6,151 research outputs found
Semidefinite Representation of the -Ellipse
The -ellipse is the plane algebraic curve consisting of all points whose
sum of distances from given points is a fixed number. The polynomial
equation defining the -ellipse has degree if is odd and degree
if is even. We express this polynomial equation as
the determinant of a symmetric matrix of linear polynomials. Our representation
extends to weighted -ellipses and -ellipsoids in arbitrary dimensions,
and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
Global optimization of polynomials using gradient tentacles and sums of squares
In this work, the combine the theory of generalized critical values with the
theory of iterated rings of bounded elements (real holomorphy rings).
We consider the problem of computing the global infimum of a real polynomial
in several variables. Every global minimizer lies on the gradient variety. If
the polynomial attains a minimum, it is therefore equivalent to look for the
greatest lower bound on its gradient variety. Nie, Demmel and Sturmfels proved
recently a theorem about the existence of sums of squares certificates for such
lower bounds. Based on these certificates, they find arbitrarily tight
relaxations of the original problem that can be formulated as semidefinite
programs and thus be solved efficiently.
We deal here with the more general case when the polynomial is bounded from
belo w but does not necessarily attain a minimum. In this case, the method of
Nie, Demmel and Sturmfels might yield completely wrong results. In order to
overcome this problem, we replace the gradient variety by larger semialgebraic
sets which we call gradient tentacles. It now gets substantially harder to
prove the existence of the necessary sums of squares certificates.Comment: 22 page
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