1,653 research outputs found
On the gap between positive polynomials and SOS of polynomials
This note investigates the gap existing between positive polynomials and sum of squares (SOS) of polynomials, which affects several analysis and synthesis tools in control systems based on polynomial SOS relaxations, and about which almost nothing is known. In particular, a matrix characterization of the PNS, that is the positive homogeneous forms that are not SOS, is proposed, which allows to show that any PNS is the vertex of an unbounded cone of PNS. Moreover, a complete parametrization of the set of PNS is introduced. Β© 2007 IEEE.published_or_final_versio
Characterizing the positive polynomials which are not SOS
Several analysis and synthesis tools in control systems are based on polynomial sum of squares (SOS) relaxations. However, almost nothing is known about the gap existing between positive polynomials and SOS of polynomials. This paper investigates such a gap proposing a matrix characterization of PNS, that is homogeneous forms that are not SOS. In particular, it is shown that any PNS is the vertex of an unbounded cone of PNS. Moreover, a complete parameterization of the set of PNS is introduced. Β© 2005 IEEE.published_or_final_versio
A Complete Characterization of the Gap between Convexity and SOS-Convexity
Our first contribution in this paper is to prove that three natural sum of
squares (sos) based sufficient conditions for convexity of polynomials, via the
definition of convexity, its first order characterization, and its second order
characterization, are equivalent. These three equivalent algebraic conditions,
henceforth referred to as sos-convexity, can be checked by semidefinite
programming whereas deciding convexity is NP-hard. If we denote the set of
convex and sos-convex polynomials in variables of degree with
and respectively, then our main
contribution is to prove that if and
only if or or . We also present a complete
characterization for forms (homogeneous polynomials) except for the case
which is joint work with G. Blekherman and is to be published
elsewhere. Our result states that the set of convex forms in
variables of degree equals the set of sos-convex forms if
and only if or or . To prove these results, we present
in particular explicit examples of polynomials in
and
and forms in
and , and a
general procedure for constructing forms in from nonnegative but not sos forms in variables and degree .
Although for disparate reasons, the remarkable outcome is that convex
polynomials (resp. forms) are sos-convex exactly in cases where nonnegative
polynomials (resp. forms) are sums of squares, as characterized by Hilbert.Comment: 25 pages; minor editorial revisions made; formal certificates for
computer assisted proofs of the paper added to arXi
Certifying Convergence of Lasserre's Hierarchy via Flat Truncation
This paper studies how to certify the convergence of Lasserre's hierarchy of
semidefinite programming relaxations for solving multivariate polynomial
optimization. We propose flat truncation as a general certificate for this
purpose. Assume the set of global minimizers is nonempty and finite. Our main
results are: i) Putinar type Lasserre's hierarchy has finite convergence if and
only if flat truncation holds, under some general assumptions, and this is also
true for the Schmudgen type one; ii) under the archimedean condition, flat
truncation is asymptotically satisfied for Putinar type Lasserre's hierarchy,
and similar is true for the Schmudgen type one; iii) for the hierarchy of
Jacobian SDP relaxations, flat truncation is always satisfied. The case of
unconstrained polynomial optimization is also discussed.Comment: 18 page
Polynomial Optimization with Real Varieties
We consider the optimization problem of minimizing a polynomial f(x) subject
to polynomial constraints h(x)=0, g(x)>=0. Lasserre's hierarchy is a sequence
of sum of squares relaxations for finding the global minimum. Let K be the
feasible set. We prove the following results: i) If the real variety V_R(h) is
finite, then Lasserre's hierarchy has finite convergence, no matter the complex
variety V_C(h) is finite or not. This solves an open question in Laurent's
survey. ii) If K and V_R(h) have the same vanishing ideal, then the finite
convergence of Lasserre's hierarchy is independent of the choice of defining
polynomials for the real variety V_R(h). iii) When K is finite, a refined
version of Lasserre's hierarchy (using the preordering of g) has finite
convergence.Comment: 12 page
Limitations of semidefinite programs for separable states and entangled games
Semidefinite programs (SDPs) are a framework for exact or approximate
optimization that have widespread application in quantum information theory. We
introduce a new method for using reductions to construct integrality gaps for
SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy
in approximating two particularly important sets in quantum information theory,
where previously no -round integrality gaps were known: the set of
separable (i.e. unentangled) states, or equivalently, the
norm of a matrix, and the set of quantum correlations; i.e. conditional
probability distributions achievable with local measurements on a shared
entangled state. In both cases no-go theorems were previously known based on
computational assumptions such as the Exponential Time Hypothesis (ETH) which
asserts that 3-SAT requires exponential time to solve. Our unconditional
results achieve the same parameters as all of these previous results (for
separable states) or as some of the previous results (for quantum
correlations). In some cases we can make use of the framework of
Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not
only the SoS hierarchy. Our hardness result on separable states also yields a
dimension lower bound of approximate disentanglers, answering a question of
Watrous and Aaronson et al. These results can be viewed as limitations on the
monogamy principle, the PPT test, the ability of Tsirelson-type bounds to
restrict quantum correlations, as well as the SDP hierarchies of
Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published
versio
Sparse sum-of-squares (SOS) optimization: A bridge between DSOS/SDSOS and SOS optimization for sparse polynomials
Optimization over non-negative polynomials is fundamental for nonlinear
systems analysis and control. We investigate the relation between three
tractable relaxations for optimizing over sparse non-negative polynomials:
sparse sum-of-squares (SSOS) optimization, diagonally dominant sum-of-squares
(DSOS) optimization, and scaled diagonally dominant sum-of-squares (SDSOS)
optimization. We prove that the set of SSOS polynomials, an inner approximation
of the cone of SOS polynomials, strictly contains the spaces of sparse
DSOS/SDSOS polynomials. When applicable, therefore, SSOS optimization is less
conservative than its DSOS/SDSOS counterparts. Numerical results for
large-scale sparse polynomial optimization problems demonstrate this fact, and
also that SSOS optimization can be faster than DSOS/SDSOS methods despite
requiring the solution of semidefinite programs instead of less expensive
linear/second-order cone programs.Comment: 9 pages, 3 figure
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