# Pareto Efficiency and Nonconvex Vector Optimization

Zapraszamy na Seminarium Centrum Technik Informatycznych WIT i Zakładu Wspomagania Decyzji w Warunkach Ryzyka IBS PAN.

W dniu 9 lutego (wtorek) o godzinie 10:00, w ramach Seminarium Centrum Technik Informatycznych WIT, Zakładu Wspomagania Decyzji w Warunkach Ryzyka IBS PAN, Masoud Karimi przedstawi referat „Pareto Efficiency and Nonconvex Vector Optimization”.

Seminarium odbedzie się za pomocą aplikacji Skype [Link do spotkania]

### Abstract

The concept of Pareto efficiency plays a major role in vector optimization. Some types of Pareto efficiency, based on the notion of trade-off, have been introduced and characterized in terms of scalarization and stability.

The concepts of proper efficiency and substantial efficiency, derived from the concept of Pareto efficiency, have been introduced in order to exclude certain Pareto efficient solutions that display an undesirable anomaly and to provide a more satisfactory characterization.

Likewise, the concepts of quasi-proper efficiency and quasi-substantial efficiency have been introduced to distinguish Pareto efficient solutions which admit in a certain growth rate in their trade-offs, whenever trade-offs are bounded or unbounded.

In the presentation, we concentrate on nonconvex optimization. We give characterizations of the above mentioned types of Pareto efficient solutions and show how to derive them by scalarization.

In the nonconvex vector optimization, separation results based on the Hahn-Banach theorem do not apply. Instead, one has to make use of conic separation theorems which provide means to construct nonlinear scalarization functions. Such functions, parametrized by augmented dual cone elements, guarantees that almost all efficient elements can be obtained by a series of conic scalarizations. In the literature, conic separation theorems are applied to study optimality conditions and duality, with ordering closed convex cones replaced by the Bishop-Phelps cones.

We argue that the use of Bishop-Phelps cones in nonconvex vector optimization is too restrictive. We propose to represent Bishop-Phelps cones as collections of closed convex cones. This, with a suitable norm, provides for the required properties of relevant scalarizations.

### References

[1] R. Kasimbeyli, M. Karimi. Hahn-Banach type separation theorems for nonconvex sets and application in optimization. Operations Research Letters, 47(6), 569-573 (2019).

[2] L. Pourkarimi, M. Karimi. Characterization of substantially and quasi substantially efficient solutions in multi-objective optimization problems. Turkish Journal of Mathematics, 41, 293-304 (2017).

### About presenter

Masoud Karimi, PhD, is Assistant Professor at Kharazmi University, Tehran, Iran. He also lectures at Azad University-Shahr-Qods Branch in Tehran, and Kermanshah University of Technology in Kermanshah.
His current research interests are Stochastistic Data Envelopment Analysis and Nonconvex Optimization.

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