427 research outputs found
Level sets and non Gaussian integrals of positively homogeneous functions
We investigate various properties of the sublevel set
and the integration of on this sublevel set when and are positively
homogeneous functions. For instance, the latter integral reduces to integrating
on the whole space (a non Gaussian integral) and when is
a polynomial, then the volume of the sublevel set is a convex function of the
coefficients of . In fact, whenever is nonnegative, the functional is a convex function of for a large class of functions
. We also provide a numerical approximation scheme to compute
the volume or integrate (or, equivalently to approximate the associated non
Gaussian integral). We also show that finding the sublevel set of minimum volume that contains some given subset is a
(hard) convex optimization problem for which we also propose two convergent
numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian
integrals for homogeneous polynomials that are sums of squares and critical
points of a specific function
Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane
We complete the complexity classification by degree of minimizing a
polynomial over the integer points in a polyhedron in . Previous
work shows that optimizing a quadratic polynomial over the integer points in a
polyhedral region in can be done in polynomial time, while
optimizing a quartic polynomial in the same type of region is NP-hard. We close
the gap by showing that this problem can be solved in polynomial time for cubic
polynomials.
Furthermore, we show that the problem of minimizing a homogeneous polynomial
of any fixed degree over the integer points in a bounded polyhedron in
is solvable in polynomial time. We show that this holds for
polynomials that can be translated into homogeneous polynomials, even when the
translation vector is unknown. We demonstrate that such problems in the
unbounded case can have smallest optimal solutions of exponential size in the
size of the input, thus requiring a compact representation of solutions for a
general polynomial time algorithm for the unbounded case
Simple Approximations of Semialgebraic Sets and their Applications to Control
Many uncertainty sets encountered in control systems analysis and design can
be expressed in terms of semialgebraic sets, that is as the intersection of
sets described by means of polynomial inequalities. Important examples are for
instance the solution set of linear matrix inequalities or the Schur/Hurwitz
stability domains. These sets often have very complicated shapes (non-convex,
and even non-connected), which renders very difficult their manipulation. It is
therefore of considerable importance to find simple-enough approximations of
these sets, able to capture their main characteristics while maintaining a low
level of complexity. For these reasons, in the past years several convex
approximations, based for instance on hyperrect-angles, polytopes, or
ellipsoids have been proposed. In this work, we move a step further, and
propose possibly non-convex approximations , based on a small volume polynomial
superlevel set of a single positive polynomial of given degree. We show how
these sets can be easily approximated by minimizing the L1 norm of the
polynomial over the semialgebraic set, subject to positivity constraints.
Intuitively, this corresponds to the trace minimization heuristic commonly
encounter in minimum volume ellipsoid problems. From a computational viewpoint,
we design a hierarchy of linear matrix inequality problems to generate these
approximations, and we provide theoretically rigorous convergence results, in
the sense that the hierarchy of outer approximations converges in volume (or,
equivalently, almost everywhere and almost uniformly) to the original set. Two
main applications of the proposed approach are considered. The first one aims
at reconstruction/approximation of sets from a finite number of samples. In the
second one, we show how the concept of polynomial superlevel set can be used to
generate samples uniformly distributed on a given semialgebraic set. The
efficiency of the proposed approach is demonstrated by different numerical
examples
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
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