gary yen: "multi-objective optimization and performance metrics ensemble"

MULTIOBJECTIVE OPTIMIZATION AND

PERFORMANCE METRICS ENSEMBLE

Gary G. Yen, FIEEE

[email protected]

Professor, Oklahoma State University

Past President, IEEE Computational Intelligence Society

mailto:[email protected]

ieee-wcci2014.org

Multiobjective Optimization

Optimization problems involve more than one objective

functions

Very common, yet difficult problems in the field of science,

engineering, and business management

Nonconflicting objectives: achieve a single optimal solution

satisfies all objectives simultaneously SOPs

Competing objectives: cannot be optimized simultaneously

MOP– search for a set of “acceptable”– maybe only

suboptimal for one objective– solutions is our goal

In operation research/management terms - multiple criterion

decision making (MCDM) (International Society on MCDM; http://www.terry.uga.edu/mcdm/)

Why MOP? Buying an Automobile

Objective = reduce cost, while maximize comfort

Which solution (1, A, B, C, 2) is best ???

No solution from this set makes both objectives look better than any other solution from the set

No single optimal solution

Trade off between conflicting objectives- cost and comfort

Mathematical Definition

• Mathematical model to formulate the optimization

problem

))x(f(min)x(fmax

o Design Variables: decision and objective vector

o Constraints: equality and inequality

o Greater-than-equal-to inequality constraint can be converted to

less-than-equal-to constraint by multiplying -1

o Objective Function: maximization can be converted to

minimization due to the duality principle

},0,,0,:),({min UL xxxeee

)g(x)h(xxfynx

Objective

vectors

Decision

vectors

Equality

constraints

Inequality

constraints

Variable

bounds

Environment

states

Pareto Optimality

• Formal Definition: the minimization of the n components

of a vector function f of a vector variable x in a universe , where

• Then a decision vector is said to be Pareto-optimal if and

only if there is no for which

dominates , that is, there is no such that

nkfk ,,1,

μ

))(,),(),(()( 21 xxxxf nfff

ux

vx ),,()( 1 nv vv xfv

),,()( 1 nu uu xfu vx

ii uvni },,,1{ and ii uvni |},,1{

When encounter problems with many objectives (more

than five), nearly all algorithms performs poorly because

of loss of selection pressure in fitness evaluation solely

based upon Pareto domination.

Distinctions from SOP

• Multiple conflicting objectives as opposed to single one

• Multiple optima vs. single optimum

• Two goals instead of one

o Progressing towards the Pareto front

o Maintaining a diverse set of solutions in the non-dominated front

• Dealing with two search spaces

o A decision variable space plus an objective space

o A proximity of two solutions in one space does not mean a

proximity in the other space

o Search is performed in the decision space

Disadvantages of Classical Methods

• We need prior knowledge of the problem domain to result in to a single objective optimization problem (e.g., weight vector, constraints)

• Results in a single solution for each run

• Non-uniformity in Pareto-optimal solution

• Require fitness function to be linear, continuous and differentiable

• Cannot deal with MOPs having discontinuous and concave Pareto fronts

• An unorthodox, stochastic, and population based parallel

searching algorithm maybe more suitable for MOPs

• Classification of EA’s–

oGenetic Algorithm;

oGenetic Programming;

oEvolutionary Strategy;

oAnt Colony;

oArtificial Immune System;

oParticle Swarm Optimization;

oDifferential Evolution;

oMemetic Algorithm

Why Population-Based Heuristics?

Efforts in Enhancing a PSO for MOPs

• Modifying the fitness assignment

• Improving PSO flight mechanism

• Enhancing the convergence

• Preserving the diversity

• Managing the population

• Constraints and uncertainty handling

• Knowledge Management through Culture/Meme

Performance Metrics

• To quantify the

performance of

evolutionary multiobjective

algorithms according two

essential metrics dictated

by Pareto Optimality

Convergence measure

Diversity measure

Current Practice

In literature, when an MOEA is proposed,

a number of benchmark problems are chosen to

quantify the performance, and

based on a set of heuristically chosen performance

metrics, the proposed MOEA and some competitive

representatives are evaluated statistically given a

large number of independent trials.

The conclusion, if any been drawn, is often

indecisive and reveals no additional insight

pertaining to the specific problem characteristics that

the proposed MOEA would do the best

By the No Free Lunch theorem, any algorithm’s

elevated performance over one class of problems is

exactly paid for in loss over another class.

Our Goal is to rank the MOEAs considered based on

a more comprehensive measure (hybrid

performance metric),

revealing specific problem characteristics that the

underlying MOEA could perform the best.

Case Study

Five state-of-the-art MOEAs – SPEA 2, NSGA-II, PESA-II, IBEA, and MOEA/D

Six Benchmark Problems – 2-objective ZDT1, ZDT2, ZDT3, ZDT4, ZDT6

– 3-objective DTLZ2, 5-objective WFG1, WFG2, and

– 10-objective DTLZ1

Five Performance Metrics – Inverted Generational Distance (IGD),

– Pareto Dominance Indicator (NR),

– Maximum Spread (MS),

– Spacing, and

– Hypervolume Indicator

For the same initial population, all five MOEAs will

generate a non-dominated front for a given benchmark

function with specific problem characteristics.

A randomly chosen performance metric is used to

identify the winner of the non-dominated front and its

associated MOEA.

This process will be repeated 50 times to gain

meaningful statistics.

These 50 non-dominated fronts could come from either

one of five MOEAs and each of five performance

metrics could be used for multiple times.

Performance Metrics Ensemble

Generates 50 non-dominated fronts as the initial

population of Double Elimination Tournament Selection:

SPEA 2 NSGA-II IBEA PESA-II MOEA/D

19 11 3 5 12

IGD NR Spacing S-metric MS

11 10 12 10 7

ZDT1

Input:

MOEAs

50

Approximation

fronts

Double

Elimination to

obtain best front

Identify the Winner

Algorithm and

Assign Its Rank

Value

Eliminate All Fronts

from Winner

Algorithm

No. Remain

fronts is 0?

NO

Output:

Rank Value of All

MOEAs

YES

Specific

Benchmark

Problem

Flow Chart

50 Winners from 50 Running Times

Winner Bracket (25) Loser Bracket (25)

13 Winners 13 Winners 13 Losers 13 Losers

13 Winners

Reserved as Winner Bracket in the Next Round

Reserved as Loser Bracket in the Next Round

Eliminate

13 Losers

Double Tournament Elimination

50 fronts are competed to down select to 26 fronts

(13 in winner bracket and 13 in loser bracket) going

through 25 + 12 +12 + 13 = 62 binary tournaments:


9 8 0 1 8

Round 1


13 13 11 13 12



7 Winners



Eliminate

7 Losers

Round 2

26 fronts are competed to down select to14 fronts


through 6 + 6 + 7 = 19 binary tournaments:


6 2 0 1 5


5 4 5 2 3



4 Winners



Eliminate

4 Losers

Round 3





3 2 0 0 3


3 1 3 3 0



2 Winners



Eliminate

2 Losers

Round 4





2 0 0 0 2


1 0 2 1 2



1 Winners



Eliminate

1 Losers

Round 5



through 1 + 1 + 1 = 3 binary tournaments : :


1 0 0 0 1


1 0 0 1 1


1 Final Winners

Round 6

In the final that 2 fronts are competed to generate the

final winner.

About 152 binary tournaments were held to decide a

final winner.

Removing 18 fronts generated by SPEA 2, the

remaining 32 fronts will go through the process again…


1 0 0 0 0


0 0 0 1 0

Final Ranking

Ranking

Order

2-obj

ZDT1

2-obj

ZDT2

2-obj

ZDT3

2-obj

ZDT4

2-obj

ZDT6

3-obj

DTLZ2

5-obj

WFG1

5-obj

WFG2

10-obj

DTLZ1

1 SPEA 2 SPEA 2 NSGA-II MOEA/D MOEA/D IBEA IBEA IBEA IBEA

2 MOEA/D MOEA/D MOEA/D NSGA-II IBEA MOEA/D MOEA/D MOEA/D NSGA-II

3 NSGA-II NSGA-II IBEA PESA-II NSGA-II SPEA 2 SPEA 2 NSGA-II MOEA/D

4 PESA-II IBEA SPEA 2 IBEA SPEA 2 NSGA-II NSGA-II SPEA 2 SPEA 2

5 IBEA PESA-II PESA-II SPEA 2 PESA-II PESA-II PESA-II PESA-II PESA-II

• 35 repeated and independent experiments are done for each

function and the findings have been consistent

It is the final winner in problem ZDT1 and ZDT2.

ZDT1 and ZDT2 do not have local Pareto-optimal

fronts and their global Pareto-optimal fronts are

continuous.

IBEA and PESA-II dropped out of competition in the

first round.

SPEA2, MOEA/D and NSGA-II compete fiercely till

round 4.

SPEA 2 will perform well in problems having

continuous Pareto-optimal fronts and do not have local

Pareto-optimal fronts.

Observations on SPEA2

In ZDT1, SPEA 2 is the final winner and it wins under all four

metrics but is inferior to NSGA-II in S-metric.

In ZDT2, SPEA 2 is the final winner and it wins under all four

metrics but it is a little bit worse than NSGA-II in Spacing

metric.

In ZDT3, NSGA-II is the final winner and it wins under all four

metrics but is inferior to MOEA/D in S-metric.

In ZDT4, MOEA/D is the final winner and it wins under all four

metrics but it is a little bit worse than NSGA-II in NR metric.

In ZDT6, MOEA/D is the final winner but is inferior to IBEA in

MS metric and a little bit worse than NSGA-II in Spacing metric.

In DTLZ 2, IBEA is the final winner and it wins under all four

metrics but is inferior to MOEA/D in Spacing metric.

It has the best performance in ZDT3.

ZDT3 has the discreteness feature and has a

Pareto-optimal front consisting of several non-

contiguous convex parts.

MOEA/D is comparable in performance.

NSGA-II will perform well in problems having a

Pareto-optimal front consisting of several

noncontiguous convex parts.

Observations on NSGA-II

It wins in both ZDT4 and ZDT6.

ZDT4 has many local Pareto-optimal fronts, make EAs

exhibit their ability to deal with multi-modality.

ZDT6’s Pareto-optimal solutions are non-uniformly

distributed.

For ZDT4, SPEA2 was eliminated in early stage of

competition. For ZDT6, SPEA2 and PESA-II were

eliminated very early.

MOEA/D will exhibit its good performance in problems

with lots of local Pareto-optimal fronts or Pareto-

optimal solutions are not uniformly distributed its

global Pareto front.

Observations on MOEA/D

It wins all in DTLZ 2, WFG1, WFG2 and DTLZ 1

which are the test problem having more than two

objectives.

Many credible publications support the ranking for

higher-dimensional benchmark problems.

We can make a comparatively conclusion that IBEA

can perform better than others in some test

problems with high-dimension objectives.

Observations on IBEA

Double elimination design allows specific

characteristic-poor performance of a quality algorithm

under the special environment still to be able to

survive through competitions and win it all.

It gives every individual two chances to take part in the

competition. This is helpful to reserve good individual,

especially in some special conditions.

Overall Findings

Remarks

knowing no single metric alone can faithfully quantify the performance of a given MOEA under real-world scenarios, this study is intended to reveal the insight pertaining to specific problem characteristics that the underlying MOEA could perform the best.

For a given real-world problem, if we know its problem characteristics (e.g., a Pareto front with a number of disconnected segments and a high number of local optima), we may make an educated judgment to choose the specific MOEA for its superior performance given the problem characteristics.

Groundbreaking applications with smashing success

Toward Many-Objective Optimization under

constraints and uncertainties

Universal fundamentals in all algorithm formulations

Publicity in Interdisciplinary World

Education for the next Generations

Grand Challenges in EMO

gary yen: "multi-objective optimization and performance metrics ensemble"

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