Evolutionary Multiobjective Optimization A Big Picture Karthik Sindhya

Evolutionary Multi-objective Optimization – A Big Picture Karthik Sindhya, Ph. D Postdoctoral Researcher Industrial Optimization Group Department of Mathematical Information Technology Karthik. sindhya@jyu. fi http: //users. jyu. fi/~kasindhy/

Objectives The objectives of this lecture are to: 1. Discuss the transition: Single objective optimization to Multi-objective optimization 2. Review the basic terminologies and concepts in use in multi-objective optimization 3. Introduce evolutionary multi-objective optimization 4. Goals in evolutionary multi-objective optimization 5. Main Issues in evolutionary multi-objective optimization

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

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

Basic terminologies and concepts • Multi-objective problem is usually of the form: Minimize/Maximize f(x) = (f 1(x), f 2(x), …, fk(x)) subject to gj(x) ≥ 0 Multiple objectives, hk(x) = 0 constraints and decision variables x. L ≤ x. U Decision space Objective space

Basic terminologies and concepts – 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. 5 f 2 (minimize) • Concept of nondominated solutions: 1 2 3 4 2 3 2 4 5 6 f 1 (minimize) • 3 dominates 2 and 4 • 1 does not dominate 3 and 4 • 1 dominates 2

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.

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(MN 2). • Kung et al. approach – O(Nlog. N)

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

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

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 = PP’. – 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.

Basic terminologies and concepts f 2 (minimize) 1 4 5 3 f 1 (minimize) Front 2 f 2 (minimize) Front 3 Front 1 2 f 1 (minimize)

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

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

Basic terminologies and concepts • Ideal point: – Non-existent – lower bound of the Pareto front. Objective space – Upper bound of the Pareto front. f 2 • Nadir point: Znadir Min-Min • Normalization of objective vectors: – fnormi = (fi - ziutopia )/(zinadir - ziutopia ) • Max point: – A vector formed by the maximum objective ε 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. Zmaximum Zideal Zutopia ε f 1

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

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.

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.

Goal in evolutionary multi-objective optimization Objective space Convergence Diversity

Goal in evolutionary multi-objective optimization Objective space Convergence

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

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 Min-Min Decision space

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)

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.