Classic Methods - for Multi-Objective Optimizationgn387/glp/lec2.pdfWeighted Sum Method Advantages and disadvantages Advantages Simple and easy to use. For convex problems it guarantees

Classic Methodsfor Multi-Objective Optimization

Giuseppe Narzisi

Courant Institute of Mathematical SciencesNew York University

31 January 2008

Outline

1 Introduction

2 Weighted Sum Method

3 Non-convex problems

4 ε-Constraint method

5 Weighted Metric MethodsTchebycheff problem

6 Rotated Weighted Metric Method

7 Dynamically Changing the Ideal Solution

8 Benson’s Method

9 Value Function Method

Introduction

We will talk about classical methods for handlingmulti-objective optimization problems.We refer to this methods as classical to distinguish themfrom the evolutionary methods (which we will cover later).

Two approaches to Multi-Objective Optimization

1 PREFERENCE-BASED PROCEDURE:Composite objective function as the weighted sum of theobjectives

Only useful if a relative preference factor of the objectives isknown in advance.

2 IDEAL PROCEDURE:1 Find multiple trade-off solutions with a wide range of values

for the objectives.2 Choose one of the obtained solution using higher-level

information

Preference-Based methodIllustration

M

Single−Objective

optimization problemOptimizier

One optimum solution

MOOPHigher−levelInformation

Estimation of the importance vector

(w ,w ,...,w )1 2

Ideal Multi-Objective procedureIllustration

solutions

Choose one solution

Higher−levelinformation

MOOPIdeal Multi−Objective

Optimizier

Multiple trade−off

Classificationof the classical methods

Classical method have been around for the past fourdecades.They can be classified in the following classes:

Non-preference methodsThese methods do not assume any information about theimportance of objectives, but an heuristic is used to find asingle optimum solution. They do not make any attempt tofind multiple Pareto-optimal solutions.

Posteriori methodsPosteriori methods use preference information of eachobjective and iteratively generate a set of Pareto-optimalsolutions.

A priori methodsOn the other hand, a priori methods use more informationabout the preferences of objectives and they usually find onepreferred Pareto-optimal solution.

Interactive methodsThese methods use the preference information progressivelyduring the optimization process.

Weighted Sum Method

This method is the simplest approach and probably themost widely used classical method.This method scalarizes the set of objectives into a singleobjective by multiplying each objective with a user suppliedweight.Although simple, it introduces a non-simple question: Whatvalue of the weights must be used? The answer depend onthe relative importance of each objective.

Weighted Sum Methodformulation

min F(x) =∑M

m=1 wmfm(x)subject to G(x) = [g1(x), g2(x), . . . , gJ(x)] ≥ 0

H(x) = [h1(x), h2(x), . . . , hK (x)] = 0x (L)

i ≤ xi ≤ x (U)i , i = 1, . . . , N

(1)

where the objectives are normalized.wm ∈ [0, 1] is the weight of the m-th objective function.It is usual practice to chose weights such that∑M

m=1 wm = 1.

Weighted Sum MethodProperties

TheoremThe solution to the problem presented in equation (1) isPareto-optimal if the weight is positive for all objectives:wm > 0, m = 1, . . . , M

The proof is achieved by contradiction considering asolution with all positive weights and that is notPareto-optimal and showing that this brings to acontradiction.Note that the theorem does not imply that anyPareto-optimal solution can be obtained using a positiveweight vector.

Weighted Sum MethodProperties

Theoremif x∗ is a Pareto-optimal solution of a convex multi-objectiveoptimization problem, then there exists a non-zero positiveweight vector w such that x∗ is a solution of problem (1)

The theorem suggests that for a convex MOOP any Paretosolution can be found using the weighted sum method (seeMiettinen’s book on Nonlinear Multiobjective Optimization[2] for the proof).

Weighted Sum MethodIllustration

��

��

A

2

1 a

b

w

w

f

f

c

1

2

Weighted Sum MethodAdvantages and disadvantages

AdvantagesSimple and easy to use.For convex problems it guarantees to find solutions on theentire Pareto-optimal set.

DisadvantagesFor mixed optimization problems (min-max), we need toconvert all the objectives into one type.Uniformly distributed set of weights does not guarantee auniformly distributed set of Pareto-optimal solutions.Two different set of weight vectors not necessarily lead totwo different Pareto-optimal solutions.There may exists multiple minimum solutions for a specificweight vector that represents different solutions in thePareto-optimal front (wasting the search effort).

Non-convex problemsDifficulties

��

��

C

b

a

f

f2

1

A

B

Feasible space

ε-Constraint methodDefinition

Idea: keep only one of the objectives and restrict the restof the objectives within some user-specified values.The modified problem is the following:

min fu(x)subject to [f1(x), f2(x), . . . , fu−1(x), fu+1(x) . . . , fM(x)] ≤ ε

G(x) = [g1(x), g2(x), . . . , gJ(x)] ≥ 0H(x) = [h1(x), h2(x), . . . , hK (x)] = 0x (L)

i ≤ xi ≤ x (U)i , i = 1, . . . , N

(2)

ε-Constraint methodIllustration

��

��

1 ε2 3 4

f

f2

1

A

B

Cε ε ε

Feasible space

ε-Constraint methodProperties

TheoremThe unique solution of the ε-constraint problem stated in (2) isPareto-optimal for any given upper bound vector:ε = (ε1, . . . , εu−1, εu+1, . . . , εm)T

The proof is again achieved by contradiction assuming thata unique solution of the ε-constraint problem is notPareto-optimal and than showing that this assumptionviolates the definition of Pareto-optimality [2].

ε-Constraint methodAdvantages and disadvantages

AdvantagesDifferent Pareto-optimal solution can be found usingdifferent εm valuesthe method can be used also for non-convexmulti-objective optimization problems

DisadvantagesThe solution to the problem largely depends on theselection of the ε vector. In particular, it must be chosensuch that it lies between the minimum and maximum valueof each objective function.As the number of objectives increase more informationfrom the user is required (εm).

Weighted Metric MethodsDefinition

Idea: instead of using a weighted sum of the objectives, wecan consider other ways of combining multiple objectives.

min Lp(x) =(∑M

m=1 wm|fm(x)− z∗m|p) 1

p

subject to G(x) = [g1(x), g2(x), . . . , gJ(x)] ≥ 0H(x) = [h1(x), h2(x), . . . , hK (x)] = 0x (L)

i ≤ xi ≤ x (U)i , i = 1, . . . , N

(3)

Weights are non-negative.p ∈ [1,∞]

z∗ is called reference point.When p = 1 is used, (3) is equivalent to the weighted summethod.

Weighted Metric Methods(p = 1) Taxicab or Manhattan norm

��

��

A

D

f

f2

1

B

Cz *

Feasible space

Weighted Metric Methods(p = 2) Euclidean norm

��

��

f

f2

1

Dz *

BA

C

Feasible space

Tchebycheff problem(p =∞)

min L∞(x) = maxMm=1wm|fm(x)− z∗m|

subject to G(x) = [g1(x), g2(x), . . . , gJ(x)] ≥ 0H(x) = [h1(x), h2(x), . . . , hK (x)] = 0x (L)

i ≤ xi ≤ x (U)i , i = 1, . . . , N

(4)

If the Tchebycheff metric is used, any Pareto optimalsolution can be found

TheoremLet x be a Pareto-optimal solution, then there exists a positiveweighting w vector such that x is a solution of the weightedTchebycheff problem (4) where the reference point is theutopian objective vector.

Weighted Metric Methods(p =∞)

��

��

f

f2

1

B

z *

A

Feasible space

Weighted Metric MethodsAdvantages and disadvantages

AdvantagesThe Tchebycheff metric allows to find each and everyPareto-optimal solution when z∗ is the utopian objectivevector [2].

DisadvantagesIt is advisable to normalize the objective functions, whichrequires knowledge of the minimum and maximum valuesof each objective.It requires the knowledge of the ideal solution z∗. So weneed to optimize each objective function before we cancompute the Lp metric.

Rotated Weighted Metric Method

Instead of using directly the Lp metric as it is stated inequation (3), the Lp metric can be applied with an arbitraryrotation from the ideal point.Let us say that the relation between the rotated objectivevector f̂ and the original objective vector f is : f̂ = Rf, whereR is the rotation matrix of size M ×M.The modified Lp metric is:

L̂p(x) =

(M∑

m=1

wm |̂fm(x)− z∗m|p) 1

p

(5)

Rotated Weighted Metric MethodIllustration

��

��

f

f2

1

A

f^

1

2

z *

f^

Dynamically Changing the Ideal Solution

Idea: update the reference point z∗ every time that aPareto-optimal solution is found.The Lp distance of the ideal solution comes closer to thePareto-optimal front and new previous undiscoveredsolution can now be found.

Dynamically Changing the Ideal SolutionIllustration

��

��

f

f2

1

o

AC

B

z *

Feasible space

Benson’s Method

Idea: take the reference solution z∗ randomly from thefeasible region. Let us call it z0.The non-negative difference (z0

m − fm(x)) for each objectiveis calculated and their sum is maximized

max B(x) =∑M

m=1 max(0, (z0m − fm(x)))

subject to F(x) = [f1(x), f2(x), . . . , fM(x)] ≤ z0

G(x) = [g1(x), g2(x), . . . , gJ(x)] ≥ 0H(x) = [h1(x), h2(x), . . . , hK (x)] = 0x (L)

i ≤ xi ≤ x (U)i , i = 1, . . . , N

(6)

Benson’s MethodIllustration

��

��

0z

f

f2

1

f(x)

Feasible space

Benson MethodAdvantages and disadvantages

AdvantagesAvoid scaling problems: individual differences can benormalized before the summation.If the point z0 is chosen appropriately than this method canbe used also for non-convex multi-objective problems.

DisadvantagesIt has an additional number of constraints.The objective function is non-differentiable, causingdifficulties for gradient based methods.

Value Function Method(or Utility Function Method)

Idea: the user provides an utility function U : RM → Rrelating all M objectives.The utility function must be valid over the entire feasiblespace.Among two solutions i and j , i is preferred to j ifU(f(xi)) > U(f(xj)).

max U(f(x))subject to G(x) = [g1(x), g2(x), . . . , gJ(x)] ≥ 0

H(x) = [h1(x), h2(x), . . . , hK (x)] = 0x (L)

i ≤ xi ≤ x (U)i , i = 1, . . . , N

(7)

where f(x) = [f1(x), f2(x), . . . , fM(x)]T .

Value Function MethodProperties

The utility function must be strongly decreasing before itcan be used in multi-objective optimization.This means that the preference of a solution must increaseif one of the objective function values is decreased whilekeeping the other objective function values the same.

TheoremLet the utility function U : RM → R be strongly decreasing. LetU attain its maximum at f∗. Then f∗ is Pareto-optimal.

Value Function MethodIllustration

��

��

f

f2

1

Utility function increases

A

Feasible space

Value Function MethodAdvantages and disadvantages

AdvantagesThe idea is simple and ideal, if adequate value functioninformation is available.Mainly used for multi-attribute decision analysis problemswith a discrete set of feasible solutions.

DisadvantagesThe solution entirely depends on the chosen valuefunction.It requires the users to come up with a value function thatis globally applicable over the entire search space.

For Further Reading

Kalyanmoy DebMulti-Objective Optimization Using EvolutionaryAlgorithms.John Wiley & Sons, Inc, New York, NY, USA, 2001.

Kaisa MiettinenNonlinear Multiobjective Optimization.Kluwer Academic Publishers, 1999. International Series inOperations Research & Management Science, Volume 12.

Documents

Classic Methods - for Multi-Objective Optimizationgn387/glp/lec2.pdfWeighted Sum Method Advantages and disadvantages Advantages Simple and easy to use. For convex problems it guarantees