16 research outputs found
Copositive certificates of non-negativity for polynomials on semialgebraic sets
A certificate of non-negativity is a way to write a given function so that
its non-negativity becomes evident. Certificates of non-negativity are
fundamental tools in optimization, and they underlie powerful algorithmic
techniques for various types of optimization problems. We propose certificates
of non-negativity of polynomials based on copositive polynomials. The
certificates we obtain are valid for generic semialgebraic sets and have a
fixed small degree, while commonly used sums-of-squares (SOS) certificates are
guaranteed to be valid only for compact semialgebraic sets and could have large
degree. Optimization over the cone of copositive polynomials is not tractable,
but this cone has been well studied. The main benefit of our copositive
certificates of non-negativity is their ability to translate results known
exclusively for copositive polynomials to more general semialgebraic sets. In
particular, we show how to use copositive polynomials to construct structured
(e.g., sparse) certificates of non-negativity, even for unstructured
semialgebraic sets. Last but not least, copositive certificates can be used to
obtain not only hierarchies of tractable lower bounds, but also hierarchies of
tractable upper bounds for polynomial optimization problems.Comment: 27 pages, 1 figur
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
The Membership Problem for quadratic modules with focus on the one dimensional case
The Membership Problem for a subset Q of the polynomial ring R[X] over a real closed field R asks: Is there an algorithm to decide whether a given polynomial f lies in Q or not?
For the case of a finitely generated quadratic module Q of IR[X] in dimension 1 we succeed and solve the Membership Problem affirmatively. We achieve the solution by first showing that Q is definable or equivalently weakly semialgebraic. The positive solution of the Membership Problem then follows by Tarski�s result about the decidability of the theory of real closed fields in the language of ordered rings. If the basic closed semialgebraic set associated to Q is bounded we explicitly describe the algorithm.
Under the additional assumption that the basic closed semialgebraic set associated to the finitely generated quadratic module Q is finite, we obtain a positive solution of the Membership Problem as well as an explicit algorithm also over arbitrary real closed fields R.
Furthermore we generalize the model theoretic concept of heirs which plays an important role in the solution of the Membership Problem for orderings. We define the heir of an arbitrary subset Q of R[X] on a real closed extension field R� of R as a certain subset of R�[X] such that the definability of Q becomes equivalent to the existence of a unique heir on every real closed extension field of R. This is a main tool for a possible affirmative answer to the Membership Problem in arbitrary dimension.
For the case of a finitely generated quadratic module Q of IR[X] in dimension 1 we explicitly compute the heirs on real closed extension fields of IR, if the basic closed semialgebraic set associated to Q is not empty and bounded
Approximate volume and integration for basic semi-algebraic sets
Given a basic compact semi-algebraic set \K\subset\R^n, we introduce a
methodology that generates a sequence converging to the volume of \K. This
sequence is obtained from optimal values of a hierarchy of either semidefinite
or linear programs. Not only the volume but also every finite vector of moments
of the probability measure that is uniformly distributed on \K can be
approximated as closely as desired, and so permits to approximate the integral
on \K of any given polynomial; extension to integration against some weight
functions is also provided. Finally, some numerical issues associated with the
algorithms involved are briefly discussed
Weyl-von Neumann Theorem and Borel Complexity of Unitary Equivalence Modulo Compacts of Self-Adjoint Operators
Weyl-von Neumann Theorem asserts that two bounded self-adjoint operators
on a Hilbert space are unitarily equivalent modulo compacts, i.e.,
for some unitary and compact self-adjoint
operator , if and only if and have the same essential spectra:
. In this paper we consider to what
extent the above Weyl-von Neumann's result can(not) be extended to unbounded
operators using descriptive set theory. We show that if is separable
infinite-dimensional, this equivalence relation for bounded self-adjoin
operators is smooth, while the same equivalence relation for general
self-adjoint operators contains a dense -orbit but does not admit
classification by countable structures. On the other hand, apparently related
equivalence relation $A\sim B\Leftrightarrow \exists u\in \mathcal{U}(H)\
[u(A-i)^{-1}u^*-(B-i)^{-1}$ is compact], is shown to be smooth. Various Borel
or co-analytic equivalence relations related to self-adjoint operators are also
presented.Comment: 36 page
Tsirelson's problem and Kirchberg's conjecture
Tsirelson's problem asks whether the set of nonlocal quantum correlations
with a tensor product structure for the Hilbert space coincides with the one
where only commutativity between observables located at different sites is
assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products
of C*-algebras would imply a positive answer to this question for all bipartite
scenarios. This remains true also if one considers not only spatial
correlations, but also spatiotemporal correlations, where each party is allowed
to apply their measurements in temporal succession; we provide an example of a
state together with observables such that ordinary spatial correlations are
local, while the spatiotemporal correlations reveal nonlocality. Moreover, we
find an extended version of Tsirelson's problem which, for each nontrivial Bell
scenario, is equivalent to the QWEP conjecture. This extended version can be
conveniently formulated in terms of steering the system of a third party.
Finally, a comprehensive mathematical appendix offers background material on
complete positivity, tensor products of C*-algebras, group C*-algebras, and
some simple reformulations of the QWEP conjecture.Comment: 57 pages, to appear in Rev. Math. Phy