67 research outputs found
Dual characterizations of set containments with strict convex inequalities
Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a finite union of convex sets (i.e., the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classification problems where they are used to describe the containments as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments.MCYT of Spain and FEDER of UE, Grant BMF2002-04114-CO201
Set containment characterization and mathematical programming
Recently, many researchers studied set containment characterizations. In this paper, we introduce some set containment characterizations for quasiconvex programming. Furthermore, we show a duality theorem for quasiconvex programming by using set containment characterizations. Notions of quasiconjugate for quasiconvex functions, especially 1, -1-quasiconjugate, 1-semiconjugate, H-quasiconjugate and R-quasiconjugate, play important roles to derive characterizations of the set containments
Characterizing the Strong Maximum Principle
In this paper we characterize the degenerate elliptic equations F(D^2u)=0
whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum
principle. We introduce an easily computed function f(t) for t > 0, determined
by F, and we show that the strong maximum principle holds depending on whether
the integral \int dy / f(y) near 0 is infinite or finite. This complements our
previous work characterizing when the (ordinary) maximum principle holds. Along
the way we characterize radial subsolutions.Comment: Minor expository revision
Even convexity, subdifferentiability, and Γ-regularization in general topological vector spaces
In this paper we provide new results on even convexity and extend some others to the framework of general topological vector spaces. We first present a characterization of the even convexity of an extended real-valued function at a point. We then establish the links between even convexity and subdifferentiability and the Γ-regularization of a given function. Consequently, we derive a sufficient condition for strong duality fulfillment in convex optimization problems.MICINN of Spain, Grant MTM2011-29064-C03-02
On Evenly Convex Sets and Functions
A subset of R^n is said to be evenly convex (e-convex, in breaf) if it is the intersection of some family (possibly empty) of open halfspaces. In this paper, we collect some published results which show that this large class of convex sets enjoys a lot of the well-known properties of the subclass of closed convex sets. We also consider functions whose epigraphs are e-convex sets, the so-called e-convex functions, and we show the main properties of this class of convex functions that contains the important class of lower semicontinuous convex functions.Esta publicación se ha realizado con el apoyo financiero del Ministerio de Economía y Competitividad, a través del proyecto MTM2011-29064-C03-02
Schubert calculus and shifting of interval positroid varieties
Consider k x n matrices with rank conditions placed on intervals of columns.
The ranks that are actually achievable correspond naturally to upper triangular
partial permutation matrices, and we call the corresponding subvarieties of
Gr(k,n) the _interval positroid varieties_, as this class lies within the class
of positroid varieties studied in [Knutson-Lam-Speyer]. It includes Schubert
and opposite Schubert varieties, and their intersections, and is Grassmann dual
to the projection varieties of [Billey-Coskun].
Vakil's "geometric Littlewood-Richardson rule" [Vakil] uses certain
degenerations to positively compute the H^*-classes of Richardson varieties,
each summand recorded as a (2+1)-dimensional "checker game". We use his same
degenerations to positively compute the K_T-classes of interval positroid
varieties, each summand recorded more succinctly as a 2-dimensional "K-IP pipe
dream". In Vakil's restricted situation these IP pipe dreams biject very simply
to the puzzles of [Knutson-Tao].
We relate Vakil's degenerations to Erd\H os-Ko-Rado shifting, and include
results about computing "geometric shifts" of general T-invariant subvarieties
of Grassmannians.Comment: 35 pp; this subsumes and obviates the unpublished
http://arxiv.org/abs/1008.430
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