## Bernstein

A formalization in PVS of an efficient representation of multivariate
Bernstein polynomials has been developed. The formalization includes formally verified
algorithms for finding lower and upper bounds of the minimum and maximum values of a
polynomial. The algorithms are used in the definition
of automated proof strategies for solving polynomial
global optimization problems such as:

- Formally checking universally quantified multivariate polynomial inequalities on closed ranges.
- Finding witnesses that formally prove existentially quantified multivariate polynomial inequalities on open and closed ranges.
- Computing boxes (hyper-rectangles) that correctly under and over approximate regions defined as conjunction or disjunctions
of multivariate polynomial inequalities.

### Downloads

- The Bernstein library is available as part of the
NASA PVS Library.
- Grizzly is a prototype client-server tool for solving global optimization problems.

### Examples

The following examples are

*automatically* proved in PVS using the strategy

`(bernstein)`

, which is included in
the Bernstein library.

FORALL (x,y:real):
-0.5 <= x AND x <= 1 AND -2 <= y AND y <= 1 IMPLIES
4*x^2 - (21/10)*x^4 + (1/3)*x^6 + (x-3)*y - 4*y^2 + 4*y^4 > -3.4

EXISTS (x,y:real):
-0.5 <= x AND x <= 1 AND -2 <= y AND y <= 1 AND
4*x^2 - (21/10)*x^4 + (1/3)*x^6 + (x-3)*y - 4*y^2 + 4*y^4 < -3.39

More challenging

`problems`

are also included in the distribution.

### Publications

- César Muñoz and Anthony Narkawicz,
*Formalization of a Representation of Bernstein Polynomials and Applications to Global Optimization*, Journal of
Automated Reasoning, Volume 51, Issue 2, pp. 151-196, August 2013. BibTeX Reference.
- Anthony Narkawicz and César Muñoz,
*Formal
Verification of Conflict Detection Algorithms for Arbitrary
Trajectories*,
Reliable
Computing, Volume 17, pp. 209-237, December, 2012. BibTeX Reference.
- Luis Crespo, César Muñoz, Anthony Narkawicz, Sean Kenny, and Daniel Giesy,
*Uncertainty Analysis via Failure Domain Characterization: Polynomial Requirement Functions*,
European Safety and Reliability Conference, September, 2011. BibTeX Reference.

The tag

identifies links that are outside
the NASA domain