235,185 research outputs found
On Minimal Valid Inequalities for Mixed Integer Conic Programs
We study disjunctive conic sets involving a general regular (closed, convex,
full dimensional, and pointed) cone K such as the nonnegative orthant, the
Lorentz cone or the positive semidefinite cone. In a unified framework, we
introduce K-minimal inequalities and show that under mild assumptions, these
inequalities together with the trivial cone-implied inequalities are sufficient
to describe the convex hull. We study the properties of K-minimal inequalities
by establishing algebraic necessary conditions for an inequality to be
K-minimal. This characterization leads to a broader algebraically defined class
of K- sublinear inequalities. We establish a close connection between
K-sublinear inequalities and the support functions of sets with a particular
structure. This connection results in practical ways of showing that a given
inequality is K-sublinear and K-minimal.
Our framework generalizes some of the results from the mixed integer linear
case. It is well known that the minimal inequalities for mixed integer linear
programs are generated by sublinear (positively homogeneous, subadditive and
convex) functions that are also piecewise linear. This result is easily
recovered by our analysis. Whenever possible we highlight the connections to
the existing literature. However, our study unveils that such a cut generating
function view treating the data associated with each individual variable
independently is not possible in the case of general cones other than
nonnegative orthant, even when the cone involved is the Lorentz cone
A Minimal Set of Shannon-type Inequalities for Functional Dependence Structures
The minimal set of Shannon-type inequalities (referred to as elemental
inequalities), plays a central role in determining whether a given inequality
is Shannon-type. Often, there arises a situation where one needs to check
whether a given inequality is a constrained Shannon-type inequality. Another
important application of elemental inequalities is to formulate and compute the
Shannon outer bound for multi-source multi-sink network coding capacity. Under
this formulation, it is the region of feasible source rates subject to the
elemental inequalities and network coding constraints that is of interest.
Hence it is of fundamental interest to identify the redundancies induced
amongst elemental inequalities when given a set of functional dependence
constraints. In this paper, we characterize a minimal set of Shannon-type
inequalities when functional dependence constraints are present.Comment: 5 pagers, accepted ISIT201
Tropical polar cones, hypergraph transversals, and mean payoff games
We discuss the tropical analogues of several basic questions of convex
duality. In particular, the polar of a tropical polyhedral cone represents the
set of linear inequalities that its elements satisfy. We characterize the
extreme rays of the polar in terms of certain minimal set covers which may be
thought of as weighted generalizations of minimal transversals in hypergraphs.
We also give a tropical analogue of Farkas lemma, which allows one to check
whether a linear inequality is implied by a finite family of linear
inequalities. Here, the certificate is a strategy of a mean payoff game. We
discuss examples, showing that the number of extreme rays of the polar of the
tropical cyclic polyhedral cone is polynomially bounded, and that there is no
unique minimal system of inequalities defining a given tropical polyhedral
cone.Comment: 27 pages, 6 figures, revised versio
On minimal immersions in Finsler spaces
We explore a connection between the Finslerian area functional and
well-investigated Cartan functionals to prove new Bernstein theorems,
uniqueness and removability results for Finsler-minimal graphs, as well as
enclosure theorems and isoperimetric inequalities for minimal immersions in
Finsler spaces. In addition, we establish the existence of smooth
Finsler-minimal immersions spanning given extreme or graphlike boundary
contours.Comment: 26 pages, changed numbering of equation
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