# Semidefinite representations for finite varieties

@article{Laurent2007SemidefiniteRF, title={Semidefinite representations for finite varieties}, author={Monique Laurent}, journal={Mathematical Programming}, year={2007}, volume={109}, pages={1-26} }

We consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. When the polynomial equations have a finite set of complex solutions, we can reformulate this problem as a semidefinite programming problem. Our semidefinite representation involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space ℝ[x1, . . . ,xn]/I, where I is the ideal generated by the polynomial equations in the problem. Moreover, we…

## 98 Citations

Sums of Squares, Moment Matrices and Optimization Over Polynomials

- Mathematics, Computer Science
- 2009

This work considers the problem of minimizing a polynomial over a semialgebraic set defined byPolynomial equations and inequalities, which is NP-hard in general and reviews the mathematical tools underlying these properties.

Semidefinite Approximations for Global Unconstrained Polynomial Optimization

- Computer ScienceSIAM J. Optim.
- 2005

This work proposes here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming, and is applied to several examples.

Exact relaxation for polynomial optimization on semi-algebraic sets

- Mathematics, Computer Science
- 2014

This paper shows that when the infimum of a real polynomial function f on a closed basic semialgebraic set and the points where this infimum is reached, a relaxation hierarchy constructed from the Karush-Kuhn-Tucker ideal is always exact and that the vanishing ideal of the KKT minimizer points is generated by the kernel of the associated moment matrix in that degree.

Semidefinite Programming in Combinatorial and Polynomial Optimization

- Mathematics, Computer Science
- 2008

Some of the main mathematical tools that underlie semidefinite programming are sketched and their application to some graph problems dealing with maximum cuts, stable sets and graph coloring is illustrated.

Certified relaxation for polynomial optimization on semi-algebraic sets

- Mathematics, Computer Science
- 2013

In this paper, we describe a relaxation method to compute the minimal critical value of a real polynomial function on a semialgebraic set S and the ideal defining the points at which the minimal…

Fe b 20 05 Minimizing Polynomials Over Semialgebraic Sets ∗

- Mathematics, Computer Science
- 2005

This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in R defined by finitely many polynomial equations and inequalities, using the…

Exact Moment Representation in Polynomial Optimization

- Mathematics
- 2020

We investigate the problem of representing moment sequences by measures in the context of Polynomial Optimization Problems. This consists in ﬁnding the inﬁmum of a real polynomial on a real…

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

- Mathematics, Computer ScienceFound. Comput. Math.
- 2008

An algorithm is proposed, using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety Vℝ(I) as well as a set of generators of theReal radical ideal, obtained in the form of a border or Gröbner basis.

Converging Semidefinite Bounds for Global Unconstrained Polynomial Optimization

- Computer Science, Mathematics
- 2007

It is proposed here a method for computing a converging sequence of upper bounds using semidefinite programming based on perturbing the original polynomial and the method is applied to several examples.

Convex Hulls of Algebraic Sets

- Mathematics
- 2012

This article describes a method to compute successive convex approximations of the convex hull of the solutions to a system of polynomial equations over the reals. The method relies on sums of…

## References

SHOWING 1-10 OF 54 REFERENCES

Semidefinite programming relaxations for semialgebraic problems

- Mathematics, Computer ScienceMath. Program.
- 2003

It is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility and provide a constructive approach for finding bounded degree solutions to the Positivstellensatz.

Semidefinite Approximations for Global Unconstrained Polynomial Optimization

- Computer ScienceSIAM J. Optim.
- 2005

This work proposes here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming, and is applied to several examples.

Converging Semidefinite Bounds for Global Unconstrained Polynomial Optimization

- Computer Science, Mathematics
- 2007

It is proposed here a method for computing a converging sequence of upper bounds using semidefinite programming based on perturbing the original polynomial and the method is applied to several examples.

Global Optimization with Polynomials and the Problem of Moments

- MathematicsSIAM J. Optim.
- 2001

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.

Optimization of Polynomials on Compact Semialgebraic Sets

- MathematicsSIAM J. Optim.
- 2005

It is proved that every sequence of "nearly" optimal solutions of the successive relaxations gives rise to a sequence of points in $\R^n$ converging to $x^\ast$.

The Moment Problem on Compact Semi-Algebraic Sets

- Mathematics
- 1991

In this chapter we begin the study of the multidimensional moment problem. The passage to dimensions d ≥ 2 brings new difficulties and unexpected phenomena. In Sect. 3.2 we derived solvability…

Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

- Mathematics
- 2000

In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the…

Minimizing Polynomial Functions

- Computer Science, MathematicsAlgorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science
- 2001

It is demonstrated that existing algebraic methods are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming.

Polynomials nonnegative on a grid and discrete optimization

- Mathematics
- 2001

We characterize the real-valued polynomials on R n that are nonnegative (not necessarily strictly positive) on a grid K of points of R n , in terms of a weighted sum of squares whose degree is…

Squared Functional Systems and Optimization Problems

- Mathematics
- 2000

It is proved that such cones can be always seen as a linear image of the cone of positive semidefinite matrices, and a description of the cones of univariate and non-negative trigonometric polynomials is given.