Evolutionary Multi-objective Optimization – A Big Picture

1. EvolutionaryMulti-objectiveOptimization – A Big Picture KarthikSindhya, PhD Postdoctoral Researcher Industrial Optimization Group Department of Mathematical Information Technology Karthik.sindhya@jyu.fi http://users.jyu.fi/~kasindhy/

2. Objectives The objectives of this lecture are to: • Discuss the transition: Single objective optimization toMulti-objective optimization • Review the basic terminologies and concepts in use in multi-objective optimization • Introduce evolutionary multi-objective optimization • Goals in evolutionary multi-objective optimization • Main Issues in evolutionary multi-objective optimization

3. Reference • Books: • K. Deb. Multi-Objective Optimization using Evolutionary Algorithms. Wiley, Chichester, 2001. • K. Miettinen. Nonlinear MultiobjectiveOptimization. Kluwer, Boston, 1999.

4. Transition Single objective: Maximize Performance Minimize: Cost Maximize: Performance

5. Basic terminologies and concepts • Multi-objective problem is usually of the form: Minimize/Maximize f(x) = (f1(x), f2(x),…,fk(x)) subject to gj(x) ≥ 0 hk(x) = 0 xL ≤ x ≤ xU Multiple objectives, constraints and decision variables Objective space Decision space

6. Basic terminologies and concepts • Concept of non-dominated solutions: • solution a dominates solution b, if • a is no worse than b in all objectives • a is strictly better than b in at least one objective. 1 2 5 3 f2 (minimize) 4 2 3 2 6 4 5 • 3 dominates 2 and 4 • 1 does not dominate 3 and 4 • 1 dominates 2 f1 (minimize)

7. Basic terminologies and concepts • Properties of dominance relationship • Reflexive: The dominance relation is not reflexive. • Since solution a does not dominate itself. • Symmetric: The dominance relation is not symmetric. • Solution a dominates b does not mean b dominated a. • Dominance relation is asymmetric. • Dominance relation is not antisymmetric. • Transitive: The dominance relation is transitive. • If a dominates b and b dominates c, then a dominates c. • If a does not dominate b, it does not mean b dominates a.

8. Basic terminologies and concepts • Finding Pareto-optimal/non-dominated solutions • Among a set of solutions P, the non-dominated set of solutions P’ are those that are not dominated by any member of the set P. • If the set of solutions considered is the entire feasible objective space, P’ is Pareto optimal. • Different approaches available. They differ in computational complexities. • Naive and slow • Worst time complexity is 0(MN2). • Kung et al. approach • O(NlogN)

9. Basic terminologies and concepts • Kung et al. approach • Step 1: Sort objective 1 based on the descending order of importance. • Ascending order for minimization objective 1 2 5 3 f2 (minimize) 4 2 3 5 2 6 4 5 P = {5,1,3,2,4} f1 (minimize)

10. Basic terminologies and concepts P = {5,1,3,2,4} Front(P) = {5} Front = {5} T = {5,1,3} B = {2,4} Front = {2,4} {5,1} {3} {2} {4} Front = {5} {5} {1}

11. Basic terminologies and concepts • Non-dominated sorting of population • Step 1: Set all non-dominated fronts Pj, j = 1,2,… as empty sets and set non-domination level counter j = 1 • Step 2: Use any one of the approaches to find the non-dominated set P’ of population P. • Step 3: Update Pj = P’ and P = P\P’. • Step 4: If P ≠ φ, increment j = j + 1 and go to Step 2. Otherwise, stop and declare all non-dominated fronts Pi, i = 1,2,…,j.

12. Basic terminologies and concepts 1 2 f2 (minimize) 4 3 5 f1 (minimize) Front 2 Front 3 f2 (minimize) Front 1 f1 (minimize)

13. Basic terminologies and concepts • Pareto optimal fronts (objective space) • For a K objective problem, usually Pareto front is K-1 dimensional Min-Max Max-Max Min-Min Max-Min

14. Basic terminologies and concepts • Local and Global Pareto optimal front • Local Pareto optimal front: Local dominance check. • Global Pareto optimal front is also local Pareto optimal front. Objective space Decision space Locally Pareto optimal front

15. Basic terminologies and concepts • Ideal point: • Non-existent • lower bound of the Pareto front. • Nadir point: • Upper bound of the Pareto front. • Normalization of objective vectors: • fnormi = (fi -ziutopia)/(zinadir-ziutopia) • Max point: • A vector formed by the maximum objective function values of the entire/part of objective space. • Usually used in evolutionary multi-objective optimization algorithms, as nadir point is difficult to estimate. • Used as an estimate of nadir point and updated as and when new estimates are obtained. Objective space Zmaximum Min-Min f2 Znadir Zideal ε Zutopia ε f1

16. Basic terminologies and concepts • What are evolutionary multi-objective optimization algorithms? • Evolutionary algorithms used to solve multi-objective optimization problems. • EMO algorithms use a population of solutions to obtain a diverse set of solutions close to the Pareto optimal front. Objective space

17. Basic terminologies and concepts • EMO is a population based approach • Population evolves to finally converge on to the Pareto front. • Multiple optimal solutions in a single run. • In classical MCDM approaches • Usually multiple runs necessary to obtain a set of Pareto optimal solutions. • Usually problem knowledge is necessary.

18. Goal in evolutionary multi-objective optimization • Goals in evolutionary multi-objective optimization algorithms • To find a set of solutions as close as possible to the Pareto optimal front. • To find a set of solutions as diverse as possible. • To find a set of satisficing solutions reflecting the decision maker’s preferences. • Satisficing: a decision-making strategy that attempts to meet criteria for adequacy, rather than to identify an optimal solution.

19. Goal in evolutionary multi-objective optimization Objective space Convergence Diversity

20. Goal in evolutionary multi-objective optimization Objective space Convergence

21. Goal in evolutionary multi-objective optimization • Changes to single objective evolutionary algorithms • Fitness computation must be changed • Non-dominated solutions are preferred to maintain the drive towards the Pareto optimal front (attain convergence) • Emphasis to be given to less crowded or isolated solutions to maintain diversity in the population

22. Goal in evolutionary multi-objective optimization • What are less-crowded solutions ? • Crowding can occur in decision space and/or objective phase. • Decision space diversity sometimes are needed • As in engineering design problems, all solutions would look the same. Objective space Decision space Min-Min

23. Main Issues in evolutionary multi-objective optimization • How to maintain diversity and obtain a diverse set of Pareto optimal solutions? • How to maintain non-dominated solutions? • How to maintain the push towards the Pareto front ? (Achieve convergence)

24. EMO History • 1984 – VEGA by Schaffer • 1989 – Goldberg suggestion • 1993-95 - Non-Elitist methods • MOGA, NSGA, NPGA • 1998 – Present – Elitist methods • NSGA-II, DPGA, SPEA, PAES etc.