5,219 research outputs found
Border Basis relaxation for polynomial optimization
A relaxation method based on border basis reduction which improves the
efficiency of Lasserre's approach is proposed to compute the optimum of a
polynomial function on a basic closed semi algebraic set. A new stopping
criterion is given to detect when the relaxation sequence reaches the minimum,
using a sparse flat extension criterion. We also provide a new algorithm to
reconstruct a finite sum of weighted Dirac measures from a truncated sequence
of moments, which can be applied to other sparse reconstruction problems. As an
application, we obtain a new algorithm to compute zero-dimensional minimizer
ideals and the minimizer points or zero-dimensional G-radical ideals.
Experimentations show the impact of this new method on significant benchmarks.Comment: Accepted for publication in Journal of Symbolic Computatio
Border Basis for Polynomial System Solving and Optimization
International audienceWe describe the software package borderbasix dedicated to the computation of border bases and the solutions of polynomial equations. We present the main ingredients of the border basis algorithm and the other methods implemented in this package: numerical solutions from multiplication matrices, real radical computation, polynomial optimization. The implementation parameterized by the coefficient type and the choice function provides a versatile family of tools for polynomial computation with modular arithmetic, floating point arithmetic or rational arithmetic. It relies on linear algebra solvers for dense and sparse matrices for these various types of coefficients. A connection with SDP solvers has been integrated for the combination of relaxation approaches with border basis computation. Extensive benchmarks on typical polynomial systems are reported, which show the very good performance of the tool
Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
The purpose of this note is to survey a methodology to solve systems of
polynomial equations and inequalities. The techniques we discuss use the
algebra of multivariate polynomials with coefficients over a field to create
large-scale linear algebra or semidefinite programming relaxations of many
kinds of feasibility or optimization questions. We are particularly interested
in problems arising in combinatorial optimization.Comment: 28 pages, survey pape
A polyhedral approach to computing border bases
Border bases can be considered to be the natural extension of Gr\"obner bases
that have several advantages. Unfortunately, to date the classical border basis
algorithm relies on (degree-compatible) term orderings and implicitly on
reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow
for calculating border bases for arbitrary degree-compatible order ideals,
which is \emph{independent} from term orderings. Moreover, the algorithm also
supports calculating degree-compatible order ideals with \emph{preference} on
contained elements, even though finding a preferred order ideal is NP-hard.
Effectively we retain degree-compatibility only to successively extend our
computation degree-by-degree. The adaptation is based on our polyhedral
characterization: order ideals that support a border basis correspond
one-to-one to integral points of the order ideal polytope. This establishes a
crucial connection between the ideal and the combinatorial structure of the
associated factor spaces
Exact relaxation for polynomial optimization on semi-algebraic sets
In this paper, we study the problem of computing by relaxation hierarchies
the infimum of a real polynomial function f on a closed basic semialgebraic set
and the points where this infimum is reached, if they exist. We show that when
the infimum is reached, a relaxation hierarchy constructed from the
Karush-Kuhn-Tucker ideal is always exact and that the vanishing ideal of the
KKT minimizer points is generated by the kernel of the associated moment matrix
in that degree, even if this ideal is not zero-dimensional. We also show that
this relaxation allows to detect when there is no KKT minimizer. We prove that
the exactness of the relaxation depends only on the real points which satisfy
these constraints.This exploits representations of positive polynomials as
elementsof the preordering modulo the KKT ideal, which only involves
polynomials in the initial set of variables. Applications to global
optimization, optimization on semialgebraic sets defined by regular sets of
constraints, optimization on finite semialgebraic sets, real radical
computation are given
Domain Decomposition for Stochastic Optimal Control
This work proposes a method for solving linear stochastic optimal control
(SOC) problems using sum of squares and semidefinite programming. Previous work
had used polynomial optimization to approximate the value function, requiring a
high polynomial degree to capture local phenomena. To improve the scalability
of the method to problems of interest, a domain decomposition scheme is
presented. By using local approximations, lower degree polynomials become
sufficient, and both local and global properties of the value function are
captured. The domain of the problem is split into a non-overlapping partition,
with added constraints ensuring continuity. The Alternating Direction
Method of Multipliers (ADMM) is used to optimize over each domain in parallel
and ensure convergence on the boundaries of the partitions. This results in
improved conditioning of the problem and allows for much larger and more
complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201
A path following algorithm for the graph matching problem
We propose a convex-concave programming approach for the labeled weighted
graph matching problem. The convex-concave programming formulation is obtained
by rewriting the weighted graph matching problem as a least-square problem on
the set of permutation matrices and relaxing it to two different optimization
problems: a quadratic convex and a quadratic concave optimization problem on
the set of doubly stochastic matrices. The concave relaxation has the same
global minimum as the initial graph matching problem, but the search for its
global minimum is also a hard combinatorial problem. We therefore construct an
approximation of the concave problem solution by following a solution path of a
convex-concave problem obtained by linear interpolation of the convex and
concave formulations, starting from the convex relaxation. This method allows
to easily integrate the information on graph label similarities into the
optimization problem, and therefore to perform labeled weighted graph matching.
The algorithm is compared with some of the best performing graph matching
methods on four datasets: simulated graphs, QAPLib, retina vessel images and
handwritten chinese characters. In all cases, the results are competitive with
the state-of-the-art.Comment: 23 pages, 13 figures,typo correction, new results in sections 4,5,
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