Optim Lett (2018) 12:399–410
A weighting subgradient algorithm for multiobjective
G. C. Bento
· J. X. Cruz Neto
· P. S. M. Santos
S. S. Souza
Received: 5 April 2016 / Accepted: 17 March 2017 / Published online: 23 March 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract We propose a weighting subgradient algorithm for solving multiobjective
minimization problems on a nonempty closed convex subset of an Euclidean space.
This method combines weighting technique and the classical projected subgradient
method, using a divergent series steplength rule. Under the assumption of convexity,
we show that the sequence generated by this method converges to a Pareto optimal
point of the problem. Some numerical results are presented.
Keywords Pareto optimality · Multiobjective optimization ·
Projected subgradient method · Weighting method
The multiobjective optimization, also known as multicriteria optimization, refers to
the process of simultaneously optimizing two or more real-valued objective functions.
The multiobjective optimization problem has applications in the economy, industry,
agriculture, and others ﬁelds. For more details see, for example, Luc , Miettinen
, and Pappalardo . Among the strategies that can be used to ﬁnd one solution
of a smooth multiobjective optimization problem, we mention the weighting methods,
steepest descent methods and Newton methods. On weighting methods, which are
very simple and easy to implement, we refer the reader to Graña Drummond et al.
P. S. M. Santos
IME/UFG, Goiania, Brazil
DM/UFPI, Teresina, Brazil
CMRV/UFPI, Parnaíba, Brazil