Progressively Interactive Evolutionary Multi-Objective Optimization ...

towards the most preferred solution. Incorporating preferences from the decision maker in the
optimization run makes the search process more efficient in terms of function evaluations as
well as accuracy. The integrated methodology proposed in this paper, interacts with the decision
maker after every few generations of an evolutionary algorithm and is different from an a posteriori
approach, as it explores only the most preferred point. An a posteriori approach like the
HBLEMO and other evolutionary multi-objective optimization algorithms [5, 26] produce the
entire efficient frontier as the final solution and then a decision maker is asked to pick up the
most preferred point. However, an a posteriori approach is not a viable methodology for problems
which are computationally expensive and/or involve high number of objectives (more than
three) where EMOs tend to suffer in convergence as well as maintaining diversity.
In this paper, the bilevel multi-objective problem has been described initially and then the
integrated procedure, Progressively Interactive Hybrid Bilevel Evolutionary Multi-objective Optimization
(PI-HBLEMO) algorithm, has been discussed. The performance of the PI-HBLEMO
algorithm has been shown on a set of five DS test problems [9, 6] and a comparison for the
savings in computational cost has been done with a posteriori HBLEMO approach.
2 Recent Studies
In the context of bilevel single-objective optimization problems a number of studies exist, including
some useful reviews [3, 21], test problem generators [2], and some evolutionary algorithm
(EA) studies [18, 17, 24, 16, 23]. Stackelberg games [13, 22], which have been widely
studied, are also in principle similar to a single-objective bilevel problem. However, not many
studies can be found in case of bilevel multi-objective optimization problems. The bilevel multiobjective
problems have not received much attention, either from the classical researchers or
from the researchers in the evolutionary community.
Eichfelder [12] worked on a classical approach on handling multi-objective bilevel problems,
but the nature of the approach made it limited to handle only problems with few decision
variables. Halter et al. [15] used a particle swarm optimization (PSO) procedure at both the levels
of the bilevel multi-objective problem but the application problems they used had linearity in
the lower level. A specialized linear multi-objective PSO algorithm was used at the lower level,
and a nested strategy was utilized at the upper level.
Recently, Deb and Sinha have proposed a Hybrid Bilevel Evolutionary Multi-objective
Optimization algorithm (HBLEMO) [9] using NSGA-II to solve both level problems in a synchronous
manner. Former versions of the HBLEMO algorithm can be found in the conference
publications [6, 8, 19, 7]. The work in this paper extends the HBLEMO algorithm by allowing
the decision maker to interact with the algorithm.
3 Multi-Objective Bilevel Optimization Problems
In a multi-objective bilevel optimization problem there are two levels of multi-objective optimization
tasks. A solution is considered feasible at the upper level only if it is a Pareto-optimal
member to a lower level optimization problem [9]. A generic multi-objective bilevel optimization
problem can be described as follows. In the formulation there are M number of objectives
at the upper level and m number of objectives at the lower level:
Minimize (xu,xl) F(x) = (F1(x), . . . , FM (x)) ,
subject to xl ∈ argmin (xl) f(x) = (f1(x), . . . , fm(x))
g(x) ≥ 0, h(x) = 0 ,
G(x) ≥ 0, H(x) = 0,
≤ xi ≤ x (U)
i , i = 1, . . . , n.
x (L)
i
In the above formulation, F1(x), . . . , FM (x) are upper level objective functions which are
M in number and f1(x), . . . , fm(x) are lower level objective functions which are m in number.
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