Some Interactive Multiobjective Optimization Methods with Hybrid Ideas

Prof. Kaisa Miettinenkaisa.miettinen@jyu.fi

Department of Mathematical Information Technology

University of Jyväskylä, Finlandhttp://www.mit.jyu.fi/miettine/

ContentsConcepts4 classes of methodsHybridsAdaptive approximation method and reference pointsPareto NavigatorEMO and reference pointsNautilusSynchronous NIMBUS®

– Software – Applications

Conclusions

Concepts

where fi: S → R are conflicting (possibly nonlinear/ nonconvex) objective functions and S consists of linear, nonlinear and/or boxconstraints for variables x ∈ Rn

S ⊂ Rn is feasible regionk objective functions; objective function valueszi = fi(x) and objective vectors z = (z1,…, zk) ∈ Rk

Feasible objective region Z ⊂ Rk is image of S. Thus z ∈ Z

OptimalityFor minimization: point x*∈ S (and z) is Pareto optimal(PO) if there exists no other point x∈S such that fi(x) ≤fi(x*) for all i =1,…,k and fj(x) < fj(x*) for at least one jRanges in the PO set:

Ideal objective vector z* (Utopian objective vector z**) Nadir objective vector znad

Solution = best possible compromiseDecision maker (DM) responsible for final solutionMultiobjective optimization = help DM in finding most preferred (PO) solutionWe need preference information from DMScalarization = combine preferences and original problem ⇒ scalarized single objective subproblemPreferences e.g. reference point consisting of aspiration levels

BackgroundHow to support DM?Four types of methodsNo decision maker – some neutral compromise solutionA priori methods: DM sets hopes and closest solution is found

– Expectations may be too optimistic or pessimistic– Hard to express preferences without knowing the problem well

A posteriori methods: generate representation of PO set+ Gives information about variety of PO solutions– Expensive, computationally demanding– Difficult to represent the PO set if k > 2o Example: evolutionary multiobjective optimization methods

Interactive methods: iterative search process+ Avoid difficulties above+ Solution pattern is formed and repeated iteratively+ Move around Pareto optimal set+ What can we expect DMs to be able to say?+ Goal: easiness of use + Cognitively valid approaches: classification and

reference point consisting of aspiration levelsFurther information: Kaisa Miettinen: Nonlinear Multiobjective Optimization, Kluwer (Springer), 1999

Why not Weighting Method

beauty cooking house-wifery

tidi-ness

Mary 1 10 10 10Jane 5 5 5 5Carol 10 1 1 1

Selecting a wife (maximization problem):

Idea originally from Prof. Pekka Korhonen

Why not Weighting Method

beauty cooking house-wifery

tidi-ness

Mary 1 10 10 10Jane 5 5 5 5Carol 10 1 1 1weights 0.4 0.2 0.2 0.2

Selecting a wife (maximization problem):

Why not Weighting Method

beauty cooking house-wifery

tidi-ness

results

Mary 1 10 10 10 6.4Jane 5 5 5 5 5Carol 10 1 1 1 4.6weights 0.4 0.2 0.2 0.2

Selecting a wife (maximization problem):

Hybrid MethodsPut together ideas of different methods to form new onesAim: at the same time

combine strengths and benefitsavoid weaknesses

A posteriori methodsinformation of whole PO set – possibilities and limitations

Interactive methodsDM can learn about the problem, its interdependencies and adjust preferencesDM can concentrate on interesting solutionscomputationally less costly

J. Branke, K. Deb, K. Miettinen, R. Slowinski (Eds.): Multiobjective Optimization: Interactive and Evolutionary Approaches, Springer, 2008

f1

f2

Achievement Scalarizing Function: Nonlinear Case

AA’

B

B’

Adaptive Approximation Method + Reference Points

Klamroth & Miettinen, Oper Res (2008)Rough approximation of PO solutions (i.e. small representative set - not too many solutions)Piecewise linear approximations provide powerful and computationally efficient tool to give overview of PO setApproximations give information about problem and serve as basis for refinements of DM’s preferences – (s)he learnsWith reference point DM indicates preference regions where to refine approximation Reference point = least acceptable values for each objectiveDM iteratively directs search towards best PO solutionAfter DM has identified most interesting region of PO set, final solution can be fine-tuned with interactive methods We support DM in updating reference point

Selecting one Solution

1 2

1

2

3

0 1 2

1

2

3

0 1 2

1

2

3

Here DM wanted to zoom in at each iteration

0 1 2

1

2

3

ClassificationChange current minimal acceptable levels– increase from current level– decrease from current level– keep the same

0 1 2

1

2

3

Pareto NavigatorEskelinen et al., OR Spectrum (2010)

Background & motivation– How to support DM?– I Learning phase II Decision phase– Challenges of computationally costly problems

Pareto optimal set = actual Pareto optimal setLearning-oriented interactive methodHybrid method which combines a posteriori and interactive methodsInstead of approximating objective functions (response surfaces or kriging etc.) we directly approximate PO set

Pareto Navigator, cont.Initialization phase

(relatively small) set of Pareto optimal solutionspolyhedral approximation of Pareto optimal set (convex hull of PO solutions available) in objective space –approximated PO set

Navigation phase dynamic real-time movement into desired directioninformation of whole PO set – possibilities and limitationsactive participation of DM: DM can learn about problem, trade-offs, interdependencies and adjust hopes DM can direct the search and concentrate on interesting solutionscomputationally inexpensiveunderstandable concepts for DM

Progress of Method

Because k=3, we can see what happened in objective space during the solution process(polyhedral approximation and actual PO set)

In 3D

Two Worlds

Multiple criteria decision making

– Role of DM and decision support emphasized

– Role of preference information important

– Different types of methods - interactive ones widely developed

– Solid theoretical background (we can prove Pareto optimality etc.)

– Scalarization combining objective and preferences into real-valued functions

Evolutionary multiobjective optimization (EMO)

–Idea to approximate the set of Pareto optimal solutions

–Criteria: minimize distance to real PO set and maximize diversity within the approximation

–Not too much emphasis on DM’s preferences so far

–Guaranteeing actual optimality not always clear

–E.g. nonconvexity and discontinuity cause no difficulties

–Background in applications–Many benchmark problems

EMO + Reference PointsThiele et al., Evol. Comp. (2009)

New interactive evolutionary algorithm including preference information in reference pointFitness function includes achievement scalarizing function (based on reference point)DM can direct search – not whole PO set approximately equally accuratelyPopulation size can be kept relatively smallSolution(s) with best achievement function value shown to DMWorks also for mathematically challenging problems

Interactive Preference-based Evolutionary Algorithm

1. Initialization: find rough approximation of PO set with small population size using IBEA. Select set of solutions characterizing approximation and show to DM

2. Reference point: Ask DM to specify desired aspiration levels to form reference point

3. Local approximation: Use reference point in quality indicator to generate a local approximation of PO set

4. Projection of reference point: Among solutions generated show to DM the one giving smallest value for achievement function or more nondominated solutions if desired

5. Termination: if DM wants to continue, go to Step 1. Otherwise, stop.

Examples ZDT1:

α= 20, N=500

g=(0.6, 1.0)

▲: without reference point

: PBEA with δ=0.1

: PBEA with δ=0.02

Circled: best

Possible Interaction

NAUTILUSMiettinen et al. (submitted)

Motivation:– Moving among PO solutions requires trading off– People do not react to gains and losses symmetrically– Also anchoring may prohibit movements

Most preferred solution may not be foundIn Nautilus, we start from the nadir objective vector and move towards PO set– Possible to gain at every iteration – no need for impairment

At each iteration, objective vector obtained dominates the previous oneTwo possibilities of specifying preference information: – Rank the relative importance of improving each current objective

value– Specify percentages how DM would like to improve the current obj.

valuesEach solution improves the previous one. Only in the last iteration we get PO objective vector. Distance to PO set shownThe method allows the DM to approach the part of the PO set (s)he wishes

nadzz =0

lo,1zz =∗∗

Z=f (S)

( )louploup zzzz ,12

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211

),(21,1−−−=⎟⎟

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lo,2z

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louploup zzzz −−−=⎟⎟⎠

⎞⎜⎜⎝

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−2f

lo,3z

Example

Background for NIMBUS®

Solution = best possible compromiseDM is responsible for the final solutionDifficult to present the Pareto optimal set, expectations may be too highInteractive approach avoids these difficultiesMove around Pareto optimal setWhat can we expect DMs to be able to say?How can we support the learning process?DM should be able to direct the solution processGoal: easiness of use ⇒ no difficult questions & possibility to change one’s mind

NIMBUS® MethodDealing with objective function values is understandable and straightforwardForm of interaction: Classification of objective functions –cognitively valid approachClassification: desirable changes in the current PO objective function values fi(xc) Classes: functions fi whose values

– should be decreased (i∈I<)– should be decreased till some aspiration level < fi(xc) (i∈I≤)– are satisfactory at the moment (i∈I=)– are allowed to increase up till some upper bound > fi(xc) (i∈I>) and – are allowed to change freely (i∈I◊)

DM must be willing to give up somethingDM moves around the Pareto optimal set(Miettinen, Mäkelä: Optim. 1995, JORS 1999, Comp&OR 2000, EJOR 2006)

Synchronous NIMBUS®

Scalarization is important and contains preference informationBut scalarizations based on same input give different solutions – Which is the best? ⇒Synchronous NIMBUS®

Different solutions are obtained using different scalarizations(Miettinen, Mäkelä: OR Spec. 2002)Show them to the DM & let her/him choose the best(Miettinen, Mäkelä: EJOR 2006)NB: DM is assumed to have knowledge about the problem in question, no deep understanding of optimization process or theory

The first, unique interactive optimization system on the Internet

• Centralized computing (server in Jyväskylä) & distributed interface

• No special requirements for computers: No computing capacity nor compilers needed

• Latest version always available• Graphical user-interface via WWW• Even for nonconvex and nondifferentiable problems

and integer-valued variables• Available to any academic Internet user for free

Tutorial and online help(Miettinen, Mäkelä: Comp & OR 2000, EJOR 2006)

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

WWW-NIMBUS® since 1995

IND-NIMBUS®

For MS-Windows and Linux operating systems Synchronous algorithmMinimize/maximize objective functionsLinear/nonlinear inequality/equality and/or box constraintsContinuous or integer-valued variablesLocal solvers and global solvers and their hybridUser can change solver at every iterationUser can change parameters of solvers

IND-NIMBUS® cont.Possibility to

– see history of generated PO solutions– save interesting solutions and return to them

(visualize, intermediate solutions)– connect with BALAS® process simulator by

(VTT Processes) and GPS-X simulator– connect with different modelling and simulation

tools (e.g. MOTA, Matlab, GAMS etc.)

ApplicationsContinuous casting of steel

Miettinen et al., Comput Opt & Appl (1998)Miettinen, Mater & Manuf Processes (2007)

Headbox design for paper machinesHämäläinen et al., JOTA (2003)

Paper machine design (paper quality)Madetoja et al., Eng with Comp (2006)

Ultrasonic transducer design Heikkola et al., Ultrasonics (2006)

Chemical process designHakanen et al., JMCDA (2005)Hakanen et al., Appl Therm Eng (2006)

Simulated moving bed processes Hakanen et al., Cont & Cyb (2007)

Optimal shape design of exhaust pipe Aittokoski & Miettinen, Eng Opt (2008)

Intensity modulated radiotherapy planning Ruotsalainen et al., Contemp Eng Sciences (2009)

Wastewater treatment system planningHakanen et al., EngOPT2008

Continuous Casting of SteelControl of secondary cooling; intensity of water sprays affects solidification rate of steelQuality of steel depends on behavior of surface temperature and solidification front in timeOriginally, empty feasible regionConstraints into objectives: minimize constraint violations

– Keep the surface temperature near a desired temperature

– Keep the surface temperature between some upper and lower bounds

– Avoid excessive cooling or reheating on the surface

– Restrict the length of the liquid pool– Avoid too low temperatures at the yield

point

Paper Machine100-150 meters long, width up to 11 metersFour main components

– headbox – former– press – drying

In addition, finishing

Objectives– qualitative properties – save energy– use cheaper fillers and

fibres– produce as much as

possible– save environment

First design problem: Headbox outlet height controlThen chain of unit process models: virtual paper machineOptimize e.g. gloss, roughness, basis weight, fibre orientat.

Process Simulation in Chemical Engineering

Using BALAS® process simulator (product of VTT Finland) Flowsheet of process designed with BALAS® provides a simulation model to be optimized with IND-NIMBUS– Heat recovery: organize heat management

taking seasonal changes in climate into account (typically single objective of annualized energy and investment costs, estimated amortization time and interest rate for capital)

– Water allocation (recycle water in the process)

Simulated Moving Bed Processes

Periodic adsorption process for separation of chemical productsUtilizes difference in migration speeds of different chemical components in liquidDynamic process operating on periodic cycles –challenging optimization problemSeparation of glucose/fructose (fructose used in soft drinks & candies, price depends on purity)Single objective solver IPOPT4 objective functions– Max throughput– Min desorbent consumption– Max purity– Max recovery

Radiotherapy treatment planningAim of radiotherapy treatment

– destroy tumour with radiation– healthy tissue, usually divided into dose sensitive

organs, i.e. organs at risk, and normal tissue should get as low dose as possible (earlier maximal amounts of dosage were used)

Conflicting: increasing dose in tumour increases unwanted dose in healthy tissueRadiotherapy treatment planning is a multiobjective optimization problemAdvantages

– Enables comparing treatment plans and considering several treatment goals simultaneously

– No artificial tools like weighting coefficients– Emphasis on optimal not only feasible solutions

ConclusionsHybridization of different methods offers a lot of potentialPlenty of real-life applications are waiting for us!Experiences with NIMBUS– DM learned about the conflicting

qualitative properties– DM obtained new insight into complex

and conflicting phenomena– DM could consider several objectives

simultaneously– DM found the method easy to use– DM found a satisfactory solution and was

convinced of its goodness

Acknowledgements

Collaboration: coauthors and Industrial Optimization Group http://www.mit.jyu.fi/optgroup/Funding: Partly Academy of Finland, Tekes: Finnish Funding Agency for Technology and Innovation & companiesPhotos: Markku Könkkölä, http://www.kuvaaja.fi

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