Progressively Interactive Evolutionary Multi-Objective Optimization ...

lems having asingle objective ineachlevel. Thelower levelproblemwas solvedusing
a standard linear programming method, whereas the upper level was solved using
a GA. Thus, this early GA study used a nested optimization strategy, which may be
computationally too expensive to extend for nonlinear and large-scale problems. Yin
(2000)proposed another GA based nested approach in which the lower level problem
was solved using the Frank-Wolfe gradient based linearizedoptimization method and
claimedtosolvenon-convexbileveloptimizationproblemsbetterthananexistingclassical
method. Oduguwa and Roy (2002) suggested a coevolutionary GA approach in
whichtwodifferentpopulationsareusedtohandlevariablevectors xu and xl independently.
Thereafter, a linking procedure is used to cross-talk between the populations.
For single-objective bilevel optimization problems, the final outcome is usually a single
optimal solution in each level. The proposed coevolutionary approach is viable
for finding corresponding single solution in xu and xl spaces. But in handling multiobjectivebilevelprogrammingproblems,multiplesolutionscorrespondingtoeachupperlevelsolution
mustbefoundandmaintainedduring thecoevolutionary process. It
is not clear how such a coevolutionary algorithm can be designed effectively for handling
multi-objective bilevel optimization problems. We do not address this issue in
thispaper,butrecognizethatOduguwaandRoy’s study(2002)wasthefirsttosuggest
a coevolutionary procedure for single-objective bilevel optimization problems. Since
2005, a surge in research in this area can be found in algorithm development mostly
using the nested approachand the explicit KKT conditions of the lower level problem,
and in various application areas (Hecheng and Wang, 2007; Li and Wang, 2007; Dimitriou
et al., 2008;Yin, 2000;Mathieu et al., 1994;Sun et al., 2006;Wang et al., 2007;Koh,
2007;Wang etal., 2005,2008).
Li et al. (2006) proposed particle swarm optimization (PSO) based procedures for
both lower and upper levels, but instead of using a nested approach, they proposed
a serial application of upper and lower levels iteratively. This idea is applicable in
solving single-objective problemsineachlevelduetothesoletargetoffinding asingle
optimalsolution. Asdiscussedabove,inthepresenceofmultipleconflicting objectives
in each level, multiple solutions need to be found and preserved for each upper level
solution and then a serial application of upper and lower level optimization does not
makesense formulti-objective bileveloptimization. HalterandMostaghim(2006)also
usedPSOonbothlevels,butsincethelowerlevelproblemintheirapplicationproblem
was linear, they used a specialized linear multi-objective PSO algorithm and used an
overallnested optimization strategyatthe upperlevel.
Recently, we have proposed a number of EMO algorithms through conference
publications (Deb and Sinha, 2009a,b; Sinha and Deb, 2009) using NSGA-II to solve
both level problems in a synchronous manner. First, our methodologies were generic
so that they can be used to linear/nonlinear, convex/non-convex, differentiable/nondifferentiableandsingle/multi-objective
problemsatbothlevels. Second,ourmethodologies
did not use the nested approach, nor did they use a serial approach, but employed
a structured intertwined evolution of upper and lower level populations. But
they werecomputationally demanding. However,these initial studies madeus understand
the complex intricacies by which both level problems can influence each other.
Basedonthisexperience,inthispaper,wesuggestaless-structural,self-adaptive,computationallyfast,andahybridevolutionaryalgorithmcoupledwithalocalsearchprocedurefor
handling multi-objective bilevelprogramming problems.
Bilevel programming problems, particularly with multiple conflicting objectives,
should have been paid more attention than what has been made so far. As more and
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