Name
Abstract | ||
|---|---|---|
For a proper cone $${{\mathcal K}\subset\mathbb{R}^n}$$ and its dual cone $${{\mathcal K}^*}$$ the complementary slackness condition $${\langle{\rm {\bf x}},{\rm {\bf s}}\rangle=0}$$ defines an n-dimensional manifold $${C({\mathcal K})}$$ in the space $${{\mathbb R}^{2n}}$$. When $${{\mathcal K}}$$ is a symmetric cone, points in $${C({\mathcal K})}$$ must satisfy at least n linearly independent bilinear identities. This fact proves to be useful when optimizing over such cones, therefore it is natural to look for similar bilinear relations for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for points in $${C({\mathcal K})}$$. We examine several well-known cones, in particular the cone of positive polynomials $${{\mathcal P}_{2n+1}}$$ and its dual, and show that there are exactly four linearly independent bilinear identities which hold for all $${({\rm {\bf x}},{\rm {\bf s}})\in C({\mathcal P}_{2n+1})}$$, regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential polynomials. We prove similar results for Müntz polynomials. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1007/s10107-011-0458-y | Math. Program. |
Keywords | Field | DocType |
bilinear optimality constraint,n linearly independent bilinear,similar bilinear relation,proper cone,positive polynomial,non-symmetric cone,symmetric cone,well-known cone,mathcal p,dual cone,linearly independent bilinear identity,mathcal k | Discrete mathematics,Mathematical optimization,Combinatorics,Linear independence,Polynomial,Symmetric cone,Exponential polynomial,Dual cone and polar cone,Mathematics,Manifold,Bilinear interpolation | Journal |
Volume | Issue | ISSN |
129 | 1 | 1436-4646 |
Citations | PageRank | References |
6 | 0.91 | 2 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| GáBor Rudolf | 1 | 96 | 7.98 |
| Nilay Noyan | 2 | 184 | 13.93 |
| Dávid Papp | 3 | 53 | 9.21 |
| Farid Alizadeh | 4 | 895 | 139.10 |