Sums of Squares, Moment Matrices and Optimization Over Polynomials

Laurent, Monique

doi:10.1007/978-0-387-09686-5_7

Cited by 643 publications

(732 citation statements)

References 117 publications

(121 reference statements)

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“…It is well-known (and at the heart of current developments in optimization of polynomial functions, see [4,15] or the survey [5]) that SOS conditions with degree constraints of the form (3.3) can be phrased as semidefinite programs. Finding an (optimal) positive semidefinite matrix within an affine linear variety is known as semidefinite programming (see e.g.…”

Section: Approximations Based On the Real Nullstellensatzmentioning

confidence: 99%

“…Using a degree truncation approach, this allows to find sumof-squares-based polynomial identities which certify that a certain point is located outside of an amoeba or coamoeba. In particular, it is well known from recent lines of research in computational semialgebraic geometry (see, e.g., [4,5,15]) that these certificates can be computed via semidefinite programming (SDP).…”

Section: Introductionmentioning

confidence: 99%

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Approximating amoebas and coamoebas by sums of squares

Theobald

Wolff

2014

Math. Comp.

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Abstract. Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming.Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstellensatz, this allows to tackle the problem by sums of squares techniques and semidefinite programming. Our method yields polynomial identities as certificates of non-containment of a point in an amoeba or coamoeba. As the main theoretical result, we establish some degree bounds on the polynomial certificates. Moreover, we provide some actual computations of amoebas based on the sums of squares approach.

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Section: Approximations Based On the Real Nullstellensatzmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximating amoebas and coamoebas by sums of squares

Theobald

Wolff

2014

Math. Comp.

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“…This result will play a crucial role in our approach. We give a proof, based on [20], although some details are simplified.…”

Section: Positive Linear Forms and Real Radical Idealsmentioning

confidence: 99%

“…, p r be interpolation polynomials at the v i 's, i.e., such that p j (v i ) = δ i,j . An easy but crucial observation (made in [20]) is that we may assume that each p j has degree at most s − d. Indeed, we can replace each interpolation polynomial p j by its normal form modulo J with respect to a basis of R[x]/J. As such a basis can be obtained by picking a column basis of M s−d (Λ), its members are monomials of degree at most s − d, and the resulting normal forms of the p j 's are again interpolation polynomials at the v i 's but now with degree at most s−d.…”

Section: Theorem 11 [17] Letmentioning

confidence: 99%

The Approach of Moments for Polynomial Equations

Laurent

Rostalski

2011

International Series in Operations Research &Amp; Management Science

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Summary. In this chapter we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and the dual theory of moments. A crucial ingredient is a semidefinite characterization of the real radical ideal, consisting of all polynomials with the same real zero set as the system of polynomials to be solved. Combining this characterization with ideas from commutative algebra, (numerical) linear algebra and semidefinite optimization yields a new class of real algebraic algorithms. This chapter sheds some light on the underlying theory and the link to polynomial optimization.

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“…The last decade has seen several developments in polynomial optimization [1,2,3]. In particular, a systematic procedure has been established to solve Polynomial Optimization Problems (POP) on compact basic semi-algebraic sets.…”

Section: Introductionmentioning

confidence: 99%

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

Jeyakumar

Lasserre

2014

J Optim Theory Appl

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The optimal value of a polynomial optimization over a compact semialgebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semialgebraic feasible sets, for which the associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. We show that for such problems, the corresponding hierachy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.

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Sums of Squares, Moment Matrices and Optimization Over Polynomials

Cited by 643 publications

References 117 publications

Approximating amoebas and coamoebas by sums of squares

Approximating amoebas and coamoebas by sums of squares

The Approach of Moments for Polynomial Equations

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

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