340 research outputs found
GloptiPoly 3: moments, optimization and semidefinite programming
We describe a major update of our Matlab freeware GloptiPoly for parsing
generalized problems of moments and solving them numerically with semidefinite
programming
Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations
Combining recent moment and sparse semidefinite programming (SDP) relaxation
techniques, we propose an approach to find smooth approximations for solutions
of problems involving nonlinear differential equations. Given a system of
nonlinear differential equations, we apply a technique based on finite
differences and sparse SDP relaxations for polynomial optimization problems
(POP) to obtain a discrete approximation of its solution. In a second step we
apply maximum entropy estimation (using moments of a Borel measure associated
with the discrete solution) to obtain a smooth closed-form approximation. The
approach is illustrated on a variety of linear and nonlinear ordinary
differential equations (ODE), partial differential equations (PDE) and optimal
control problems (OCP), and preliminary numerical results are reported
Stokes, Gibbs and volume computation of semi-algebraic sets
We consider the problem of computing the Lebesgue volume of compact basic
semi-algebraic sets. In full generality, it can be approximated as closely as
desired by a converging hierarchy of upper bounds obtained by applying the
Moment-SOS (sums of squares) methodology to a certain innite-dimensional linear
program (LP). At each step one solves a semidenite relaxation of the LP which
involves pseudo-moments up to a certain degree. Its dual computes a polynomial
of same degree which approximates from above the discon-tinuous indicator
function of the set, hence with a typical Gibbs phenomenon which results in a
slow convergence of the associated numerical scheme. Drastic improvements have
been observed by introducing in the initial LP additional linear moment
constraints obtained from a certain application of Stokes' theorem for
integration on the set. However and so far there was no rationale to explain
this behavior. We provide a rened version of this extended LP formulation. When
the set is the smooth super-level set of a single polynomial, we show that the
dual of this rened LP has an optimal solution which is a continuous function.
Therefore in this dual one now approximates a continuous function by a
polynomial, hence with no Gibbs phenomenon, which explains and improves the
already observed drastic acceleration of the convergence of the hierarchy.
Interestingly, the technique of proof involves classical results on Poisson's
partial dierential equation (PDE)
Polynomial argmin for recovery and approximation of multivariate discontinuous functions
We propose to approximate a (possibly discontinuous) multivariate function f
(x) on a compact set by the partial minimizer arg miny p(x, y) of an
appropriate polynomial p whose construction can be cast in a univariate sum of
squares (SOS) framework, resulting in a highly structured convex semidefinite
program. In a number of non-trivial cases (e.g. when f is a piecewise
polynomial) we prove that the approximation is exact with a low-degree
polynomial p. Our approach has three distinguishing features: (i) It is
mesh-free and does not require the knowledge of the discontinuity locations.
(ii) It is model-free in the sense that we only assume that the function to be
approximated is available through samples (point evaluations). (iii) The size
of the semidefinite program is independent of the ambient dimension and depends
linearly on the number of samples. We also analyze the sample complexity of the
approach, proving a generalization error bound in a probabilistic setting. This
allows for a comparison with machine learning approaches
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