Optimality Conditions for Metrically Consistent Approximate Solutions in Vector Optimization

In this paper, approximate solutions of vector optimization problems are analyzed via a metrically consistent ε-efficient concept. Several properties of the ε-efficient set are studied. By scalarization, necessary and sufficient conditions for approximate solutions of convex and nonconvex vector optimization problems are provided; a characterization is obtained via generalized Chebyshev norms, attaining the same precision in the vector problem as in the scalarization. This research was partially supported by the Ministerio de Educación y Ciencia (Spain), Project MTM2006-02629 and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees for helpful comments and suggestions.     

End-to-end availability-dependent pricing of network services

We discuss how a new pricing scheme can be integrated within a communication network. The pricing scheme is based on the availability of end-to-end communications, and is an alternative to congestion pricing, which is not applicable when communication capacity is higher than demand (as happens in most communication backbone networks). We also investigate how, based on this scheme, an optimization algorithm for updating the network topology can be applied. The network update problem is modeled as a combinatorial optimization problem, which is approximately solved using a Genetic Algorithm. The good results obtained in a case study show that the method is robust and can be applied even when end-to-end availability measures can only be computed approximately (in this case, using a Monte Carlo method). This research is part of the PAIR associated research project, supported by the INRIA, France, and has also received the support of ECOS-Sud, under Action U03E02. The participation of Pablo Rodríguez was supported by the French Embassy in Uruguay as part of the French Ministère des Affaires étrangères scientific cooperation program; and by the “Programa de Jóvenes Investigadores” of CSIC, UDELAR, Uruguay.     

Obtaining an outer approximation of the efficient set of nonlinear biobjective problems

A new method for obtaining an outer approximation of the efficient set of nonlinear biobjective optimization problems is presented. It is based on the well known ‘constraint method’, and obtains a superset of the efficient set by computing the regions of δ-optimality of a finite number of single objective constraint problems. An actual implementation, which makes use of interval tools, shows the applicability of the method and the computational studies on a set of competitive location problems demonstrate its efficiency. An extended version of this paper, with more comments, details, examples, and references, can be found in Fernández and Tóth [5]. This paper has been supported by the Ministry of Education and Science of Spain under the research project SEJ2005-06273/ECON, in part financed by the European Regional Development Fund (ERDF). Boglárka Tóth—On leave from the Research Group on Artificial Intelligence of the Hungarian Academy of Sciences and the University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1., Hungary.     

Jacobi transplantation revisited

A transplantation theorem for Jacobi series proved by Muckenhoupt is reinvestigated by means of a suitable variant of Calderón–Zygmund operator theory. An essential novelty of our paper is weak type (1,1) estimate for the Jacobi transplantation operator, located in a fairly general weighted setting. Moreover, L ^p estimates are proved for a class of weights that contains the class admitted in Muckenhoupt’s theorem. Research of ó. Ciaurri and K. Stempak was supported by the grant MTM2006-13000-C03-03 of the DGI. Research of A. Nowak and K. Stempak was supported by MNiSW Grant N201 054 3214285.     

Characterizing efficiency without linear structure: a unified approach

In this paper, we study a general optimization problem without linear structure under a reflexive and transitive relation on a nonempty set E, and characterize the existence of efficient points and the domination property for a subset of E through a generalization of the order-completeness condition introduced earlier. Afterwards, we study the abstract optimization problem by using generalized continuity concepts and establish various existence results. As an application, we extend and improve several existence results given in the literature for an optimization problem involving set-valued maps under vector and set criteria. The work of the first author was partially supported by CONICYT-Chile through FONDECYT 107-0689 and FONDAP-Matemáticas Aplicadas II; whereas that of the second and third author were supported in part by Ministerio de Educación y Ciencia (Spain), project MTM2006-02629 and by Junta de Castilla y León (Spain) Project VA027B06.     

A Sharp Form of the Cramér–Wold Theorem

The Cramér–Wold theorem states that a Borel probability measure P on ℝ ^d is uniquely determined by its one-dimensional projections. We prove a sharp form of this result, addressing the problem of how large a subset of these projections is really needed to determine P. We also consider extensions of our results to measures on a separable Hilbert space. First author partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant BFM2002-04430-C02-02. Second author partially supported by Instituto de Cooperación Iberoamericana, Programa de Cooperación Interuniversitaria AL-E 2003. Third author partially supported by grants from NSERC and the Canada research chairs program.     

Lagrangian Duality in Set-Valued Optimization

In this paper, we study optimization problems where the objective function and the binding constraints are set-valued maps and the solutions are defined by means of set-relations among all the images sets (Kuroiwa, D. in Takahashi, W., Tanaka, T. (eds.) Nonlinear analysis and convex analysis, pp. 221–228, 1999). We introduce a new dual problem, establish some duality theorems and obtain a Lagrangian multiplier rule of nonlinear type under convexity assumptions. A necessary condition and a sufficient condition for the existence of saddle points are given. The authors thank the two referees for valuable comments and suggestions on early versions of the paper. The research of the first author was partially supported by Ministerio de Educación y Ciencia (Spain) Project MTM2006-02629 and by Junta de Castilla y León (Spain) Project VA027B06.     

Strict approximation on closed convex sets

The concept of strict approximation over subspaces of an euclidean space, introduced by John R. Rice, is extended to closed convex sets. It is proved that the best p-approximants converge as p→∞ to the strict approximant not generally but when the closed convex set satisfies certain approximative property. Finally, a similar problem is considered in the space c₀ of real sequences tending to 0. This paper was partially supported by the Consojo de Investigacions Cient/ficas y Tecnológicas de la Provincia de Córdoba.     

Compromise solutions for bankruptcy situations with references

This paper deals with bankruptcy situations in which in addition to the claims, an exogenously given reference point for the allocation of the estate is present. We introduce and analyse two types of compromise solutions and show that they coincide with the τ value of two corresponding TU games. We apply our solutions to a real-life case of allocating university money to degree courses. Research supported by Generalitat Valenciana, through ACOMP06/040, ACOMP07/136 and GV05/189, and by the Government of Spain and FEDER fund under project MTM2005-09184-C02-02. M. Pulido’s research is partially supported by Fundación Séneca of Comunidad Autónoma de la Región de Murcia, through 02911/PI/05. Ruud Hendrickx author acknowledges financial support from the Netherlands Organisation for Scientific Research (NWO).     

Polar Conic Set-Valued Map in Vector Optimization. Continuity and Derivability

In the context of vector optimization, several results are stated mainly about the continuity and the derivability of a conic set-valued map (the polar conic function) having a close relation with the positive efficient points, the ideal points and other distinguished elements of the efficient line. The contingent cone to the set of the general positive quasiefficient points at a point x ₀ is also related with the frontier of the dual cone of the image at x ₀ of the polar conic function. This work was partially supported by Grant SEJ2006–15401–C04–02 of Spanish Ministerio de Ciencia y Tecnología and Grant S-0505/tic/0230-D3 of Comunidad Autónoma de Madrid. The authors are grateful to the referees for suggestions which led to improving the paper.     

New Second-Order Directional Derivative and Optimality Conditions in Scalar and Vector Optimization

We present a new second-order directional derivative and study its properties. Using this derivative and the parabolic second-order derivative, we establish second-order necessary and sufficient optimality conditions for a general scalar optimization problem by means of the asymptotic and parabolic second-order tangent sets to the feasible set. For the sufficient conditions, the initial space must be finite dimensional. Then, these conditions are applied to a general vector optimization problem obtaining second-order optimality conditions that generalize the differentiable case. For this aim, we introduce a scalarization, and the relationships between the different types of solutions to the vector optimization problem and the scalarized problem are studied. This research was partially supported by the Ministerio de Educación y Ciencia (Spain), under projects MTM2006-02629 and Ingenio Mathematica (i-MATH) CSD2006-00032 (Consolider-Ingenio 2010), and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees for valuable comments and suggestions.     

A vector valued transference of Fourier-Bessel multipliers onto Hankel multipliers with applications

In this note we prove a vector valued transference theorem relating Fourier-Bessel multipliers and Hankel multipliers. An application of such a transference theorem allows to show that results of Córdoba [3] and Romera [4] can be deduced from a recent result of Balodis and Córdoba [1, Theorem 3]. Partially supported by DGICYT Grant PB 97-1489 (Spain). Partially supported by KBN grant # 2 PO3A 034 20.     

On the convergence of the Weiszfeld algorithm

In this work we analyze the paper “Brimberg, J. (1995): The Fermat-Weber location problem revisited. Mathematical Programming 71, 71–76” which claims to close the question on the conjecture posed by Chandrasekaran and Tamir in 1989 on the convergence of the Weiszfeld algorithm. Some counterexamples are shown to the proofs showed in Brimberg’s paper. Received: January 1999 / Accepted: December 2001?Published online April 12, 2002 RID="*" ID="*"Partially supported by PB/11/FS/97 of Fundación Séneca of the Comunidad Autónoma de la Región de Murcia RID="**" ID="**"Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (I+I+D), project TIC2000-1750-C06-06 RID="*" RID="**"     

Gorenstein quivers

In this paper we characterize when the path ring associated to a quiver is Gorenstein (in the sense of Iwanaga [9]). Then, by using the notion of a Gorenstein category (cf. [2]), we extend the classes of quivers whose corresponding category of representations has finite Gorenstein global dimension. This extension includes non-noetherian quivers. E. E., S.E., and J.R.G.R., partially supported by the DGI MTM2005-03227. Estrada’s work was supported by a MEC/Fulbright grant from the Spanish Secretaría de Estado de Universidades e Investigación del Ministerio de Educación y Ciencia. Received: 28 February 2006     

Index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints

 Multidimensional optimization problems where the objective function and the constraints are multiextremal non-differentiable Lipschitz functions (with unknown Lipschitz constants) and the feasible region is a finite collection of robust nonconvex subregions are considered. Both the objective function and the constraints may be partially defined. To solve such problems an algorithm is proposed, that uses Peano space-filling curves and the index scheme to reduce the original problem to a H?lder one-dimensional one. Local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search. The method neither uses penalty coefficients nor additional variables. Convergence conditions are established. Numerical experiments confirm the good performance of the technique. Received: April 2002 / Accepted: December 2002 Published online: March 21, 2003 RID="⋆" ID="⋆" This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, and RFBR 01–01–00587. Key Words. global optimization – multiextremal constraints – local tuning – index approach     

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant Members of VBAC (Vector Bundles on Algebraic Curves). Second and Third authors partially supported by Ministerio de Educación y Ciencia and Conselho de Reitores das Universidades Portuguesas through Acción Integrada Hispano-Lusa HP2002-0017 (Spain)/E–30/03 (Portugal). First and Second authors partially supported by Ministerio de Educación y Ciencia (Spain) through Project MTM2004-07090-C03-01. Third author partially supported by the Centro de Matemática da Universidade do Porto and the project POCTI/MAT/58549/2004, financed by FCT (Portugal) through the programmes POCTI and POSI of the QCA III (2000–2006) with European Community (FEDER) and national funds. The second author visited the IHES with the partial support of the European Commission through its 6th Framework Programme “Structuring the European Research Area” and the Contract No. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action     

Towards a goldie theory for jordan pairs

A Goldie theory for Jordan pairs is started in this paper. We introduce a notion of order in linear Jordan pairs and study orders in nondegenerate linear Jordan pairs with descending chain condition on principal inner ideals. This work has been supported by DGICYT Grant PB93-0990 and by the “Convenio Marco de Cooperación Hispano-Marroquí”     

The Erdős–Pósa Property for Odd Cycles in Highly Connected Graphs

Dedicated to the memory of Paul Erdős In [9] Thomassen proved that a -connected graph either contains k vertex disjoint odd cycles or an odd cycle cover containing at most 2k-2 vertices, i.e. he showed that the Erdős–Pósa property holds for odd cycles in highly connected graphs. In this paper, we will show that the above statement is still valid for 576k-connected graphs which is essentially best possible. Received November 17, 1999 RID="*" ID="*" This work was supported by a post-doctoral DONET grant. RID="†" ID="†" This work was supported by an NSF-CNRS collaborative research grant. RID="‡" ID="‡" This work was performed while both authors were visiting the LIRMM, Université de Montpellier II, France.     

Wall Rational Functions and Khrushchev’s Formula for Orthogonal Rational Functions

The Nevalinna–Pick algorithm yields a continued fraction expansion of every Schur function, whose approximants are identified. These approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants permit us to obtain a Khrushchev’s formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is presented. This work was partially realized during two stays of the second author at the Norwegian University of Science and Technology (NTNU) financed respectively by Secretaría de Estado de Universidades e Investigación from the Ministry of Education and Science of Spain and by the Department of Mathematical Sciences of NTNU. The work of the second author was also partially supported by the Spanish grants from the Ministry of Education and Science, project code MTM2005-08648-C02-01, and the Ministry of Science and Innovation, project code MTM2008-06689-C02-01, and by Project E-64 of Diputación General de Aragón (Spain).     

The Maximal Graded Left Quotient Algebra of a Graded Algebra1)

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.