EXPLORATORY RESEARCH PROJECT ID_379 13/ 28.09.2007


Director: Prof. dr. Constantin ZALINESCU

  

  Proposal's title

  Research Team

  Objectives

  Publications


1. PROPOSAL'S TITLE

Proposal's Title : Scalarization functions and Lagrange multipliers in optimization problems
Acronym: ID_379

1.1 Thematic fields :
  • 11 Main Sciences: Mathematics
1.2 Abstract

  It is well known the role of the Minkowski functional associated to a convex set in Functional Analysis. It is less known another functional associated to a subset of a linear space, called in the sequel scalarization function, due to its important role in vectorial programming problems. In this context this function has been introduced by C. Tammer and E. Iwanow in 1985, and some of its properties were studied and used by C. Zalinescu (1987) and D.T. Luc (1989). In the framework of mathematical economics such a function was introduced by D. Luenberger (1992) under the name of shortage function and by P. Artzner et. al. (1999) under the name of coherent measure of risk. In the framework of this project we propose ourselves to continue the study of the scalarization function: 1) to establish local properties of continuity and lipschitzianity and to obtain optimality conditions in vector programming, 2) to construct some dual problems as well as to find necessary or/and sufficient conditions for no duality gap, 3) to construct and study some similar functions with non-scalar values. We propose ourselves also to study some problems concerning Lagrange multipliers for nonsmooth vectorial problems on general Banach spaces partially ordered by closed convex cones with possibly empty interior. The envisaged problems in this area are: 4) the existence of Lagrange multipliers, 5) to obtain some Mangasarian-Fromovitz type conditions to guarantee the boundedness of the set of Lagrange multipliers, 6) to develop some corresponding algorithms for general nonsmooth optimization problems.

2. RESEARCH TEAM

Crt. Nr. Family and given names Birth year Scientific title PhD
1 Durea Marius 1975 Lector Yes
2 Strugariu Claudiu Radu 1978 Doctorand No
3 Blanariu Florentiu 1985 Researcher No


3. OBJECTIVES

The aims of the project are:
  • (1a) To put in evidence some local continuity and lipschitzianity properties of the scalarization function. Mainly, the Lipschitz properties are useful for the calculus of Clarke and Mordukhovich types. Even results published by economists use such properties.
  • (1b) The construction and study of some dual problems using scalarization functions.It is well known the fact that the dual problems oftenly have a simpler structure; even when they are not simpler the dual problems are useful for estimating bounds for the values of the primal problems. The problems under consideration appear in economics.
  • (1c) The construction and study of some vector functions with similar properties as those of scalarization functions. The study of such problems is natural for the development of this field (passing from scalar to vector problems is natural in mathematical programming); moreover, vector "coherent measures of risk" were already considered.
  • 2a)  To obtain existence results for Lagrange multipliers in the case of general vector optimization problems on infinite dimensional spaces when the partial order on the output space is given by a cone with empty interior. For the case when the interior of the cone is nonempty there exist many results for weak Pareto solutions but for the case we envisage there are no results in the literature. We consider that this kind of results could be of real interest in the domain of multicriteria optimization.
  • 2b) To obtain some Mangasarian-Fromovitz conditions for vector optimization problems in order to ensure the boundedness of the set of Lagrange multipliers. This problem was been considered in the literature only in finite dimension, up to now. Taking into account the technical difficulties we expect to face in infinite dimension, we consider  that such results could be important for further studies concerning the numerical aspects of the vector optimization problems.
  • 2c) To obtain numerical results for the problems we study. In the scalar case, the boundedness of the set of Lagrange multipliers is an important property in the numerical study of the optimization problems. We consider that a similar situation can occur in the case (not studied up to now) of nonsmooth vectorial problems. 
  • 3a) The study of the mathematical literature in the field of the thesis "Duality and triality in optimization theory".
  • 3b) The elaboration of the results related to the PhD thesis mentioned above.
  • 3c) The valorisation of the results envisaged at 3b).
    The theme of drd. C. Strugariu's PhD thesis is strongly related to the field of this project. The triality theory introduced by D. Y. Gao represents a domain which is not deeply studied, and so a critical study of it in a PhD thesis could be useful for its author as well as for the mathematical community.

4. PUBLICATIONS

  1. C. Tammer, C. Zalinescu: Lipschitz properties of the scalarization function and applications, Optimization 59 (2) (2010), 305-319 .

  2. A. Hantoute, M.A. Lopez, C. Zalinescu: Subdifferential calculus rules in convex analysis: A unifying approach via pointwise supremum functions, SIAM Journal on Optimization 19 (2008), 863-882.

  3. C. Zalinescu: Duality results involving functions associated to nonempty subsets of locally convex spaces, Rev. R. Acad. Cien. Serie A. Mat. VOL. 103 (2), 2009, pp. 219-234.

  4. M. Durea: Optimality conditions for weak and firm efficiency in set-valued optimization, Journal of Mathematical Analysis and Applications, 344 (2008), 1018-1028.

  5. M. Durea, J. Dutta: Lagrange multipliers for Pareto minima in general Banach spaces, Pacific Journal of Optimization, 4 (2008), 447-463.

  6. M. Durea, J. Dutta, Chr. Tammer: Lagrange multipliers for e-Pareto solutions in vector optimization with non solid cones in Banach spaces, Journal of Optimization Theory and Aplications, 145 (2010), 196-211.

  7. M. Durea, J. Dutta, Chr. Tammer: Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension , Journal of Mathematical Analysis and Applications, 348 (2008), 589-606.

  8. M. Durea, Remarks on strict efficiency in scalar and vector optimization, Journal of Global Optimization 47 (2010), 13-27.

  9. M. Durea, R. Strugariu, On some Fermat rules for set-valued optimization problems, Optimization, DOI: 10.1080/02331930903531527 .

  10. C. Zalinescu: On the duality between the profit and the indirect distance functions in production theory, European Journal of Operational Research, 207 (2010), 30-36.

  11. R. Strugariu, M.D. Voisei, C. Zalinescu: Counter-examples in bi-duality, triality and tri-duality, Communications on Pure and Applied Analysis (va apare).

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