An additive subfamily of enlargements of a maximally monotone operator
- Authors: Burachik, Regina , Martinez-Legaz, Juan , Rezaie, Mahboubeh , Thera, Michel
- Date: 2015
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 643-665
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- Description: We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by Svaiter. When the operator under consideration is the subdifferential of a convex lower semicontinuous proper function, we prove that some members of the subfamily are smaller than the classical epsilon-subdifferential enlargement widely used in convex analysis. We also recover the epsilon-subdifferential within the subfamily. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the epsilon-subdifferential enlargement.
Special Issue on recent advances in continuous optimization on the occasion of the 25th European conference on Operational Research (EURO XXV 2012)
- Authors: Weber, Gerhard-Wilhelm , Kruger, Alexander , Martinez-Legaz, Juan , Mordukhovich, Boris , Sakalauskas, Leonidas
- Date: 2014
- Type: Text , Journal article
- Relation: Optimization Vol. 63, no. 1 (2014), p. 1-5
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Monotonic analysis over cones : III
- Authors: Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 15, no. 3 (2008), p. 561-579
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- Description: This paper studies the class of increasing and co-radiant (ICR) functions over a cone equipped with an order relation which agrees with the conic structure. In particular, a representation of ICR functions as abstract convex functions is provided. This representation suggests the introduction of some polarity notions between sets. The relationship between ICR functions and increasing positively homogeneous functions is also shown.
- Description: C1
Increasing quasiconcave co-radiant functions with applications in mathematical economics
- Authors: Martinez-Legaz, Juan , Rubinov, Alex , Schaible, Siegfried
- Date: 2005
- Type: Text , Journal article
- Relation: Mathematical Methods of Operations Research Vol. 61, no. 2 (2005), p. 261-280
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- Description: We study increasing quasiconcave functions which are co-radiant. Such functions have frequently been employed in microeconomic analysis. The study is carried out in the contemporary framework of abstract convexity and abstract concavity. Various properties of these functions are derived. In particular we identify a small "natural" infimal generator of the set of all coradiant quasiconcave increasing functions. We use this generator to examine two duality schemes for these functions: classical duality often used in microeconomic analysis and a more recent duality concept. Some possible applications to the theory of production functions and utility functions are discussed. © Springer-Verlag 2005.
- Description: C1
- Description: 2003001423
Monotonic analysis over cones : I
- Authors: Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex
- Date: 2004
- Type: Text , Journal article
- Relation: Optimization Vol. 53, no. 2 (2004), p. 129-146
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- Description: In this article, we study increasing and positively homogeneous functions defined on convex cones of locally convex spaces. This work is the first part in a series of studies to have a general view of the emerging area of Monotonic Analysis. We develop a general notion of so-called elementary functions, so that the generalized increasing and positively homogeneous functions can be represented as upper-envelopes of families of such functions. We also study many other associated properties like the description of support sets and normal and co-normal sets in a very general setting.
- Description: C1
- Description: 2003000930
Monotonic analysis over cones : II
- Authors: Dutta, J. , Martinez-Legaz, Juan , Rubinov, Alex
- Date: 2004
- Type: Text , Journal article
- Relation: Optimization Vol. 53, no. 5-6 (2004), p. 529-547
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- Description: In this article, we study the class of increasing and convex along rays (ICAR) functions over a cone. Apart from studying its basic properties, we study them from the point of view of Abstract Convexity. Further, we study the relation between the ICAR and Lipschitz functions and the properties under which an ICAR function has a Lipschitz behaviour. We also study the class of decreasing and convex along rays functions (DCAR).
- Description: C1
- Description: 2003000931
Downward sets and their separation and approximation properties
- Authors: Martinez-Legaz, Juan , Rubinov, Alex , Singer, Ivan
- Date: 2002
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 23, no. 2 (Jun 2002), p. 111-137
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- Description: We develop a theory of downward subsets of the space R-I, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x is an element of R-I, f(t)(x) less than or equal to 0 (t is an element of T), where T is an arbitrary index set and each f(t) (t is an element of T) is an increasing function defined on R-I. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.
- Description: 2003000119