥′ = 𝑓 2 𝑥 𝛽
Def. Let 𝛱 be a bi-objective minimization problem with objectives 𝑓 1 and 𝑓 2 . Solution x is an 𝛼,𝛽 -approximation solution for 𝛱 if there is no solution y such that 𝑓 1 𝑥 >𝛼 𝑓 1 𝑦 and 𝑓 2 𝑥 ≥𝛽 𝑓 2 𝑦 , or 𝑓 1 𝑥 ≥𝛼 𝑓 1 𝑦 and 𝑓 2 𝑋 >𝛽 𝑓 2 𝑦 , f2 f1
𝑓 1 𝑥′ = 𝑓 1 𝑥 𝛼
𝑓 2 𝑥′ = 𝑓 2 𝑥 𝛽 x 𝛽 𝛼 x'

19. Approximation of MOO
D𝐞𝐟. 𝜖-approximate Pareto curve is a set of solutions which are not dominated by any other by a ratio of more than 1+ 𝜖. f2 f1
𝜖-approximate Pareto curve P𝜖 is a set of solutions s such that there is no other solution 𝑠 ′ such that,
for all s∈P𝜖, 𝑓 𝑖 𝑠 ′ ≥ 𝑓 𝑖 𝑠 1+𝜖 for some 𝑖
PType equation here.

20. Multi-objective optimization (MOO)

21. Multi-objective optimization (MOO)
Optimal algorithm for Manhattan and
2 , 2 -approximation
for the Euclidean metric
𝜃(n log n)

22. Bi-objective Path Planning

23. Bi-objective Path Planning

24. Bi-objective Path Planning

25. Bi-objective Path Planning

26. Bi-objective Path Planning
2 ,1 -approximation of the Euclidean path

27. Challenges … . Clustering Locating Planning Networking
The real world is a trade-off world and it is full of conflict objectives. Any application of MOO approximation is welcome.
Basic definitions of approximation Pareto-optimal curve.
Complexity of MOO approximation.
… .
Clustering
Locating
Planning
Networking

28. References
I. Diakonikolas, Approximation of Multiobjective Optimization Problems, PhD thesis, Columbia University, 2011.
M. Yannakakis, CH. Papadimitriou, On the approximability of offs - trade and optimal access of web source,
V. Roostapour, I. Kiarazm, M. Davoodi, Deterministic Algorithm for 1-Median 1- Center Two-Objective Optimization Problem. TTCS, Springer, 2016.
M. Davoodi, Bi-objective Path Planning using Deterministic Algorithms, submitted to Robotics and Autonomous Systems, March 2016.
C. C. A. Coello, D. A. Van Veldhuizen, G. B. Lamont, Evolutionary algorithms for solving multi-objective problems. New York, 2002.

29. Thanks…
؟؟؟ Any Question ???