11,446 research outputs found
On semidefinite representations of plane quartics
This note focuses on the problem of representing convex sets as projections
of the cone of positive semidefinite matrices, in the particular case of sets
generated by bivariate polynomials of degree four. Conditions are given for the
convex hull of a plane quartic to be exactly semidefinite representable with at
most 12 lifting variables. If the quartic is rationally parametrizable, an
exact semidefinite representation with 2 lifting variables can be obtained.
Various numerical examples illustrate the techniques and suggest further
research directions
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
Optimal designs for rational function regression
We consider optimal non-sequential designs for a large class of (linear and
nonlinear) regression models involving polynomials and rational functions with
heteroscedastic noise also given by a polynomial or rational weight function.
The proposed method treats D-, E-, A-, and -optimal designs in a
unified manner, and generates a polynomial whose zeros are the support points
of the optimal approximate design, generalizing a number of previously known
results of the same flavor. The method is based on a mathematical optimization
model that can incorporate various criteria of optimality and can be solved
efficiently by well established numerical optimization methods. In contrast to
previous optimization-based methods proposed for similar design problems, it
also has theoretical guarantee of its algorithmic efficiency; in fact, the
running times of all numerical examples considered in the paper are negligible.
The stability of the method is demonstrated in an example involving high degree
polynomials. After discussing linear models, applications for finding locally
optimal designs for nonlinear regression models involving rational functions
are presented, then extensions to robust regression designs, and trigonometric
regression are shown. As a corollary, an upper bound on the size of the support
set of the minimally-supported optimal designs is also found. The method is of
considerable practical importance, with the potential for instance to impact
design software development. Further study of the optimality conditions of the
main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory
and additional example
Computation of sum of squares polynomials from data points
We propose an iterative algorithm for the numerical computation of sums of
squares of polynomials approximating given data at prescribed interpolation
points. The method is based on the definition of a convex functional
arising from the dualization of a quadratic regression over the Cholesky
factors of the sum of squares decomposition. In order to justify the
construction, the domain of , the boundary of the domain and the behavior at
infinity are analyzed in details. When the data interpolate a positive
univariate polynomial, we show that in the context of the Lukacs sum of squares
representation, is coercive and strictly convex which yields a unique
critical point and a corresponding decomposition in sum of squares. For
multivariate polynomials which admit a decomposition in sum of squares and up
to a small perturbation of size , is always
coercive and so it minimum yields an approximate decomposition in sum of
squares. Various unconstrained descent algorithms are proposed to minimize .
Numerical examples are provided, for univariate and bivariate polynomials
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
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
Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations
It is well-known that any sum of squares (SOS) program can be cast as a
semidefinite program (SDP) of a particular structure and that therein lies the
computational bottleneck for SOS programs, as the SDPs generated by this
procedure are large and costly to solve when the polynomials involved in the
SOS programs have a large number of variables and degree. In this paper, we
review SOS optimization techniques and present two new methods for improving
their computational efficiency. The first method leverages the sparsity of the
underlying SDP to obtain computational speed-ups. Further improvements can be
obtained if the coefficients of the polynomials that describe the problem have
a particular sparsity pattern, called chordal sparsity. The second method
bypasses semidefinite programming altogether and relies instead on solving a
sequence of more tractable convex programs, namely linear and second order cone
programs. This opens up the question as to how well one can approximate the
cone of SOS polynomials by second order representable cones. In the last part
of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201
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