Saddlepoint optimality criteria in vector optimization
Giancarlo Bigi
Summary
In mathematical programming most optimality criteria require that the
involved functions satisfy some assumptions, generally smoothness or at
least locally Lipschitz conditions so that nonsmooth analysis tools can be
used to replace differentiability. In the case of single-objective
optimization the main criteria, which do not require any such assumption
and do not involve any kind of derivative, are well-known since the Fifties
and the analysis is based upon the saddlepoints of the classical Lagrangian
function of a nonlinear program.
Different generalizations for vector optimization problems have been
proposed and developed in the last decades. At least two different
approaches can be considered: the first one is based on vector-valued
Lagrangians and suitable concepts of
saddlepoint for vector-valued functions, the other one consists in
considering a real-valued Lagrangian also in the vector framework
and relies on the ordinary concept of saddlepoint.
Starting from pioneering papers, the most
studied vector-valued Lagrangians do not have multipliers corresponding to
the objective function; Lagrangians with such multipliers have been considered
only in a few recent papers. Actually,
in the single-objective case this difference is not really meaningful at
least when sufficient optimality criteria are developed but in the
multiobjective case it is not so.
Theferore, we deepen the analysis of the approach based upon a vector-valued
Lagrangian with matrices of multipliers corresponding to both the objective
and constraining functions. Another advantage of this approach is that the
saddlepoints of the real-valued Lagrangian can be recovered as a particular
class of saddlepoints of the vector-valued one; this allows to develop
optimality criteria based on the saddlepoints of both Lagrangians in a unified
framework. However, some meaningful differences exist all the same,
since the real-valued Lagrangian is deeply related to scalarization
techniques.
On the contrary, the simpler structure of the saddlepoints of the
real-valued Lagrangian allows to deepen the analysis of regularity
conditions, which guarantee that the assumptions on saddlepoints in
the sufficient optimality criteria are satisfied (see also this
paper for the Pareto case).
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