---

Exploratory Research Project PN2-IDEI-PCE-2008-2, Code CNCSIS 2261, Contract No. 543

Project duration: 3 years (2009, 2010, 2011)

 

ADVANCED RESEARCHES ON VECTOR AND SET-VALUED OPTIMIZATION PROBLEMS, AND VARIATIONAL INEQUALITIES, UNDER GENERALIZED CONVEXITY ASSUMPTIONS

 

Project manager: Assoc. prof. Nicolae Popovici, Ph.D.

---

 

The research team

Project summary

Publications

 

 

The research team

 

Assoc. prof. Nicolae POPOVICI, Ph.D. - project manager

Prof. Wolfgang Werner BRECKNER, Ph.D.

Prof. Stefan COBZAS, Ph.D.

Prof. Gabor KASSAY, Ph.D.

Asist. prof. Anca GRAD, Ph.D.

 

 

Project summary

 

VECTOR AND SET-VALUED OPTIMIZATION, VARIATIONAL INEQUALITIES HAVE ENCOUNTERED A DYNAMIC DEVELOPMENT OFFERING A FRUCTUOUS FRAMEWORK FOR THE APPLICABILITY OF ADVANCED THEORETICAL ACQUISITIONS FROM CONVEX, NONLINEAR AND SET-VALUED ANALYSIS.

 

FROM SOME RECENT THEORETICAL OPEN PROBLEMS, CONSIDERING THEIR PRACTICAL APPLICABILITY, TWELVE OBJECTIVES ARE PROPOSED:

 

-          STUDY OF THE PARETO REDUCIBILITY FOR MULTICRITERIA OPTIMIZATION PROBLEMS WHOSE OBJECTIVE FUNCTIONS ARE DEFINED ON ARCWISE CONNECTED DOMAINS;

-          DEVELOPMENT OF SOME SEQUENTIAL OPTIMALITY CONDITIONS;

-          EXTENSION OF SOME RECENT RESULTS WITH RESPECT TO THE WELL-POSED VECTOR OPTIMIZATION PROBLEMS UNDER GENERALIZED CONVEXITY;

-          INTRODUCTION OF AN APPROPRIATE REDUCIBILITY RESULT FOR VECTOR VARIATIONAL INEQUALITIES;

-          STUDY OF SET-VALUED DUALITY CONCEPTS;

-          SYSTEMATIZATION OF GENERALIZED CONVEXITY CONCEPTS FOR FUNCTIONS WITH VALUES IN VECTOR TOPOLOGICAL ORDERED SPACES;

-          DEVELOPMENT OF A SOLUTION METHOD FOR VARIATIONAL INEQUALITIES GOVERNED BY GENERALIZED MONOTONE OPERATORS;

-          EXTENSION OF THE WELL-POSEDNESS CONCEPT TO SET-VALUED OPTIMIZATION;

-          INTRODUCTION OF GENERALIZED PARETO REDUCIBILITY FOR VECTOR OPTIMIZATION PROBLEMS THROUGH EXTREMAL SUBSETS OF THE ORDER CONE;

-          STUDY OF THE TOPOLOGICAL STRUCTURE OF THE EFFICIENT SOLUTIONS SET IN VECTOR OPTIMIZATION PROBLEMS;

-          EXTENSION OF SOME REMARKABLE RESULTS WITH RESPECT TO OPTIMIZATION PROBLEMS TO LINEAR SPACES WITH ASYMMETRIC NORM;

-          OBTAINMENT OF LAGRANGE MULTIPLIER RULES IN SET-VALUED OPTIMIZATION USING DERIVED SETS. THE RESEARCH RELIES ON ARTICLES BY PROJECT MEMBERS.

 

Publications

 

1.          Davide LA TORRE, Nicolae POPOVICI: Arcwise cone-quasiconvex multicriteria optimization, Operations Research Letters, Vol. 38 (2), 2010, 143-146, [ISI 2008 Impact factor: 0.830; ISI 2009 Impact factor: 0.681]

2.          Anca GRAD: Improved sequential optimality conditions for vector optimization problems with cone-epi-closed functions, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, Vol. 7, 2009, 55-72.

3.          Monica BIANCHI, Gabor KASSAY, Rita PINI: Well-posedness for vector equilibrium problems, Mathematical Methods of Operations Research, Vol. 70 (1), 2009, 171-182 [ISI 2008 Impact factor: 0.476; ISI 2009 Impact factor: 0.522]

4.          Nicolae POPOVICI, Matteo ROCCA: Pareto reducibility of vector variational inequalities, Quaderni di Ricerca, Universita dell'Insubria, Dipartimento di Economia, 2010/4

5.          Anca GRAD: Quasi-relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints, Journal of Mathematical Analysis and Applications, Vol. 361 (1), 2010, 86-95, [ISI 2008 Impact factor: 1.046; ISI 2009 Impact factor: 1.225]

6.          Davide LA TORRE, Nicolae POPOVICI, Matteo ROCCA: Scalar characterizations of weakly cone-convex and weakly cone-quasiconvex functions, Nonlinear Analysis: Theory, Methods & Applications, Vol. 72 (3-4), 2010, 1909-1915 [ISI 2008 Impact factor: 1.295; ISI 2009 Impact factor: 1.487]

7.          Anca GRAD: Converse duality for a new Fenchel dual problem in finite dimensional vector optimization, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, Vol. 8, 2010, 27-37.

8.          Adela CAPATA, Gabor KASSAY, Boglarka MOSONI: On weak multifunctions equilibrium problems, In: Burachik, Regina S.; Yao, Jen-Chih (Eds.), Variational Analysis and Generalized Differentiation in Optimization and Control. In Honor of Boris S. Mordukhovich, Springer Series in Optimization and its Applications, Vol. 47, 2010, 133-148.

9.          Giancarlo BIGI, Adela CAPATA, Gabor KASSAY: Existence results for strong vector equilibrium problems and their applications, Optimization, DOI: 10.1080/02331934.2010.528761, [ISI 2009 Impact factor: 0.616]

10.      Davide LA TORRE, Nicolae POPOVICI, Matteo ROCCA: A note on explicitly quasiconvex set-valued maps, Journal of Nonlinear and Convex Analysis, Vol.12 (1) (2011), 113-118. [ISI 2009 Impact factor: 0.585]

11.      Stefan COBZAS: Completeness in quasi-metric spaces and Ekeland variational principle, Topology and its Applications, Vol.158 (8), 2011, 1073-1084. [ISI 2009 Impact factor: 0.441]

12.      Nicolae POPOVICI, Matteo ROCCA: Decomposition of vector variational inequalities, Nonlinear Analysis: Theory, Methods and Applications, Vol.75 (3), 2012, 1516-1523 [ISI 2009 Impact factor: 1.487, ISI 2010 Impact factor: 1.279]

13.      Anca GRAD: Generalized duality and optimality conditions, Editura Mega, Cluj-Napoca, 2010 (210 pp., ISBN 978-606-543-098-3)

14.      Anca GRAD: Quasi interior-type optimality conditions in set-valued duality, Journal of Nonlinear and Convex Analysis, in press [ISI 2010 Impact factor: 0,738]

15.      Monica BIANCHI, Gabor KASSAY, Rita PINI: Conditioning for optimization problems under general perturbations, Nonlinear Analysis: Theory, Methods and Applications, Vol. 75 (1) (2012), 37-45 [ISI 2010 Impact factor: 1.279]

16.      Gabor KASSAY, Simeon REICH, Shoham SABACH: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM Journal on Optimization, Vol. 21 (4) (2011), 1319-1344 [ISI 2010 Impact factor: 2,079]

17.      Nicolae POPOVICI, Matteo ROCCA: Scalarization and decomposition of vector variational inequalities governed by bifunctions, Optimization, DOI:10.1080/02331934.2012.672984, [ISI 2010 Impact factor: 0,509]

18.      Stefan COBZAS: Ekeland variational principle in asymmetric locally convex spaces, Topology and its Applications, DOI: 10.1016/j.topol.2012.04.015, [ISI 2010 Impact factor: 0.447]

19.      Stefan COBZAS: Functional analysis in asymmetric normed spaces, submitted to Quaderni di Matematica, Caserta, Seconda Universita di Napoli

20.      Anca GRAD: Generalized interior characterizations for saddle points, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, Vol. 9 (2011), 47-57

21.      Stefan COBZAS: Functional Analysis in Asymmetric Normed Spaces, Birkhauser Mathematics Series: Frontiers in Mathematics, Springer Basel (ISBN 978-3-0348-0477-6)

22.      Wolfgang W. BRECKNER, Rational s-Convexity. A Generalized Jensen-Convexity, Presa Universitara Clujeana, Cluj-Napoca, 2011 (165 pp., ISBN: 978-973-595-325-6)

23.      Gabor KASSAY: The equilibrium problem and its applications to optimization, minimax problems, and Nash equilibria, Chapter 8 in: Q. H. Ansari (Ed.), Topics in Nonlinear Analysis and Optimization, World Education, Delhi, 2012, pp. 203-226, ISBN 978-81-909873-3-2