Saddlepoint optimality criteria in vector optimization

Giancarlo Bigi


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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