# Some Existence Results and Stability in Multi Objective Optimization

• Roberto Lucchetti
Part of the International Centre for Mechanical Sciences book series (CISM, volume 289)

## Abstract

If someone is curious about multiobjective optimization, and enters in a library to look for works in the argument, he can find an enormous amount of things, results, applications, references. For this and other reasons, it is quite obvious that the results given here are a very little part of the known theorems, mainly in the existence. So, no claims to be complete. We shall deal with a topological vector space X and a cone C ⊂ ℝm that is always assumed closed convex, pointed (C ∩ − C = {0}) and with nonempty interior. Moreover it is given a function f: X → ℝm to be maximized with respect to the ordering induced by the cone, on a general constraint set A. We are interested in existence theorems, namely the non-emptiness of the set . At first, observe that, if C1 ⊂ C2, then : this means that, in particular, the non emptiness of sC guarantees existence of solutions for the weak optimization, namely of the points . In the sequel we write s for sC and W for the weak solutions. In the scalar case, i.e. when m = 1 the most celebrated theorem says that, if f is upper semicontinuous (u.s.c.) and there is in A a relatively compact maximizing sequence, then there is a solution for the problem. We want to present here an analogous simple result, so we have to clarify the meaning of “maximizing sequences” and “upper semicontinuity”. We shall make use of the following notations:

## Keywords

Multi Objective Optimization Existence Result Multiobjective Optimization Topological Vector Space Scalar Case
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Cesari, L. Suryanarayana, M.B., Existence Theorems for Pareto Optimization in Banach Spaces, Bull. Amer. Math. Soc. (82), 1976.Google Scholar
2. 2.
Corley, H.V., An Existence Result for Maximizations with Respect to Cones, J. Optim. Th. and Appl. (31) 1980.Google Scholar
3. 3.
Borwein, J., On the Existence of Pareto Efficient Points, Math. Of Oper. Res. (8), 1983.Google Scholar
4. 4.
Caligaris, O. Oliva, P., Necessary and Sufficient Conditions for Pareto Problems, Boll. U.M. I. (17 B), 1980.Google Scholar
5. 5.
Caligaris, O. Oliva, P., Optimality in Pareto Problems, Proceedings in “Generalized Lagrangians in Systems and Economic Theory”,IIASA Luxembourg, 1979.Google Scholar
6. 6.
Caligaris, O. Oliva, P., Constrained Optimization of Infinite Dimensional Vector Functions with Application to Infinite Horizon Integrals, to appear.Google Scholar
7. 7.
Caligaris, O. Oliva, P., Semicontinuità di Funzioni a Valori Vettoriali ed Esistenza del Minimo per Problemi di Pareto, Boll. U.M.I. (II C) 1983.Google Scholar
8. 8.
Trudzik, L.I., Continuity Properties for Vector Valued Convex Functions, J. Austral. Math. Soc. (36) 1984.Google Scholar
9. 9.
Lucchetti, R., On the Continuity of the Value and of the Optimal Set in Minimum Problems, Ist. Mat. Appl. C.N.R. Genova, 1983.Google Scholar
10. 10.
Lucchetti, R., Stability in Pareto Problems, to appear.Google Scholar
11. 11.
Lucchetti, R. Patrone, F., Closure and Upper Semicontinuity Results in Mathematical Programming, Nash and Economic equilibria, to appear.Google Scholar
12. 12.
Tanino, M.J. Sawaragi, Y., Stability of Nondominated Solutions in Multicriteria Decision Making, J. Optim. Th. and Appl. (30), 1980.Google Scholar
13. 13.
Jansen, M.J. Tijs, S., Continuity of the Barganing Solutions, Intern. J. of Game Th. (12), 1983.Google Scholar
14. 14.
Jurkiewicz, E., Stability of Compromise Solution in Multicriteria Decision Making Problems, J. Optim. Th. and Appl. (31), 1980.Google Scholar
15. 15.
Peirone, R., Rlimiti e minimi Pareto, Atti Accad. Naz. Lincei (LXXIV), 1983.Google Scholar