Evolutionary Multi-objective Optimization – A Big Picture Karthik Sindhya, PhD Postdoctoral Researcher Industrial Optimization Group Department of Mathematical Information Technology

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, – K. Miettinen. Nonlinear Multiobjective Optimization. Kluwer, Boston, 1999.

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

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

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. Basic terminologies and concepts f 1 (minimize) f 2 (minimize) dominates 2 and 4 1 does not dominate 3 and 4 1 dominates 2 3 dominates 2 and 4 1 does not dominate 3 and 4 1 dominates 2

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(NlogN) 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 Basic terminologies and concepts f 1 (minimize) f 2 (minimize) P = {5,1,3,2,4} 5

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

Non-dominated sorting of population – Step 1: Set all non-dominated fronts P j, 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 P j = 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 P i, i = 1,2,…,j. Basic terminologies and concepts

f 1 (minimize) f 2 (minimize) Front 1 Front 2 Front 3 f 1 (minimize) f 2 (minimize)

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

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

Ideal point: – Non-existent – lower bound of the Pareto front. Nadir point: – Upper bound of the Pareto front. Normalization of objective vectors: – f norm i = (f i - z i utopia )/(z i nadir - z i utopia ) 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. Basic terminologies and concepts Min-Min Z ideal Z nadir Z maximum Z utopia ε ε Objective space f1f1 f2f2

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. Basic terminologies and concepts Objective space

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. Basic terminologies and concepts

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

Convergence Diversity Objective space

Goal in evolutionary multi-objective optimization Convergence Objective space

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. Goal in evolutionary multi-objective optimization Min-Min Decision space Objective space

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) Main Issues in evolutionary multi-objective optimization

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