Progressively Interactive Evolutionary Multi-Objective Optimization ...

problems, in which the upper level problem optimizes the overall cost and quality of
product,whereasthelowerleveloptimizationproblemoptimizes errormeasuresindicating
how closely the processadherestodifferenttheoretical processconditions, such
asmass balanceequations, cracking,or distillation principles (Dempe,2002).
ThebilevelproblemsarealsosimilarinprincipletotheStackelberggames(Fudenberg
and Tirole, 1993;Wang and Periaux,2001)in which a leadermakes the first move
andafollowerthenmaximizesitsmoveconsidering theleader’smove. Theleaderhas
anadvantageinthatitcancontrolthegamebymakingitsmoveinawaysoastomaximize
its own gain knowing that the follower will always maximize its own gain. For
an example (Zhang et al., 2007), a company CEO (leader) may be interested in maximizing
net profits andquality ofproducts, whereasheadsof branches (followers)may
maximize their own net profit and worker satisfaction. The CEO knows that for each
of his/her strategy,the headsof branches will optimize their ownobjectives. The CEO
mustthenadjusthis/herowndecisionvariablessothatCEO’sownobjectivesaremaximized.
Stackelberg’sgamemodelanditssolutionsareusedinmanydifferentproblem
domains,includingengineeringdesign(Pakala,1993),securityapplications(Paruchuri
et al.,2008),andothers.
3 Existing Classical andEvolutionaryMethodologies
Theimportanceofsolvingbileveloptimizationproblems,particularlyproblemshaving
a single objective in each level, has been recognized amply in the optimization literature.
The researchhas been focused in both theoretical and algorithmic aspects. However,therehas
beenalukewarminterest in handling bilevel problems having multiple
conflicting objectives in any or both levels. Here we provide a brief description of the
main research outcomes so far in both single and multi-objective bilevel optimization
areas.
3.1 TheoreticalDevelopments
Severalstudiesexistindeterminingtheoptimalityconditionsforaupperlevelsolution.
Thedifficultyarisesduetotheexistenceofanotheroptimizationproblemasahardconstraint
to the upper level problem. Usually the Karush-Kuhn-Tucker (KKT) conditions
ofthelowerleveloptimizationproblemsarefirstwrittenandusedasconstraintsinformulating
theKKT conditions ofthe upperlevelproblem,involving secondderivatives
of the lower level objectives and constraints as the necessary conditions of the upper
level problem. However,as discussed in Dempe et al. (2006),although KKT optimality
conditions can be written mathematically, the presence of many lower level Lagrange
multipliers andanabstractterminvolving coderivativesmakes the proceduredifficult
tobe appliedin practice.
Fliege and Vicente (2006) suggested a mapping concept in which a bilevel singleobjectiveoptimizationproblem(oneobjectiveeachinupperandlowerlevelproblems)
can be converted to an equivalent four-objective optimization problem with a special
cone dominance concept. Although the idea may apparently be extended for bilevel
multi-objectiveoptimizationproblems,nosuchsuggestionwithanexactmathematical
formulation is made yet. Moreover, derivatives of original objectives are involved in
the problem formulation, thereby making the approach limited to only differentiable
problems.
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