On some interconnections between strict monotonicity, globally uniquely solvable, and P properties in semidefinite linear complementarity problems
✍ Scribed by M.Seetharama Gowda; Yoon Song; G. Ravindran
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 134 KB
- Volume
- 370
- Category
- Article
- ISSN
- 0024-3795
No coin nor oath required. For personal study only.
✦ Synopsis
In the setting of semidefinite linear complementarity problems on S n , the implications strict monotonicity ⇒ P 2 ⇒ GUS ⇒ P are known. Here, P and P 2 properties for a linear transformation L : S n → S n are respectively defined by: X
to the global unique solvability in semidefinite linear complementarity problems corresponding to L. In this article, we show that the reverse implications hold for any self-adjoint linear transformation, and for normal Lyapunov and Stein transformations. By introducing the concept of a principal subtransformation of a linear transformation, we show that L : S n → S n has the P 2 -property if and only if for every n × n real invertible matrix Q, every principal subtransformation of L has the P-property where L(X) := Q T L(QXQ T )Q. Based on this, we show that P 2 , GUS, and P properties coincide for the two-sided multiplication transformation.