82,360 research outputs found
An interval version of separation by semispaces in max-min convexity
We study separation of a closed box from a max-min convex set by max-min
semispace. This can be regarded as an interval extension of known separation
results. We give a constructive proof of the separation in the case when the
box and the max-min convex set satisfy certain condition, and we show that
separation is never possible if this condition does not hold. We also study
separation of max-min convex sets by boxes and by box and semispace
Convexity and a Stone-type theorem for convex sets in abelian semigroup setting
In this paper, two parallel notions of convexity of sets are introduced in
the abelian semigroup setting. The connection of these notions to algebraic and
to set-theoretic operations is investigated. A formula for the computation of
the convex hull is derived. Finally, a Stone-type separation theorem for
disjoint convex sets is established
Ball and Spindle Convexity with respect to a Convex Body
Let be a convex body. We introduce two notions of
convexity associated to C. A set is -ball convex if it is the
intersection of translates of , or it is either , or . The -ball convex hull of two points is called a -spindle. is
-spindle convex if it contains the -spindle of any pair of its points. We
investigate how some fundamental properties of conventional convex sets can be
adapted to -spindle convex and -ball convex sets. We study separation
properties and Carath\'eodory numbers of these two convexity structures. We
investigate the basic properties of arc-distance, a quantity defined by a
centrally symmetric planar disc , which is the length of an arc of a
translate of , measured in the -norm, that connects two points. Then we
characterize those -dimensional convex bodies for which every -ball
convex set is the -ball convex hull of finitely many points. Finally, we
obtain a stability result concerning covering numbers of some -ball convex
sets, and diametrically maximal sets in -dimensional Minkowski spaces.Comment: 27 pages, 5 figure
The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra
This article investigates matrix convex sets and introduces their tracial
analogs which we call contractively tracial convex sets. In both contexts
completely positive (cp) maps play a central role: unital cp maps in the case
of matrix convex sets and trace preserving cp (CPTP) maps in the case of
contractively tracial convex sets. CPTP maps, also known as quantum channels,
are fundamental objects in quantum information theory.
Free convexity is intimately connected with Linear Matrix Inequalities (LMIs)
L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets
{ X : L(X) is positive semidefinite }, called free spectrahedra. The
Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states
that matrix convex sets are solution sets of LMIs with operator coefficients.
Motivated in part by cp interpolation problems, we develop the foundations of
convex analysis and duality in the tracial setting, including tracial analogs
of the Effros-Winkler Theorem.
The projection of a free spectrahedron in g+h variables to g variables is a
matrix convex set called a free spectrahedrop. As a class, free spectrahedrops
are more general than free spectrahedra, but at the same time more tractable
than general matrix convex sets. Moreover, many matrix convex sets can be
approximated from above by free spectrahedrops. Here a number of fundamental
results for spectrahedrops and their polar duals are established. For example,
the free polar dual of a free spectrahedrop is again a free spectrahedrop. We
also give a Positivstellensatz for free polynomials that are positive on a free
spectrahedrop.Comment: v2: 56 pages, reworked abstract and intro to emphasize the convex
duality aspects; v1: 60 pages; includes an index and table of content
Nonlinear Cone Separation Theorems in Real Topological Linear Spaces
The separation of two sets (or more specific of two cones) plays an important
role in different fields of mathematics such as variational analysis, convex
analysis, convex geometry, optimization. In the paper, we derive some new
results for the separation of two not necessarily convex cones by a (convex)
cone / conical surface in real (topological) linear spaces. We follow basically
the separation approach by Kasimbeyli (2010, SIAM J. Optim. 20) based on
augmented dual cones and normlinear separation functions. Classical separation
theorems for convex sets will be the key tool for proving our main nonlinear
cone separation theorems. Also in the setting of a real reflexive Banach space,
we are able to extend the cone separation result derived by Kasimbeyli
Equivalent Results in Minimax Theory
In this paper we review known minimax results with applications ingame theory and show that these results are easy consequences of thefirst minimax result for a two person zero sum game with finite strategysets published by von Neumann in 1928: Among these results are thewell known minimax theorems of Wald, Ville and Kneser and their generalizationsdue to Kakutani, Ky-Fan, KĂÂśnig, Neumann and Gwinner-Oettli. Actually it is shown that these results form an equivalent chainand this chain includes the strong separation result in finite dimensionalspaces between two disjoint closed convex sets of which one is compact.To show these implications the authors only use simple propertiesof compact sets and the well-known Weierstrass Lebesgue lemma.convex analysis;game theory;finite dimensional separation of convex sets;generalized convexity;minimax theory
Mixing sets linked by bidirected paths
Recently there has been considerable research on simple mixed-integer sets, called mixing sets, and closely related sets arising in uncapacitated and constant capacity lot- sizing. This in turn has led to study of more general sets, called network-dual sets, for which it is possible to derive extended formulations whose projection gives the convex hull of the network-dual set. Unfortunately this formulation cannot be used (in general) to optimize in polynomial time. Furthermore the inequalities definining the convex hull of a network-dual set in the original space of variables are known only for some special cases. Here we study two new cases, in which the continuous variables of the network-dual set are linked by a bi- directed path. In the first case, which is motivated by lot-sizing problems with (lost) sales, we provide a description of the convex hull as the intersection of the convex hulls of 2^n mixing sets, where n is the number of continuous variables of the set. However optimization is polynomial as only n + 1 of the sets are required for any given objective function. In the second case, generalizing single arc flow sets, we describe again the convex hull as an intersection of an exponential number of mixing sets and also give a combinatorial polynomial-time separation algorithm.mixing sets, extended formulations, mixed integer programming, lot-sizing with sales
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