Representation of nonnegative convex polynomials
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 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.
An important research area of real algebraic geometry is concerned with representations of polynomials positive on a basic semi-algebraic set
where , . \parAn important result in this vein is Schmüdgen’s Positivstellensatz  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 . Under a rather weak additional assumption on the ’s, Putinar’s refinement  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 , 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 . \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,
Throughout the paper we make the following assumption.
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.
Let Assumption \@setrefass1 hold and let be convex and such that for some . \parThen there exists such that
In other words, the Lagrangian defined by
is a nonnegative polynomial which satisfies
See e.g. Polyak . \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.
Let Assumption \@setrefass1 hold and let be convex and such that for some . If the nonnegative polynomial of (\@setreflagrangian) is sos then
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.
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 
Definition 2.4 (Helton and Nie ).
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
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 , .
From Proposition \@setrefprop1, let be as in (\@setreflagrangian). As and are sos convex, write
for some and some , . Hence,
with , and so is sos-convex. As (\@setrefkkt2) holds, by Lemma 3.2 in Helton and Nie , the polynomial is sos, and so, by Corollary \@setrefcoro1, the desired result (\@setrefmain) holds. ∎ Next, consider the subset defined by:
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.
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 .
Let and let . Given , let be the polynomial
For every , the polynomial is convex and nonnegative on , i.e., . In addition,
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.,
for some nonnegative vector . As on , by Corollary 3.3 in Lasserre and Netzer , there exists such that for every , is sos. That is, and so
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
This perturbation also preserves convexity. In addition, not only as , but the convergence is also uniform on compact sets! \par
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