Progressively Interactive Evolutionary Multi-Objective Optimization ...

e satisfied (Reklaitis et al., 1983; Rao, 1984). Optimization problems come in many
different forms and complexities involving the type and size of the variable vector,
objective and constraint functions, nature of problem parameters, modalities of objectivefunctions,interactionsamongobjectives,extentoffeasiblesearchregion,computational
burdenand inherent noise in evaluating solutions, etc. (Deb, 2001). While these
factorsarekeepingoptimizationresearchersandpractitionersbusyindevisingefficient
solution procedures,the practicealways seems to have more to offer thanwhat the researchers
have been able to comprehend and implement in the realm of optimization
studies.
Bilevel programming problems have twolevels of optimization problems – upper
and lower levels (Colson et al., 2007; Vicente and Calamai, 2004). In the upper level
optimization task, a solution, in addition to satisfying its own constraints, must also
be an optimal solution to another optimization problem, called the lower level optimization
problem. Although the concept is intriguing, bilevel programming problems
commonlyappearinmanypracticaloptimizationproblems(Bard,1998). Thinkingsimply,
the bilevel scenario occurs when a solution in an (upper level) optimization task
must be a physically or a functionally acceptable solution, such as being a stable solution
or being a solution in equilibrium or being a solution which must satisfy certain
conservationprinciples,etc. Satisfactionofoneormoreoftheseconditions canthenbe
posed as another (lower level) optimization task. However, often in practice (Bianco
etal.,2009;Dempe,2002;Pakala,1993),suchproblemsarenotusuallytreatedasbilevel
programming problems, instead some approximatemethodologies areused to replace
the lower level problem. In many scenarios it is observed that approximate solution
methodologies are not available or practically and functionally unacceptable. Ideally
such problems involving an assurance of a physically or functionally viable solution
must be posedas bilevel programming problemsandsolved.
Bilevel programming problems involving a single objective function inupper and
lower levels have received some attention from theory (Dempe et al., 2006), algorithm
development and application (Alexandrov and Dennis, 1994; Vicente and Calamai,
2004),andevenusing evolutionary algorithms (Yin, 2000;Wang et al., 2008). However,
apartfromafewrecentstudies(Eichfelder,2007,2008;HalterandMostaghim,2006;Shi
andXia,2001)andourrecentevolutionarymulti-objectiveoptimization(EMO)studies
(Deb and Sinha, 2009a,b; Sinha and Deb, 2009), multi-objective bilevel programming
studies are scarce in both classical and evolutionary optimization fields. The lack of
interests for handling multiple conflicting objectives in a bilevel programming context
is not due to lack of practical problems, but more due to the need for searching and
storing multiple trade-off lower level solutions for a single upper level solution and
due to the complex interactions which upper and lower level optimization tasks can
provide. Inthispaper,wemakeacloserlookattheintricaciesofmulti-objectivebilevel
programming problems, present a set of difficult test problems by using an extended
version of our earlier proposed test problem construction procedure,and propose and
evaluateahybridEMO-cum-local-searchbilevelprogrammingalgorithm(H-BLEMO).
In the remainder of this paper, we briefly outline a generic multi-objective bilevel
optimization problemand then providean overview of existing studies both on single
and multi-objective bilevel programming. Past evolutionary methods are particularly
highlighted. Thereafter, we list a number of existing multi-objective bilevel test problemsandthendiscussanextensionofourrecentsuggestion.
Theproposedhybridand
self-adaptive bilevel evolutionary multi-objective optimization algorithm (H-BLEMO)
isthendescribedindetailbyprovidingastep-by-stepprocedure. Simulationresultson
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