Appl Math Optim 50:279–293 (2004)
2004 Springer Science+Business Media, Inc.
The Scalarization Approach to Multiobjective Markov Control
Problems: Why Does It Work?
and Rosario Romera
Departamento de Matem´aticas, CINVESTAV-IPN,
A. Postal 14-740, M´exico D.F. 07000, M´exico
Departamento de Estad´ıstica, Universidad Carlos III de Madrid,
Calle Madrid 126, 28903 Getafe, Madrid, Spain
Abstract. This paper concerns discrete-time multiobjective Markov control pro-
cesses on Borel spaces and unbounded costs. Under mild assumptions, it is shown
that the usual “scalarization approach” to obtain Pareto policies for the multiobjec-
tive control problem is in fact equivalent to solving the dual of a certain multi-
objective inﬁnite-dimensional linear program. The latter program is obtained from a
multiobjective measure problem which is also used to prove the existence of strong
Pareto policies, that is, Pareto policies whose cost vector is the closest to the control
problem’s virtual minimum.
Key Words. Markov control processes, Multiobjective control problems, Pareto
optimality, (Inﬁnite-dimensional) multiobjective linear programming.
AMS Classiﬁcation. 93E20, 90C40, 90C29.
In this paper we study a discrete-time multiobjective Markov control process (MCP) on
Borel spaces and unbounded costs. The main problem is to “minimize” in the sense of
This research was partially supported by CONACYT (M´exico) Grant 37355-E for OHL, and MCyT
(Spain) Grant BEC2000-0167 for RR.