Progressively Interactive Evolutionary Multi-Objective Optimization ...

3.2 AlgorithmicDevelopments
Onesimplealgorithmforsolvingbileveloptimizationproblemsusingapoint-by-point
approachwouldbetodirectlytreatthelowerlevelproblemasahardconstraint. Every
solution (x = (xu, xl)) must be sent to the lower level problem as an initial point and
anoptimization algorithm canthen be employed to find the optimal solution x∗ l of the
lowerleveloptimizationproblem. Then,theoriginalsolution xoftheupperlevelprob-
lemmustberepairedas(xu, x ∗ l
). Theemploymentofalowerleveloptimizerwithinthe
upper level optimizer for every upper level solution makes the overall search a nested
optimization procedure, which may be computationally an expensive task. Moreover,
ifthisideaistobeextendedformultipleconflictingobjectivesinthelowerlevel,foreveryupperlevelsolution,
multiple Pareto-optimalsolutions forthelowerlevelproblem
needtobe found andstoredby asuitable multi-objective optimizer.
Anotheridea(Herskovitsetal.,2000;Biancoetal.,2009)ofhandlingthelowerlevel
optimization problemhaving differentiableobjectives and constraints is to include the
explicitKKT conditions of thelower leveloptimization problemdirectlyas constraints
to the upper level problem. This will then involve Lagrange multipliers of the lower
leveloptimizationproblemasadditionalvariablestotheupperlevelproblem. AsKKT
pointsneednotalwaysbeoptimumpoints,furtherconditionsmusthavetobeincluded
to ensure the optimality of lower level problem. For multi-objective bilevel problems,
corresponding multi-objective KKT formulations need to be used, thereby involving
furtherLagrangemultipliersandoptimalityconditionsasconstraintstotheupperlevel
problem.
Despite these apparent difficulties, there exist some useful studies, including reviews
on bilevel programming (Colson et al., 2007; Vicente and Calamai, 2004), test
problem generators (Calamai and Vicente, 1994), nested bilevel linear programming
(Gaur and Arora, 2008), and applications (Fampa et al., 2008; Abass, 2005; Koh, 2007),
mostly inthe realmof single-objective bilevel optimization.
RecentstudiesbyEichfelder(2007,2008)concentratedonhandlingmulti-objective
bilevel problems using classical methods. While the lower level problem uses a numerical
optimization technique, the upper level problemis handled using an adaptive
exhaustive search method, thereby making the overall procedure computationally expensiveforlarge-scaleproblems.
Thismethod usesthenestedoptimizationstrategyto
find and store multiple Pareto-optimal solutions for each of finitely-many upper level
variablevectors.
Another study by Shi and Xia (2001) transformed a multi-objective bilevel programming
problem into a bilevel ǫ-constraint approach in both levels by keeping one
of the objective functions and converting remaining objectives to constraints. The ǫ
values for constraints weresupplied by the decision-maker as differentlevels of ‘satisfactoriness’.
Further,thelower-levelsingle-objectiveconstrainedoptimizationproblem
wasreplacedbyequivalentKKTconditionsandavariablemetricoptimizationmethod
was usedtosolve the resulting problem.
Certainly, more efforts are needed to devise effective classical methods for multiobjective
bilevel optimization, particularly tohandle the upperlevel optimization task
inamore coordinatedwaywith the lower leveloptimization task.
3.3 EvolutionaryMethods
Several researchers have proposed evolutionary algorithm based approaches in solving
single-objective bilevel optimization problems. As early as in 1994, Mathieu et al.
(1994) proposed a GA-based approach for solving bilevel linear programming prob-
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