multiobjective optimization methods -...

Multiobjective optimization

methods

Jussi Hakanen

Post-doctoral researcher [email protected]

spring 2014 TIES483 Nonlinear optimization

mailto:[email protected]

No-preference methods

DM not available (e.g. online optimization)

No preference information available

Compute some PO solution

Do not take into account which problem is

solved

Fast methods

– One PO solution is enough

– No communication with the DM

Method of global criterion

min𝑥∈𝑆

𝑓𝑖 𝑥 − 𝑧𝑖∗ 𝑝𝑘

𝑖=1

1/𝑝

Distance to the ideal objective vector is

minimized

Different metrics can be used, e.g. Lp metric

where 1 ≤ p ≤ ∞

A single objective optimization problem is solved


L1 metric

L2 metric

L∞ metric

Ideal objective

vector


When p=∞, maximum metric → nonsmooth

optimization problem

If p < ∞, the solution obtained is PO

If p = ∞, the solution obtained is weakly PO

A posteriori methods

Idea: 1) compute different PO solutions,

2) the DM selects the most preferred one

Approximation of the PO set (or part of it) is

approximated

Benefits

– Well suited for problems with 2 objectives since the

PO solutions can be easily visualized for the DM

– Understanding of the whole PO set

A posteriori methods

Drawbacks

– Approximating the PO set often time consuming

– DM has to choose the most preferred solution

among large number of solutions

– Visualization of the solutions for high number of

objectives

Weighting method

min𝑥∈𝑆

𝑤𝑖𝑓𝑖(𝑥)𝑘𝑖=1 ,

where 𝑤𝑖𝑘𝑖=1 = 1, 𝑤𝑖 ≥ 0, 𝑖 = 1,… , 𝑘

A weighted sum of the objectives is optimized

different PO solutions can be obtained by

changing the weights wi

One of the most well-known methods

– Gass & Saaty (1955), Zadeh (1963)

Weighting method

Benefits

– Solution obtained with positive weights is PO

– Easy to solve (simple objective function, no

additional constraints)

Drawbacks

– Can’t find solutions from non-convex parts of the

PO set

– PO solution obtained does not necessarily reflect

the preferences

Convex / non-convex PO set

Weights = slope of the level set of the objective function – Slope changes by changing the weights

Non-convex part can’t be reached with any weights!

f1, min

f2,

min

f1, min

f2,

min

w1=0.5, w2=0.5

w1=1/3, w2=2/3

convex PO set Non-convex PO set

Weighting method

Result1: The solution given by the weighting method is weakly PO

Result2: The solution given by the weighting method is PO if all the weights are strictly positive

Result3: Let 𝑥∗ be a PO solution of a convex multiobjective optimization problem. Then there exists a weighting vector 𝑤 = 𝑤1, … , 𝑤𝑘

𝑇 such that 𝑥∗ is the solution obtained with the weighting method.

Example

Where to go for a vacation (adopted from Prof. Pekka Korhonen)

The place with the best value for the objective function is the worst with respect to the most important objective!

Price Hiking Fishing Surfing

A 1 10 10 10

B 5 5 5 5

C 10 1 1 1

weight

Price Hiking Fishing Surfing

A 1 10 10 10

B 5 5 5 5

C 10 1 1 1

weight 0,4 0,2 0,2 0,2

Price Hiking Fishing Surfing Max

A 1 10 10 10 6,4

B 5 5 5 5 5

C 10 1 1 1 4,6

weight 0,4 0,2 0,2 0,2

ε-constraint method

min𝑥∈𝑆

𝑓𝑗(𝑥) 𝑠. 𝑡. 𝑓𝑖 𝑥 ≤ 𝜖𝑖 , ∀ 𝑖 ≠ 𝑗

Choose one of the objectives to be

optimized, give other objectives an upper

bound and consider them as constraints

Different PO solutions can be obtained by

changing the bounds and/or the objective to be

optimized

Haimes, Lasdon & Wismer (1971)


PO solutions for

different upper

bounds for 𝑓2

– ε1: no solutions

– ε2: z2

– ε3: z3

– ε4: z4

Fro

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Benefits

– Every PO solution can be found (also for non-

convex problems)

– Easy to implement

Drawbacks

– How to choose upper bounds?

• Does not necessarily give feasible solutions

– How to choose the objective to be optimized?


Result1: A solution obtained with the ε-

constraint method is weakly PO

Result2: A unique solution obtained with the ε-

constraint method is PO

Result3: A solution 𝑥∗ ∈ 𝑆 is PO if and only if it

is the solution given by the ε-constraint method

for every 𝑗 = 1,… , 𝑘 where 𝜖𝑖 = 𝑓𝑖 𝑥∗ , 𝑖 ≠ 𝑗.

(every PO solution can be found)


PO vs. weakly PO

– ε1: weakly PO

– ε2: PO

f1, min

f2,

min

ε1

ε2=f2(x*)

weakly PO

PO

Equally spaced PO solutions?

The weighting method:

change the weights

systematically

In the figure, PO solutions

are nearer to each other

towards the minimum of 𝑓2

How to obtain equally

spaced set? f1, min

f2,

min

Normal Boundary Intersection (NBI)

Find the extreme solutions of the PO set

Construct a plane passing through the extreme solutions; fix equally spaced points in the plane

Search orthogonal to the plane

f1, min

f2,

min


Idea: produce equally spaced approximation of the PO set

Solutions are produced by solving

max𝑥∈𝑆

𝜆 𝑠. 𝑡. 𝑃𝑤 − 𝜆𝑃𝑒 = 𝑓 𝑥 − 𝑧∗,

where 𝑃 is a payoff table, 𝑤 is the vector of weights ( 𝑤𝑖 = 1,𝑤𝑖 ≥ 0𝑘

𝑖=1 ) and 𝑒 = 1,… , 1 𝑇

Das & Dennis, SIAM Journal of Optimization, 8, 1998


Properties

– Equally spaced solutions

aproximating the PO set

– Computation time increases

significantly when the number

of objectives increases

– Can produce non PO

solutions for non-convex

problems f1, min

f2,

min

Equally spaced PO solutions?

The weighting method

f1, min

f2,

min

f1, min

f2,

min

Normal Boundary Intersection

NBI gives more equally spaced solutions

A priori methods

Idea: 1) ask first the preferences of the DM, 2) optimize using the preferences

Only such PO solutions are produced that are of interest to the DM

Benefits – Computed PO solutions are based on the preferences

of the DM (no ”unnecessary” solutions)

Drawbacks – It may be difficult for the DM to express preferences

before (s)he has seen any solutions

Lexicographic ordering

Order the objectives according to their importance

Optimize first w.r.t. to the most important one and continue optimizing the second most important one in the set of optimal solutions for the first one etc.

Requires the importance order from the DM before optimization

The solution obtained is PO

Lexicographic ordering

2 objectives: 1st

more important

Optimize w.r.t. 1st:

z1 and z2 obtained

Optimize w.r.t. 2nd:

choose better → z1

In practice, some

tolerance is used

for optimal values Fro

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Interactive methods

Idea: DM is utilized actively during the solution process

Solution process is iterative: 1. Initialization: compute some PO solution(s)

2. Show PO solution(s) to the DM

3. Is the DM satisfied? If no, ask the DM to give new preferences. Otherwise, stop. A most preferred solution has been found.

4. Compute new PO solution(s) by taking into account new preferences. Go to step 2.

Solution process ends when the DM is satisfied with the PO solution obtained

Interactive methods

Benefits – Only such solutions are computed that are of interest to the

DM

– DM is able to steer the solution process with his/her preferences

– DM can learn about the interdependences between the conflilcting objectives through the solutions obtained based on the preferences → helps adjusting the preferences

Drawbacks – DM has to invest a lot of time in the solution process

– If computing PO solutions takes time, DM does not necessarily remember what happened in the early phases

Reference point method

Interactive method, based on the usage of a reference point

Reference point is an intuitive way to express preferences

DM gives a reference point that is used in scalarizing the problem

Different PO solutions are obtined by changing the reference point

Wierzbicki, ”The Use of Reference Objectives in Multiobjective Optimization”, In: Multiple Criteria Decision Making, Theory and Applications, Springer, 1980

• min𝑥∈𝑆

max𝑖=1,…,𝑘

𝑤𝑖(𝑓𝑖 𝑥 − 𝑧 𝑖)

• Reference point

– Consists of aspiration levels for the objectives

– Can be in the image of the feasible region (𝑍 = 𝑓(𝑆)) or not

• Weights

– Affect the solution obtained, are not coming from the DM


Effect of the weights

f1, min

f2,

min

nadz

1z*z

2z

*1

i

nad

iizzw

*1

iiizzw

i

nad

iizzw

1


Results:

– Reference point method produces weakly PO

solutions

– Every weakly PO solution can be found

– Scalarization of the reference point method can be

changed so that the solution obtained is PO


Scalarized problem is not differentiable due to

the min-max form

Can be reformulated in order to have

differentiable form (if the objective are

differentiable)

– An additional variable and extra constraints

min𝑥∈𝑆,𝛿∈𝑅

𝛿 𝑠. 𝑡. 𝑤𝑖 𝑓𝑖 𝑥 − 𝑧 𝑖 ≤ 𝛿 ∀ 𝑖 = 1,… , 𝑘

Satisficing Trade-Off Method (STOM)

Interactive method, based on classification of

the objective functions

Very similar to the idea of the reference point

method

Nakayama & Sawaragi, “Satisficing Trade-Off

Method for Multiobjective Programming”, In:

Interactive Decision Analysis, Springer-Verlag,

1984


DM classifies the objectives into 3 classes at the current PO solution – 𝑓𝑖, whose values should be improved

– 𝑓𝑖, whose value is satisfactory at the moment

– 𝑓𝑖 whose value is allowed to get worse

A reference point is formed based on the classification – DM gives aspiration levels for the functions in the first class

– Aspiration levels for the functions in the second class are the current values

– Aspiration levels for the functions in the third class can be computed by using automatic trade-off → help for the DM


min𝑥∈𝑆

max𝑖=1,…,𝑘

𝑓𝑖 𝑥 −𝑧𝑖∗

𝑧 𝑖−𝑧𝑖∗ + 𝜌

𝑓𝑖(𝑥)

𝑧 𝑖−𝑧𝑖∗

𝑘𝑖=1

Aspiration levels must be greater than the

components of the ideal objective vector

A solution of the scalarized problem in STOM

is PO (if the augmentation term is used)


f1, min

f2,

min

nadz

1z*z

2z

*1

iiizzw

NIMBUS method

Interactive method, based on classification of the

objectives

Classification: consider the current PO solution

and set every objective into one of the classes

Miettinen, Nonlinear Multiobjective Optimization,

Kluwer Academic Publishers, 1999

Miettinen & Mäkelä, ” Synchronous Approach in

Interactive Multiobjective Optimization”, European

Journal of Operational Research, 170, 2006

NIMBUS method

5 classes consist of objectives 𝑓𝑖 whose values

– should be improved as much as possible (i є Iimp)

– should be improved until 𝑧 𝑖 (i є Iasp)

– is satisfactory at the moment (i є Isat)

– Is allowed to get worse until 𝜖𝑖 (i є Ibound)

– Can change freely at the moment (i є Ifree)

NIMBUS method

Classification is feasible if

A scalarized problem is formed based on the

classification (𝑥𝑐 is the current PO solution)

min𝑥∈𝑆

max𝑖∈𝐼𝑖𝑚𝑝,𝑗∈𝐼𝑎𝑠𝑝

𝑓𝑖 𝑥 −𝑧𝑖∗

𝑧𝑖𝑛𝑎𝑑−𝑧𝑖

∗ ,𝑓𝑗 𝑥 −𝑧 𝑗

𝑧𝑗𝑛𝑎𝑑−𝑧𝑗

∗ + 𝜌 𝑓𝑖(𝑥)

𝑧𝑖𝑛𝑎𝑑−𝑧𝑖

∗𝑘𝑖=1

𝑠. 𝑡. 𝑓𝑖 𝑥 ≤ 𝑓𝑖 𝑥𝑐 ∀ 𝑖 ∈ 𝐼𝑖𝑚𝑝 ∪ 𝐼𝑎𝑠𝑝 ∪ 𝐼𝑠𝑎𝑡,

𝑓𝑖 𝑥 ≤ 𝜖𝑖 ∀ 𝑖 ∈ 𝐼𝑏𝑜𝑢𝑛𝑑

NIMBUS method

Results:

– Solution of the scalarized problem in the NIMBUS

method is weakly PO without the augmentation term

– It is PO if the augmentation term is used

In the synchronous NIMBUS method, 4 different

scalarizations are used

– Different solutions can be obtained for the same

preference information

– No just one way to scalarize the problem, the DM gets

to choose from the solutions obtained

WWW-NIMBUS: implementation of the NIMBUS

method operating on the Internet

1st multiobjective optimization software

operating on the Internet (2000)

All the computations are done in servers at JYU,

only a browser is needed

Always the latest version available

Graphical user interface based on forms

Freely available for academic purposes

http://nimbus.it.jyu.fi/

http://nimbus.it.jyu.fi/

http://www.mcdmsociety.org/

Newsletter

23rd International Conference on Multiple

Criteria Decision Making

3-7 August 2015, Hamburg (Germany)

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January 23-27, 2012 Dagstuhl Seminar on Learning in

Multiobjective Optimization

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multiobjective optimization methods -...

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