| | |
| | |
Stat |
Members: 3643 Articles: 2'487'895 Articles rated: 2609
28 March 2024 |
|
| | | |
|
Article overview
| |
|
S.o.s. approximation of polynomials nonnegative on a real algebraic set | Jean B. Lasserre
; | Date: |
20 Dec 2004 | Subject: | Algebraic Geometry MSC-class: 11E25; 12D15; 13P05; 12Y05; 90C22; 90C25 | math.AG | Abstract: | With every real polynomial $f$, we associate a family ${f_{epsilon r}}_{epsilon, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $epsilon>0,rin N$, and such that $Vert f-f_{epsilon r}Vert_1 o 0$ as $epsilon o 0$. Let $Vsubset R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,jin J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $epsilon>0$, there exist nonnegative scalars ${lambda_j(epsilon)}_{jin J}$ such that, for all $r$ sufficiently large, $$f_{epsilon r}+sum_{jin J} lambda_j(epsilon) g_j^2,quad is a sum of squares.$$ This representation is an obvious certificate of nonnegativity of $f_{epsilon r}$ on $V$, and very specific in terms of the $g_j$ that define the set $V$. In particular, it is valid with {it no} assumption on $V$. In addition, this representation is also useful from a computation point of view, as we can define semidefinite programing relaxations to approximate the global minimum of $f$ on a real algebraic set $V$, or a semi-algebraic set $K$, and again, with {it no} assumption on $V$ or $K$. | Source: | arXiv, math.AG/0412400 | Services: | Forum | Review | PDF | Favorites |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
| News, job offers and information for researchers and scientists:
| |