2,646 research outputs found
An error analysis for polynomial optimization over the simplex based on the multivariate hypergeometric distribution
We study the minimization of fixed-degree polynomials over the simplex. This
problem is well-known to be NP-hard, as it contains the maximum stable set
problem in graph theory as a special case. In this paper, we consider a
rational approximation by taking the minimum over the regular grid, which
consists of rational points with denominator (for given ). We show that
the associated convergence rate is for quadratic polynomials. For
general polynomials, if there exists a rational global minimizer over the
simplex, we show that the convergence rate is also of the order . Our
results answer a question posed by De Klerk et al. (2013) and improves on
previously known bounds in the quadratic case.Comment: 17 page
Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex
Let and let , be
such that and . We prove that \begin{equation*} a \mapsto \frac{\Gamma(aM +
1)}{\prod_{i=1}^{d+1} \Gamma(a \gamma_i + 1)} \prod_{i=1}^{d+1} x_i^{a\gamma_i}
\end{equation*} is completely monotonic on . This result
generalizes the one found by Alzer (2018) for binomial probabilities ().
As a consequence of the log-convexity, we obtain some combinatorial
inequalities for multinomial coefficients. We also show how the main result can
be used to derive asymptotic formulas for quantities of interest in the context
of statistical density estimation based on Bernstein polynomials on the
-dimensional simplex.Comment: 7 pages, 0 figur
On the Complexity of Optimization over the Standard Simplex
We review complexity results for minimizing polynomials over the standard simplex and unit hypercube.In addition, we show that there exists a polynomial time approximation scheme (PTAS) for minimizing Lipschitz continuous functions and functions with uniformly bounded Hessians over the standard simplex.This extends an earlier result by De Klerk, Laurent and Parrilo [A PTAS for the minimization of polynomials of fixed degree over the simplex, Theoretical Computer Science, to appear.]global optimization;standard simplex;PTAS;multivariate Bernstein approximation;semidefinite programming
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