1,171 research outputs found
Tangential Extremal Principles for Finite and Infinite Systems of Sets, I: Basic Theory
In this paper we develop new extremal principles in variational analysis that
deal with finite and infinite systems of convex and nonconvex sets. The results
obtained, unified under the name of tangential extremal principles, combine
primal and dual approaches to the study of variational systems being in fact
first extremal principles applied to infinite systems of sets. The first part
of the paper concerns the basic theory of tangential extremal principles while
the second part presents applications to problems of semi-infinite programming
and multiobjective optimization
The tropical double description method
We develop a tropical analogue of the classical double description method
allowing one to compute an internal representation (in terms of vertices) of a
polyhedron defined externally (by inequalities). The heart of the tropical
algorithm is a characterization of the extreme points of a polyhedron in terms
of a system of constraints which define it. We show that checking the
extremality of a point reduces to checking whether there is only one minimal
strongly connected component in an hypergraph. The latter problem can be solved
in almost linear time, which allows us to eliminate quickly redundant
generators. We report extensive tests (including benchmarks from an application
to static analysis) showing that the method outperforms experimentally the
previous ones by orders of magnitude. The present tools also lead to worst case
bounds which improve the ones provided by previous methods.Comment: 12 pages, prepared for the Proceedings of the Symposium on
Theoretical Aspects of Computer Science, 2010, Nancy, Franc
Rated Extremal Principles for Finite and Infinite Systems
In this paper we introduce new notions of local extremality for finite and
infinite systems of closed sets and establish the corresponding extremal
principles for them called here rated extremal principles. These developments
are in the core geometric theory of variational analysis. We present their
applications to calculus and optimality conditions for problems with infinitely
many constraints
Multivariate extremality measure
We propose a new multivariate order based on a concept that we will call extremality". Given a unit vector, the extremality allows to measure the "farness" of a point with respect to a data cloud or to a distribution in the vector direction. We establish the most relevant properties of this measure and provide the theoretical basis for its nonparametric estimation. We include two applications in Finance: a multivariate Value at Risk (VaR) with level sets constructed through extremality and a portfolio selection strategy based on the order induced by extremality.Extremality, Oriented cone, Value at risk, Portfolio selection
Support theorems in abstract settings
In this paper we establish a general framework in which the verification of
support theorems for generalized convex functions acting between an algebraic
structure and an ordered algebraic structure is still possible. As for the
domain space, we allow algebraic structures equipped with families of algebraic
operations whose operations are mutually distributive with respect to each
other. We introduce several new concepts in such algebraic structures, the
notions of convex set, extreme set, and interior point with respect to a given
family of operations, furthermore, we describe their most basic and required
properties. In the context of the range space, we introduce the notion of
completeness of a partially ordered set with respect to the existence of the
infimum of lower bounded chains, we also offer several sufficient condition
which imply this property. For instance, the order generated by a sharp cone in
a vector space turns out to possess this completeness property. By taking
several particular cases, we deduce support and extension theorems in various
classical and important settings
Computing the vertices of tropical polyhedra using directed hypergraphs
We establish a characterization of the vertices of a tropical polyhedron
defined as the intersection of finitely many half-spaces. We show that a point
is a vertex if, and only if, a directed hypergraph, constructed from the
subdifferentials of the active constraints at this point, admits a unique
strongly connected component that is maximal with respect to the reachability
relation (all the other strongly connected components have access to it). This
property can be checked in almost linear-time. This allows us to develop a
tropical analogue of the classical double description method, which computes a
minimal internal representation (in terms of vertices) of a polyhedron defined
externally (by half-spaces or hyperplanes). We provide theoretical worst case
complexity bounds and report extensive experimental tests performed using the
library TPLib, showing that this method outperforms the other existing
approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section
5 (using directed hypergraphs), detailed appendix; v3: major revision of the
article (adding tropical hyperplanes, alternative method by arrangements,
etc); v4: minor revisio
Completely positive maps on modules, instruments, extremality problems, and applications to physics
Convex sets of completely positive maps and positive semidefinite kernels are
considered in the most general context of modules over -algebras and a
complete charaterization of their extreme points is obtained. As a byproduct,
we determine extreme quantum instruments, preparations, channels, and extreme
autocorrelation functions. Various applications to quantum information and
measurement theories are given. The structure of quantum instruments is
analyzed thoroughly.Comment: 32 page
Optimal transportation, topology and uniqueness
The Monge-Kantorovich transportation problem involves optimizing with respect
to a given a cost function. Uniqueness is a fundamental open question about
which little is known when the cost function is smooth and the landscapes
containing the goods to be transported possess (non-trivial) topology. This
question turns out to be closely linked to a delicate problem (# 111) of
Birkhoff [14]: give a necessary and sufficient condition on the support of a
joint probability to guarantee extremality among all measures which share its
marginals. Fifty years of progress on Birkhoff's question culminate in Hestir
and Williams' necessary condition which is nearly sufficient for extremality;
we relax their subtle measurability hypotheses separating necessity from
sufficiency slightly, yet demonstrate by example that to be sufficient
certainly requires some measurability. Their condition amounts to the vanishing
of the measure \gamma outside a countable alternating sequence of graphs and
antigraphs in which no two graphs (or two antigraphs) have domains that
overlap, and where the domain of each graph / antigraph in the sequence
contains the range of the succeeding antigraph (respectively, graph). Such
sequences are called numbered limb systems. We then explain how this
characterization can be used to resolve the uniqueness of Kantorovich solutions
for optimal transportation on a manifold with the topology of the sphere.Comment: 36 pages, 6 figure
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