163,609 research outputs found
A Semidefinite Programming approach for minimizing ordered weighted averages of rational functions
This paper considers the problem of minimizing the ordered weighted average
(or ordered median) function of finitely many rational functions over compact
semi-algebraic sets. Ordered weighted averages of rational functions are not,
in general, neither rational functions nor the supremum of rational functions
so that current results available for the minimization of rational functions
cannot be applied to handle these problems. We prove that the problem can be
transformed into a new problem embedded in a higher dimension space where it
admits a convenient representation. This reformulation admits a hierarchy of
SDP relaxations that approximates, up to any degree of accuracy, the optimal
value of those problems. We apply this general framework to a broad family of
continuous location problems showing that some difficult problems (convex and
non-convex) that up to date could only be solved on the plane and with
Euclidean distance, can be reasonably solved with different -norms and
in any finite dimension space. We illustrate this methodology with some
extensive computational results on location problems in the plane and the
3-dimension space.Comment: 27 pages, 1 figure, 7 table
Handling convexity-like constraints in variational problems
We provide a general framework to construct finite dimensional approximations
of the space of convex functions, which also applies to the space of c-convex
functions and to the space of support functions of convex bodies. We give
estimates of the distance between the approximation space and the admissible
set. This framework applies to the approximation of convex functions by
piecewise linear functions on a mesh of the domain and by other
finite-dimensional spaces such as tensor-product splines. We show how these
discretizations are well suited for the numerical solution of problems of
calculus of variations under convexity constraints. Our implementation relies
on proximal algorithms, and can be easily parallelized, thus making it
applicable to large scale problems in dimension two and three. We illustrate
the versatility and the efficiency of our approach on the numerical solution of
three problems in calculus of variation : 3D denoising, the principal agent
problem, and optimization within the class of convex bodies.Comment: 23 page
Geometrical Insights for Implicit Generative Modeling
Learning algorithms for implicit generative models can optimize a variety of
criteria that measure how the data distribution differs from the implicit model
distribution, including the Wasserstein distance, the Energy distance, and the
Maximum Mean Discrepancy criterion. A careful look at the geometries induced by
these distances on the space of probability measures reveals interesting
differences. In particular, we can establish surprising approximate global
convergence guarantees for the -Wasserstein distance,even when the
parametric generator has a nonconvex parametrization.Comment: this version fixes a typo in a definitio
The equivariant Minkowski problem in Minkowski space
The classical Minkowski problem in Minkowski space asks, for a positive
function on , for a convex set in Minkowski space with
space-like boundary , such that is the
Gauss--Kronecker curvature at the point with normal . Analogously to the
Euclidean case, it is possible to formulate a weak version of this problem:
given a Radon measure on the generalized Minkowski problem
in Minkowski space asks for a convex subset such that the area measure of
is .
In the present paper we look at an equivariant version of the problem: given
a uniform lattice of isometries of , given a
invariant Radon measure , given a isometry group of
Minkowski space, with as linear part, there exists a unique convex set
with area measure , invariant under the action of .
The proof uses a functional which is the covolume associated to every
invariant convex set.
This result translates as a solution of the Minkowski problem in flat space
times with compact hyperbolic Cauchy surface. The uniqueness part, as well as
regularity results, follow from properties of the Monge--Amp\`ere equation. The
existence part can be translated as an existence result for Monge--Amp\`ere
equation.
The regular version was proved by T.~Barbot, F.~B\'eguin and A.~Zeghib for
and by V.~Oliker and U.~Simon for . Our method is
totally different. Moreover, we show that those cases are very specific: in
general, there is no smooth -invariant surface of constant
Gauss-Kronecker curvature equal to
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
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