The lax conjecture is true
@article{Lewis2005TheLC, title={The lax conjecture is true}, author={Adrian S. Lewis and Pablo A. Parrilo and Motakuri V. Ramana}, journal={arXiv: Optimization and Control}, year={2005}, volume={133}, pages={2495-2499} }
In 1958 Lax conjectured that hyperbolic polynomials in three variables are determinants of linear combinations of three symmetric matrices. This conjecture is equivalent to a recent observation of Helton and Vinnikov.
168 Citations
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References
SHOWING 1-10 OF 19 REFERENCES
Hyperbolic Polynomials and Convex Analysis
- MathematicsCanadian Journal of Mathematics
- 2001
Abstract A homogeneous real polynomial $p$ is hyperbolic with respect to a given vector $d$ if the univariate polynomial $t\,\mapsto \,p(x\,-\,td)$ has all real roots for all vectors $x$ . Motivated…
Relating Homogeneous Cones and Positive Definite Cones via T-Algebras
- MathematicsSIAM J. Optim.
- 2003
It is concluded that every homogeneous cone is isomorphic to a "slice" of a cone of positive definite matrices in the T-algebras defined by Vinberg.
On Nesterov's Approach to Semi-infinite Programming
- Mathematics, Computer Science
- 2002
Nesterov's construction for the reduction of various classes of semi-infinite programming problems to the semidefinite programming form is generalized to consider ‘cones of squares’ of real-valued and matrix-valued functions as rather particular cases of a unifying abstract scheme.
Hyperbolic Polynomials and Interior Point Methods for Convex Programming
- MathematicsMath. Oper. Res.
- 1997
It is shown that the long-step primal potential reduction methods of Nesterov and Todd and the surface-following methods of Wojciech Nemirovskii extend to hyperbolic barrier functions and that there exists a hyperBolic barrier function on every homogeneous cone.
Linear matrix inequality representation of sets
- Mathematics
- 2003
This article concerns the question, Which subsets of ℝm can be represented with linear matrix inequalities (LMIs)? This gives some perspective on the scope and limitations of one of the most powerful…
Interior-point polynomial algorithms in convex programming
- MathematicsSiam studies in applied mathematics
- 1994
This book describes the first unified theory of polynomial-time interior-point methods, and describes several of the new algorithms described, e.g., the projective method, which have been implemented, tested on "real world" problems, and found to be extremely efficient in practice.
Linear hyperbolic partial differential equations with constant coefficients
- Mathematics
- 1951
I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 C h a p t e r I. P r o o f of t h e o r e m I . . . . . . . . . . . . . . . . . . . . . 9 C h a p t e r 2. H y p e…
Relating homogeneous ones and positive de nite ones viaTalgebras
- 2003