258 research outputs found
Reflection positive doubles
Here we introduce reflection positive doubles, a general framework for reflection positivity, covering a wide variety of systems in statistical physics and quantum field theory. These systems may be bosonic, fermionic, or parafermionic in nature. Within the framework of reflection positive doubles, we give necessary and sufficient conditions for reflection positivity. We use a reflection-invariant cone to implement our construction. Our characterization allows for a direct interpretation in terms of coupling constants, making it easy to check in concrete situations. We illustrate our methods with numerous examples
Reflection Positive Doubles
Here we introduce reflection positive doubles, a general framework for
reflection positivity, covering a wide variety of systems in statistical
physics and quantum field theory. These systems may be bosonic, fermionic, or
parafermionic in nature. Within the framework of reflection positive doubles,
we give necessary and sufficient conditions for reflection positivity. We use a
reflection-invariant cone to implement our construction. Our characterization
allows for a direct interpretation in terms of coupling constants, making it
easy to check in concrete situations. We illustrate our methods with numerous
examples.Comment: 53 pages, 3 figure
Analysis of Control Systems on Symmetric Cones
It is well known that exploiting special structure is a powerful way to extend the reach of current optimization tools to higher dimensions. While many linear control systems can be treated satisfactorily with linear matrix inequalities (LMI) and semidefinite programming (SDP), practical considerations can still restrict scalability of general methods. Thus, we wish to work with high dimensional systems without explicitly forming SDPs. To that end, we exploit a particular kind of structure in the dynamics matrix, paving the way for a more efficient treatment of a certain class of linear systems. We show how second order cone programming (SOCP) can be used instead of SDP to find Lyapunov functions that certify stability. This framework reduces to a famous linear program (LP) when the system is internally positive, and to a semidefinite program (SDP) when the system has no special structure
Sum of squares generalizations for conic sets
In polynomial optimization problems, nonnegativity constraints are typically
handled using the sum of squares condition. This can be efficiently enforced
using semidefinite programming formulations, or as more recently proposed by
Papp and Yildiz [18], using the sum of squares cone directly in a nonsymmetric
interior point algorithm. Beyond nonnegativity, more complicated polynomial
constraints (in particular, generalizations of the positive semidefinite,
second order and -norm cones) can also be modeled through structured
sum of squares programs. We take a different approach and propose using more
specialized polynomial cones instead. This can result in lower dimensional
formulations, more efficient oracles for interior point methods, or
self-concordant barriers with smaller parameters. In most cases, these
algorithmic advantages also translate to faster solving times in practice
Bilinearity rank of the cone of positive polynomials and related cones
For a proper cone K ⊂ Rn and its dual cone K the complementary slackness condition xT s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K^* }. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. When K is a symmetric cone, this fact translates to a set of n linearly independent bilinear identities (optimality conditions) satisfied by every (x, s) ∈ C(K). This proves to be very useful when
optimizing over such cones, therefore it is natural to look for similar optimality conditions for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for the cone, and describe a linear algebraic technique to bound this quantity. We examine several well-known cones, in particular
the cone of positive polynomials P2n+1 and its dual, the closure of the moment cone M2n+1, and compute their bilinearity ranks. We show that there are exactly four linearly independent bilinear identities which hold for all (x,s) ∈ C(P2n+1), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential
polynomials
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