# Bound-Constrained Polynomial Optimization Using Only Elementary Calculations

@article{Klerk2017BoundConstrainedPO,
title={Bound-Constrained Polynomial Optimization Using Only Elementary Calculations},
author={Etienne de Klerk and Jean Bernard Lasserre and Monique Laurent and Zhao Sun},
journal={Math. Oper. Res.},
year={2017},
volume={42},
pages={834-853}
}
• Published 15 July 2015
• Computer Science, Mathematics
• Math. Oper. Res.
We provide 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. The novelty with respect to the converging sequence of upper bounds in Lasserre [Lasserre JB (2010) A new look at nonnegativity on closed sets and polynomial optimization, SIAM J. Optim. 21:864–885] is that only elementary computations are required. For optimization over the hypercube [0, 1]n, we show that the new bounds fkH have…

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## References

SHOWING 1-10 OF 38 REFERENCES
An Error Analysis for Polynomial Optimization over the Simplex Based on the Multivariate Hypergeometric Distribution
• Mathematics, Computer Science
SIAM J. Optim.
• 2015
This paper considers a rational approximation by taking the minimum over the regular grid, which consists of rational points with denominator $r$ (for given $r) and shows that the associated convergence rate is$O(1/r^2)$for quadratic polynomials. A New Look at Nonnegativity on Closed Sets and Polynomial Optimization A convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting is obtained, of the cone of nonnegative polynomials of degree at most$d$, used in polynomial optimization on certain simple closed sets. A PTAS for the minimization of polynomials of fixed degree over the simplex • Mathematics, Computer Science Theor. Comput. Sci. • 2006 Global optimization of rational functions: a semidefinite programming approach • Mathematics Math. Program. • 2006 It is shown that in the univariate case (n = 1), these problems have exact reformulations as semidefinite programming (SDP) problems, by using reformulations introduced in the PhD thesis of Jibetean [16]. Error Bounds for Some Semidefinite Programming Approaches to Polynomial Minimization on the Hypercube • Mathematics, Computer Science SIAM J. Optim. • 2010 We consider the problem of minimizing a polynomial on the hypercube$[0,1]^n\$ and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to
Representing polynomials by positive linear functions on compact convex polyhedra.
If K is a compact polyhedron in Euclidean έ/-space, defined by linear inequalities, βt > 0, and if / is a polynomial in d variables that is strictly positive on AT, then / can be expressed as a
Global Optimization with Polynomials and the Problem of Moments
It is shown that the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): R n to R, in a compact set K defined byPolynomial inequalities reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems.
Moments, Positive Polynomials And Their Applications
The semideﬁnite programming methodology to solve the generalized problem of moments is presented and several applications of the GPM are described in detail (notably in optimization, probability, optimal control andmathematical ﬁnance).
Semidefinite Programming vs. LP Relaxations for Polynomial Programming
• J. Lasserre
• Mathematics, Computer Science
Math. Oper. Res.
• 2002
Two hierarchies of relaxations are compared, namely, LP relaxations based on products of the original constraints, in the spirit of the RLT procedure of Sherali and Adams (1990), and recent semidefinite programming (SDP) relaxations introduced by the author.
Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming
• Mathematics, Computer Science
J. Glob. Optim.
• 2002
This paper shows how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's).