An Error Analysis for Polynomial Optimization over the Simplex Based on the Multivariate Hypergeometric Distribution

@article{Klerk2015AnEA,
  title={An Error Analysis for Polynomial Optimization over the Simplex Based on the Multivariate Hypergeometric Distribution},
  author={Etienne de Klerk and Monique Laurent and Zhao Sun},
  journal={SIAM J. Optim.},
  year={2015},
  volume={25},
  pages={1498-1514}
}
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 $r$ (for given $r$). We show that the associated convergence rate is $O(1/r^2)$ for quadratic polynomials. For general polynomials, if there exists a… Expand
On the convergence rate of grid search for polynomial optimization over the simplex
TLDR
It is shown that the rational minimizer condition is not necessary to obtain the O(1/r^2) bound, and the approximate minimization of a given polynomial on the standard simplex is obtained by taking the minimum value over all rational grid points with given denominator. Expand
Polynomial optimization: Error analysis and applications
Polynomial optimization is the problem of minimizing a polynomial function subject to polynomial inequality constraints. In this thesis we investigate several hierarchies of relaxations forExpand
Bound-Constrained Polynomial Optimization Using Only Elementary Calculations
TLDR
This work provides a monotone nonincreasing sequence of upper bounds fkH(k≥1) converging to the global minimum of a polynomial f on simple sets like the unit hypercube in ℝn and shows a stronger convergence rate in O(1/k) for quadratic polynomials and more generally for polynoms having a rational minimizer in the hypercube. Expand
Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization
TLDR
A hierarchy of upper bounds proposed by Lasserre is analyzed, obtained by searching for an optimal probability density function h on K which is a sum of squares of polynomials, so that the rate of convergence is no worse than O(1/r), where 2r is the degree bound on the density function. Expand
Convex Hulls, Relaxations, and Approximations of General Monomials and Multilinear Functions
Motivated by a variety of problems in global optimization and integer programming that involve multilinear expressions of discrete or continuous variables, this research derives approximations ofExpand
A survey of semidefinite programming approaches to the generalized problem of moments and their error analysis
The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has a large variety of applications, including globalExpand
Error bounds for monomial convexification in polynomial optimization
TLDR
Borders on the worst-case error for convexifying a monomial over subsets of . Expand
Inner Approximations of Completely Positive Reformulations of Mixed Binary Quadratic Optimization Problems : A Unified Analysis ∗
August 24, 2015 Abstract Every quadratic optimization problem with a mix of continuous and binary variables can be equivalently reformulated as a completely positive optimization problem, i.e., aExpand
Inner approximations of completely positive reformulations of mixed binary quadratic programs: a unified analysis
TLDR
The results imply that polyhedral inner approximations are equivalent to a finite discretization of the feasible region of the original completely positive optimization problem. Expand
Maximization of Homogeneous Polynomials over the Simplex and the Sphere: Structure, Stability, and Generic Behavior
TLDR
It is shown that generically any local maximizer is an evolutionarily stable strategy in the case of polynomial optimization over the sphere and this program is directly related to the concept of evolutionarilystable strategies in biology. Expand
...
1
2
...

References

SHOWING 1-10 OF 21 REFERENCES
A Refined Error Analysis for Fixed-Degree Polynomial Optimization over the Simplex
We consider the problem of minimizing a fixed-degree polynomial over the standard simplex. This problem is well known to be NP-hard, since it contains the maximum stable set problem in combinatorialExpand
A PTAS for the minimization of polynomials of fixed degree over the simplex
TLDR
It is shown that the bounds pΔ(k) yield a polynomial time approximation scheme for the minimization of polynomials of fixed degree d on the simplex, extending an earlier result of Bomze and De Klerk for degree d = 2. Expand
An alternative proof of a PTAS for fixed-degree polynomial optimization over the simplex
TLDR
New insight is provided into the polynomial-time approximation scheme for polynomials of fixed degree by establishing precise links with Bernstein approximation and the multinomial distribution. Expand
Random walk in a simplex and quadratic optimization over convex polytopes
In this paper we develop probabilistic arguments for justifying thequality of an approximate solution for global quadratic minimization problem, obtained as a best point among all points of a uniformExpand
Analysis of copositive optimization based linear programming bounds on standard quadratic optimization
TLDR
This paper investigates the sequences of upper and lower bounds on the optimal value of a standard quadratic optimization problem arising from these two hierarchies of inner and outer polyhedral approximations and presents several geometric and topological properties of these sets. Expand
A New Look at Nonnegativity on Closed Sets and Polynomial Optimization
  • J. Lasserre
  • Mathematics, Computer Science
  • SIAM J. Optim.
  • 2011
TLDR
A convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting is obtained, of the cone of nonnegative polynomials of degree at most $d$, used in polynomial optimization on certain simple closed sets. Expand
On the complexity of optimization over the standard simplex
TLDR
It is shown that there exists a polynomial time approximation scheme (PTAS) for minimizing Lipschitz continuous functions and functions with uniformly bounded Hessians over the standard simplex. Expand
On the accuracy of uniform polyhedral approximations of the copositive cone
TLDR
A hierarchy of increasingly better outer polyhedral approximations to the copositive cone is proposed, and it is established that the sequence of approxIMations is exact in the limit. Expand
Rounding on the standard simplex: regular grids for global optimization
TLDR
It is shown that the minimal distance to the regular grid on the standard simplex can exceed one, even for very fine mesh sizes in high dimensions. Expand
Error Bounds for Some Semidefinite Programming Approaches to Polynomial Minimization on the Hypercube
We consider the problem of minimizing a polynomial on the hypercube $[0,1]^n$ and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding toExpand
...
1
2
3
...