Representation of nonnegative convex polynomials

# Representation of nonnegative convex polynomials

[
###### Abstract

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic set . Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define .

Lasserre]Jean B. Lasserre

This work was completed with the support of the (french) ANR grant NT05-3-41612..\par\@mkboth\shortauthors\shorttitle \par Representation of nonnegative convex polynomials\par

Mathematics Subject Classification (2000). Primary 14P10; Secondary 11E25 12D15 90C25.

Keywords. Positive polynomials; sums of squares; quadratic modules; convex sets.

\par\par\@xsect

An important research area of real algebraic geometry is concerned with representations of polynomials positive on a basic semi-algebraic set

 K:={x∈Rn:gj(x)≥0,j=1,…,m}⊂Rn (1.1)

where , . \parAn important result in this vein is Schmüdgen’s Positivstellensatz [6] which states that if is compact and is positive on then belongs to the preordering generated by the ’s; bounds on the degrees in the representation are even provided in Schweighofer [7]. Under a rather weak additional assumption on the ’s, Putinar’s refinement [4] states that even belongs to the quadratic module generated by the ’s. The above mentioned representation results do not specialize when is convex and the ’s are concave (so that is convex) a highly important case, particularly in optimization. Also, as soon as is not compact any more then negative results, notably by Scheiderer [5], exclude to represent any positive on as an element of or (except perhaps in low-dimensional cases). For more details, the interested reader is referred to the nice survey [5]. \parHowever, inspired and motivated by some classical results from convex optimization, we show that specialized representation results are possible when is convex and the ’s are concave, in which case is a closed (not necessarily compact) convex basic semi-algebraic set. Namely, a specific subset of the quadratic module is such that is dense (for the -norm of coefficients) in the convex cone of convex polynomials, nonnegative on . \par\par\@xsect \par\@xsect Let be the ring of real polynomials in the variables , and let be the subset of sums of squares (sos) polynomials. If , write , and denote its -norm by ). \par\parLet be the quadratic module generated by a set of polynomials , that is,

 Q(g):={σ0+m∑j=1σjgj:σj∈Σ2,j=0,…,m}. (2.1)

Throughout the paper we make the following assumption.

###### Assumption 2.1.

is defined in (\@setrefsetk) and is such that: \par(a) is concave for every . \par(b) There exists such that for every . Assumption \@setrefass1(b), known as Slater condition, is an important regularity condition for the celebrated Karush-Kuhn-Tucker optimality conditions.

###### Proposition 2.2.

Let Assumption \@setrefass1 hold and let be convex and such that for some . \parThen there exists such that

 ∇f(x∗)−m∑j=1λj∇gj(x∗)=0;λjgj(x∗)=0,j=1,…,m. (2.2)

In other words, the Lagrangian defined by

 X↦Lf(X):=f(X)−f∗−m∑j=1λjgj(X),X∈Rn, (2.3)

is a nonnegative polynomial which satisfies

 Lf(x∗)=0;∇Lf(x∗)=0. (2.4)

See e.g. Polyak [3]. \par\par\@xsect \parIf one is interested in representation of polynomials nonnegative on , the first polynomial to consider is of course where . Indeed, any other positive polynomial is just with . And so, if belongs to some preordering or some quadratic module, then so does . From Proposition \@setrefprop1 it is easy to establish the following result.

###### Corollary 2.3.

Let Assumption \@setrefass1 hold and let be convex and such that for some . If the nonnegative polynomial of (\@setreflagrangian) is sos then

 f−f∗=σ+n∑j=1λjgj (2.5)

for some convex sos polynomial and some nonnegative scalars , . That is, , with as in (\@setrefquad). In addition, the sos weights associated with the ’s are just nonnegative constants, and is convex.

###### Proof.

Follows from the definition (\@setreflagrangian) of , and the fact that is sos. ∎ Hence in view of Corollary \@setrefcoro1, an interesting issue is to provide sufficient conditions for to be sos. For instance, consider the following definition from Helton and Nie [1]

###### Definition 2.4 (Helton and Nie [1]).

A polynomial is sos-convex if its Hessian is a sum of squares (sos), that is, there is some integer and some matrix polynomial such that

 ∇2f(X):=(∂2f(X)∂Xi∂Xj)ij=F(X)TF(X). (2.6)
###### Corollary 2.5.

Let Assumption \@setrefass1 hold, and let be convex and such that for some . \parIf is sos-convex and is sos-convex for every , then . More precisely, (\@setrefmain) holds for some convex sos polynomial and some nonnegative scalars , .

###### Proof.

From Proposition \@setrefprop1, let be as in (\@setreflagrangian). As and are sos convex, write

 ∇2f(X)=F(X)TF(X);−∇2gj(X)=Gj(X)TGj(X),j=1,…,m,

for some and some , . Hence,

 ∇2Lf=∇2f−m∑j=1λj∇2gj=FTF+m∑j=1λjGTjGj=HTH,

with , and so is sos-convex. As (\@setrefkkt2) holds, by Lemma 3.2 in Helton and Nie [1], the polynomial is sos, and so, by Corollary \@setrefcoro1, the desired result (\@setrefmain) holds. ∎ Next, consider the subset defined by:

 (2.7)

The set is a specialization of to the convex case, in that the weights asociated with the ’s are nonnegative scalars, i.e., sos polynomials of degree 0, and the sos polynomial is convex.

###### Theorem 2.6.

Let Assumption \@setrefass1 hold, and let be as in (\@setrefqc). Let be the convex cone of convex polynomials nonnegative on . \parThen is dense in for the -norm . In particular, if (so that is now the set of nonnegative convex polynomials), then is dense in .

###### Proof.

Let and let . Given , let be the polynomial

 X↦Θr(X):=1+n∑i=1X2ri. (2.8)

For every , the polynomial is convex and nonnegative on , i.e., . In addition,

 0≤f∗:=infx∈Kf(x)≤infx∈Kfϵ0(x)=fϵ0(x∗ϵ)=:f∗ϵ,

for some . Indeed, the level set is compact for every , and so, attains its minimum on . Obviously, we also have as . Next, let be as in (\@setreflagrangian), i.e.,

 Lfϵ0=f+ϵΘr0−f∗ϵ−n∑j=1λϵjgj,

for some nonnegative vector . As on , by Corollary 3.3 in Lasserre and Netzer [2], there exists such that for every , is sos. That is, and so

 fϵ:=f+ϵ(Θr0+Θr)=σ+f∗ϵ+n∑j=1λϵjgj.

Notice that by definition, is convex. Next, as , , and so, equivalently, . \parIn addition, because is convex (as ) and nonnegative on (as ), and so, . Finally, as . \parFinally, if (so that is now the set of nonnegative convex polynomials), one obtains . ∎ One may also replace in (\@setrefTheta) with the new perturbation

 X↦θr(X):=r∑k=0n∑j=1X2kjk!.

This perturbation also preserves convexity. In addition, not only as , but the convergence is also uniform on compact sets! \par

## References

• [1] J.W. Helton, J. Nie, Semidefinite representation of convex sets, Technical report, Mathematics Dept., University of California at San Diego, USA, 2007. arXiv:0705.4068.
• [2] J.B. Lasserre, T. Netzer, SOS approximations of nonnegative polynomials via simple high degree perturbations, Math. Z. 256 (2006), 99–112.
• [3] B.T. Polyak, Introduction to Optimization, Optimization Software Inc., New York, 1987.
• [4] M. Putinar, Positive polynomials on compact semi-algebraic sets, Ind. Univ. Math. J. 42 (1993), pp. 969–984.
• [5] C. Scheiderer, Positivity and sums of squares: A guide to recent results, in: Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivant (eds.), IMA Proceedings, Institute of Mathematics and Its Applications, Minneapolis, USA. To appear.
• [6] K. Schmüdgen, The -moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), pp. 203-206.
• [7] M. Schweighofer, On the complexity of Schmüdgen’s Positivstellensatz, J. of Complexity 20 (2004), 529–543.
\parLAAS-CNRS and Institute of Mathematics
University of Toulouse
LAAS, 7 avenue du Colonel Roche
31077 Toulouse Cédex 4
France\pare-mail: lasserre@laas.fr\par

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