Progressively Interactive Evolutionary Multi-Objective Optimization ...

Bilevel Multi-Objective Optimization Problem
Solving Using Progressively Interactive EMO
Ankur Sinha Ankur.Sinha@aalto.fi
Department of Business Technology, Aalto University School of Economics
PO Box 21210, FIN-00076 Aalto, Helsinki, Finland
Abstract
Bilevel multi-objective optimization problems are known to be highly complex optimization
tasks which require every feasible upper-level solution to satisfy optimality of a lower-level
optimization problem. Multi-objective bilevel problems are commonly found in practice and
high computation cost needed to solve such problems motivates to use multi-criterion decision
making ideas to efficiently handle such problems. Multi-objective bilevel problems have been
previously handled using an evolutionary multi-objective optimization (EMO) algorithm where
the entire Pareto set is produced. In order to save the computational expense, a progressively
interactive EMO for bilevel problems has been presented where preference information from
the decision maker at the upper level of the bilevel problem is used to guide the algorithm
towards the most preferred solution (a single solution point). The procedure has been evaluated
on a set of five DS test problems suggested by Deb and Sinha. A comparison for the number
of function evaluations has been done with a recently suggested Hybrid Bilevel Evolutionary
Multi-objective Optimization algorithm which produces the entire upper level Pareto-front for
a bilevel problem.
Keywords
Genetic algorithms, evolutionary algorithms, bilevel optimization, multi-objective optimization,
evolutionary programming, multi-criteria decision making, hybrid evolutionary algorithms, sequential
quadratic programming.
1 Introduction
Bilevel programming problems are often found in practice [25] where the feasibility of an upper
level solution is decided by a lower level optimization problem. The qualification for an
upper level solution to be feasible is that it should be an optimal candidate from a lower level
optimization problem. This requirement consequentially makes a bilevel problem very difficult
to handle. Multiple objectives at both the levels of a bilevel problem further adds to the
complexity. Because of difficulty in searching and defining optimal solutions for bilevel multiobjective
optimization problems [11], not many solution methodologies to such problems have
been explored. One of the recent advances made in this direction is by Deb and Sinha [9] where
the entire Pareto set at the upper level of the bilevel multi-objective problem is explored. The
method, though successful in handling complex bilevel multi-objective test problems, is computationally
expensive and requires high function evaluations, particularly at the lower level. High
computational expense associated to such problems provides a motivation to explore a different
solution methodology.
Concepts from a Progressively Interactive Evolutionary Multi-objective Optimization algorithm
(PI-EMO-VF) [10] has been integrated with the Hybrid Bilevel Evolutionary Multiobjective
Optimization algorithm (HBLEMO) [9] in this paper. In the suggested methodology,
preference information from the decision maker at the upper level is used to direct the search
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