Computers and Chemical Engineering 25 (2001) 257 – 266
Evolutionary algorithms approach to the solution of mixed integer
non-linear programming problems
Lino Costa, Pedro Oliveira *
Department of Production and Systems Engineering, Uniersity of Minho, 4710 Braga, Portugal
Received 7 January 2000; received in revised form 25 October 2000; accepted 25 October 2000
Abstract
The global optimization of mixed integer non-linear problems (MINLP), constitutes a major area of research in many
engineering applications. In this work, a comparison is made between an algorithm based on Simulated Annealing (M-SIMPSA)
and two Evolutionary Algorithms: Genetic Algorithms (GAs) and Evolution Strategies (ESs). Results concerning the handling of
constraints, through penalty functions, with and without penalty parameter setting, are also reported. Evolutionary Algorithms
seem a valid approach to the optimization of non-linear problems. Evolution Strategies emerge as the best algorithm in most of
the problems studied. © 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Evolutionary algorithms; Genetic algorithms; Evolution strategies; Mixed integer non-linear programming
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1. Introduction
Real world problems can, in general, be formulated
as mixed integer non-linear programming problems
(MINLP). These problems, due to their combinatorial
nature, are considered difficult problems. Gradient op-
timization techniques have only been able to tackle
special formulations, where continuity or convexity had
to be imposed, or by exploiting special mathematical
structures. Approaches based on stochastic algorithms
have been used. These approaches, also know as adap-
tive random search, have successfully tackle MINLP,
mostly in the area of chemical engineering (Reklaitis,
Ravindran, & Ragsdell, 1983; Salcedo, 1992; Banga &
Seider, 1996). In recent years, a vast amount of work
has been published on applications of evolutionary
algorithms (Genetic Algorithms, Evolution Strategies,
Simulated Annealing, etc.) to the solution of MINLP in
many engineering applications. These algorithms are
distinct from conventional algorithms since, in general,
only the information regarding the objective function is
required. Moreover, they start from a pool of points
that evolves over time, possibly in the direction of the
optimum. It should be stressed that the objective of this
on going research is not to find the ‘best’ algorithm for
all problem instances, but to compare different al-
gorithms, in order to find out classes of problems which
may be more suitable for certain algorithms than
others.
The general formulation of the problem is as follows:
min f (x )
subject to
h (x ) =0 (1)
g (x ) 0
x
j
integer, j I
x X ={x x R
n
, x
l
x x
u
}
where
h R
m
, g R
p
.
In this work, 7 test problems, proposed by indepen-
dent authors, are studied using Genetic Algorithms
(GAs) and Evolution Strategies (ESs). These problems
arise from the area of chemical engineering, and repre-
sent difficult non-convex optimization problems, with
continuous and discrete variables. Comparisons are
made with an M-SIMPSA algorithm (Cardoso, Sal-
cedo, & Feyo de Azevedo, 1996a,b; Cardoso, Salcedo,
* Corresponding author. Fax: +351-53-253604741.
E-mail address: pno@dps.uminho.pt (P. Oliveira).
0098-1354/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved.
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